Research ArticleConnection Formulae between Ellipsoidal andSpherical Harmonics with Applications to Fluid Dynamics andElectromagnetic Scattering

Michael Doschoris12 and Panayiotis Vafeas1

1Department of Chemical Engineering University of Patras 26504 Patras Greece2Institute of Chemical Engineering Sciences Stadiou Street PO Box 1414 Platani 26504 Patras Greece

Correspondence should be addressed to Michael Doschoris mdoschochemengupatrasgr

Received 26 July 2014 Accepted 30 October 2014

Academic Editor George Dassios

Copyright copy 2015 M Doschoris and P Vafeas This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The environment of the ellipsoidal system significantly more complex than the spherical one provides the necessary settings fortackling boundary value problems in anisotropic space However the theory of Lame functions and ellipsoidal harmonics affiliatedwith the ellipsoidal system is rather complicated A turning point would reside in the existence of expressions interlacing thesetwo different systems Still there is no simple way if at all to bridge the gap The present paper addresses this issue We provideexplicit formulas of specific ellipsoidal harmonics expressed in terms of their counterparts in the classical spherical system Theseexpressions are then put into practice in the framework of physical applications

1 Introduction

The ellipsoidal coordinate system by its very nature isdemanding concealing numerous difficulties The main rea-son can be associated with the acquisition of solutions formiscellaneous operators Even in the case of the Laplacianderiving the corresponding eigensolutions is a nontrivial taskThe French engineer andmathematician Gabriel Lame in themid nineteenth century following an ingenious argumentseparated variables for the Laplace operator arriving at thefunctions which carry nowadays his name Taking theproduct of Lame functions leads to the ellipsoidal harmonics

But the complications regarding the particular systemdo not end here In contrast to the theory of sphericalharmonics only ellipsoidal harmonics of low order have beencomputed in closed form [1 2] Why First of all a recursivetechnique in order to generate Lame functions does not existAlthough we know that Lame functions are connected bythree-term recurrence relations [3] to the authors knowledgeno procedure calculating the corresponding coefficients has

been proposed so far This particular impediment forces usto undergo an involved algorithm from which the Lamefunctions are determined This essentially two-step opera-tion requires the computation of the roots of polynomialfunctions allowing nontrivial solutions for the initial linearhomogeneous systems We note that the previously indicatedalgorithm can be applied analytically only for Lame functionsup to the seventh degree [2] Higher-order terms demandcomputational implementations [4] introducing numericalinstability which in the sequence is transferred to the calcu-lation of the corresponding Lame function

The aforementioned hurdles could in theory be avoidedon the assumption that the functions of Lame and corre-sponding ellipsoidal harmonics would be able to be expressedin terms of Legendre functions and spherical harmonicsrespectively Although in principle the possibility exists nogeneral formulae connecting these functions are availableThe absence of such relations is justified bearing in mind thatellipsoidal harmonics are not reducible in a straightforwardand unique way to the corresponding spherical harmonics

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 572458 12 pageshttpdxdoiorg1011552015572458

2 Advances in Mathematical Physics

Nonetheless distinct ellipsoidal harmonics can be repre-sented in terms of a finite set of spherical harmonics of degreeequal or less in view of the initial ellipsoidal harmonic

The present communication aims towards this directionThe section following is devoted to a brief introduction ofthe peculiarities of the ellipsoidal system Lame functionsand ellipsoidal harmonics (missing details can be found in[1]) We then continue elaborating explicit relations connect-ing ellipsoidal harmonics with the spherical ones FinallySection 3 introducing two real-world problems showcasesthe efficiency of the derived formulas

2 Mathematical Background

21 The Confocal Ellipsoidal Coordinate System The ellip-soidal coordinates (120588 120583 ]) of each point (119909

1 1199092 1199093) are

1199091=120588120583]ℎ2ℎ3

1199092=radic1205882 minus ℎ2

3radic1205832 minus ℎ2

3radicℎ23minus ]2

ℎ1ℎ3

1199093=radic1205882 minus ℎ2

2radicℎ22minus 1205832radicℎ2

2minus ]2

ℎ1ℎ2

(1)

provided that 0 le ]2 le ℎ23le 1205832 le ℎ2

2le 1205882 lt +infin

The coordinate 120588 which determines a family of confocalellipsoids is comparable to the radial variable 119903 in sphericalcoordinates On the other hand the coordinates 120583 and ]specify a family of confocal hyperboloids of one and twosheets respectively and correlate to the angular variables 120579and 120601

In the ellipsoidal coordinate system each direction isunique providing a particular perception of anisotropicspace Since any direction exhibits its own character theellipsoidal coordinate system requires the introduction ofa reference ellipsoid establishing the variations in angulardependence a direct analogy to the unit sphere This refer-ence ellipsoid defined by

3

sum119895=1

(119909119895

119886119895

)

2

= 1 (2)

is not one of a kind but must incorporate the physical realityat hand The constants

ℎ2

1= 1198862

2minus 1198862

3

ℎ2

2= 1198862

1minus 1198862

3

ℎ2

3= 1198862

1minus 1198862

2

(3)

are the squares of the semifocal distances and 119886120581 120581 = 1 2 3

with 0 lt 1198863lt 1198862lt 1198861lt +infin fixed parameters determining

the reference semiaxesA decisive aspect when concerned with boundary value

problems in ellipsoidal coordinates resides in the spectral

decomposition of the Laplacian Separating variables leadsto three identical ordinary differential equations known asLamersquos equation The corresponding solutions are the so-called Lame functions of the first kind E119898

119899 where 119899 designates

the set of nonnegative integers and119898 = 1 2 2119899+1 as wellas the matching second kind functions F119898

119899

Analytically the first second and third degree Lamefunctions of the first kind are presented below The variable119906 represents either variable 120588 isin [ℎ

2 +infin) 120583 isin [ℎ

3 ℎ2] or

] isin [0 ℎ3] Therein

E10(119906) = 1

E1205811(119906) = radic1003816100381610038161003816119906

2 minus 11988621+ 1198862120581

1003816100381610038161003816 120581 = 1 2 3

E12(119906) = 119906

2+ Λ minus 119886

2

1

E22(119906) = 119906

2+ Λ1015840minus 1198862

1

E120581+1198972

(119906) = radic10038161003816100381610038161199062 minus 11988621+ 1198862120581

1003816100381610038161003816radic10038161003816100381610038161199062 minus 11988621+ 1198862119897

1003816100381610038161003816

120581 119897 = 1 2 3 120581 = 119897

E2120581minus13

(119906) = radic10038161003816100381610038161199062 minus 11988621+ 1198862120581

1003816100381610038161003816 (1199062+ Λ120581minus 1198862

1)

120581 = 1 2 3

E21205813(119906) = radic1003816100381610038161003816119906

2 minus 11988621+ 1198862120581

1003816100381610038161003816 (1199062+ Λ1015840

120581minus 1198862

1)

120581 = 1 2 3

E73(119906) = 119906radic

10038161003816100381610038161199062 minus ℎ23

1003816100381610038161003816radic10038161003816100381610038161199062 minus ℎ22

1003816100381610038161003816

(4)

where the constants Λ and Λ1015840 as well as Λ120581and Λ1015840

120581 120581 =

1 2 3 are given as

Λ = 1198862

1minusℎ23+ ℎ22

3+radicℎ41+ ℎ22ℎ23

3

(5)

Λ1015840= 1198862

1minusℎ23+ ℎ22

3minusradicℎ41+ ℎ22ℎ23

3

(6)

Λ120581= 1198862

1minusℎ22+ (1 + 2120575

1205811) ℎ23+ ℎ2120581

5

+radicℎ41+ ℎ22ℎ23+ 3ℎ2120581

5 120581 = 1 2 3

(7)

Λ1015840

120581= 1198862

1minusℎ22+ (1 + 2120575

1205811) ℎ23+ ℎ2120581

5

minusradicℎ41+ ℎ22ℎ23+ 3ℎ2120581

5 120581 = 1 2 3

(8)

where 120575 represents the Kronecker delta The above constantssatisfy the following relations3

sum119895=1

1

Λ minus 1198862119895

= 0

3

sum119895=1

1 + 2120575120581119895

Λ120581minus 1198862119895

= 0 120581 = 1 2 3 (9)

respectively

Advances in Mathematical Physics 3

The corresponding Lame functions of the second kind are

F119898119899(119906) = E119898

119899(119906) I119898119899(119906) (10)

for every 119899 = 0 1 2 and119898 = 1 2 2119899 + 1 where

I119898119899(119906) = int

119906

1199060

d119905

(E119898119899(119905))2radic10038161003816100381610038161199052 minus ℎ23

1003816100381610038161003816radic10038161003816100381610038161199052 minus ℎ22

1003816100381610038161003816(11)

is an elliptic integralIn view of problems where the boundary consists of a

triaxial confocal ellipsoid the product of two Lame functionsbelonging to the same class defines the surface ellipsoidalharmonics S119898

119899 that is

S119898119899(120583 ]) = E119898

119899(120583)E119898119899(]) (12)

whereas

E119898

119899(120588 120583 ]) = E119898

119899(120588) S119898119899(120583 ]) (13)

designate the interior ellipsoidal harmonicsOn the other hand the exterior ellipsoidal harmonics are

specified as

F119898

119899(120588 120583 ]) = (2119899 + 1)E119898

119899(120588 120583 ]) I119898

119899(120588) (14)

where

I119898119899(120588) = int

+infin

120588

d119905

(E119898119899(119905))2radic1199052 minus ℎ2

3radic1199052 minus ℎ2

2

120588 ge ℎ2 (15)

22 Connecting Ellipsoidal and Spherical Harmonics Wealready mentioned in the introduction the paucity of generalformulas associating ellipsoidal harmonics with the samedegree or less spherical harmonics Another way to compre-hend this is the following As the triaxial ellipsoid deterioratesto a sphere the ellipsoidal harmonics reduce to the so-called spheroconal harmonics which are a form of sphericalharmonics The spheroconal system which incorporatesthe radial coordinate 119903 of the spherical system with thecoordinates of the ellipsoidal system that specify orientationover any ellipsoidal surface (120583 ]) is established on the sameparameters 119886

120581 120581 = 1 2 3 thus preserving its ellipsoidal

characteristic Nevertheless although it seems that a generalframework cannot be established it is possible to representdistinct ellipsoidal harmonics with reference to finite termsof spherical harmonics (see Figure 1 for an illustration)

Forasmuch as the spherical harmonics form a completeset any continuous function can be expanded in a series ofY119898119899(r) Therefore

E119902

119901(r) =

infin

sum119899=0

119899

sum119898=minus119899

119860119898

119899119903119899Y119898119899(r) 119901 ge 0 119902 = 1 2 2119901 + 1

(16)

and the coefficients 119860119898119899

depend solely on the referencesemiaxes 119886

1 1198862 1198863

Ellipsoidal harmonics Spherical harmonics

Cartesian coordinates

Figure 1 In order to obtain representations of ellipsoidal harmonicsin terms of spherical harmonics and vice versa one has to go throughCartesian coordinates (solid lines) A direct connection appears notto be feasible (dashed line)

Provided that on the unit sphere 1198782

∮1198782

Y119898119899(r)Y119898

1015840

1198991015840 (r) dΩ (r) = 120575

11989911989910158401205751198981198981015840 (17)

the coefficients of (16) are computed as

119860119898

119899=1

119903119899∮1198782

E119902

119901(r)Y119898119899(r) dΩ (r) (18)

Equations (16) and (18) provide the backbone of the presentedanalysis

Employing the Cartesian form of the ellipsoidal harmon-ics for degree up to three namely

E1

0(r) = 1

E120581

1(r) = ℎ

1ℎ2ℎ3

ℎ120581

119909120581 120581 = 1 2 3

E1

2(r) = L(

3

sum119895=1

1199092

119895

Λ minus 1198862119895

+ 1)

E2

2(r) = L

1015840(

3

sum119895=1

1199092

119895

Λ1015840 minus 1198862119895

+ 1)

E6minus120581

2(r) = ℎ

1ℎ2ℎ3ℎ120581

119909111990921199093

119909120581

120581 = 1 2 3

E2120581minus1

2(r) = ℎ

1ℎ2ℎ3L120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ120581minus 1198862119895

+ 1) 120581 = 1 2 3

E2120581

2(r) = ℎ

1ℎ2ℎ3L1015840

120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ1015840120581minus 1198862119895

+ 1) 120581 = 1 2 3

E7

3(r) = ℎ

2

1ℎ2

2ℎ2

3119909111990921199093

(19)

provided that r = (119909 119910 119911) and

L = (Λ minus 1198862

1) (Λ minus 119886

2

2) (Λ minus 119886

2

3)

L1015840= (Λ1015840minus 1198862

1) (Λ1015840minus 1198862

2) (Λ1015840minus 1198862

3)

L120581= (Λ120581minus 1198862

1) (Λ120581minus 1198862

2) (Λ120581minus 1198862

3) 120581 = 1 2 3

L1015840

120581= (Λ1015840

120581minus 1198862

1) (Λ1015840

120581minus 1198862

2) (Λ1015840

120581minus 1198862

3) 120581 = 1 2 3

(20)

4 Advances in Mathematical Physics

where the constants ΛΛ1015840 Λ120581 Λ1015840120581 120581 = 1 2 3 are given by

(5)ndash(8) respectively the following representations hold

E1

0(r) = radic4120587119903

0Y00(r)

E1

1(r) = ℎ

2ℎ3radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

E2

1(r) = 119894ℎ

1ℎ3radic2120587

3119903 (Yminus11(r) + Y1

1(r))

E3

1(r) = 2ℎ

1ℎ2radic120587

3119903Y01(r)

E1

2(r) = L + radic

4120587

5

L

Λ minus 11988623

1199032Y02(r)

+ radic2120587

15L(

1

Λ minus 11988621

minus1

Λ minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E2

2(r) = L

1015840+ radic

4120587

5

L1015840

Λ1015840 minus 11988623

1199032Y02(r)

+ radic2120587

15L1015840(

1

Λ1015840 minus 11988621

minus1

Λ1015840 minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E3

2(r) = 119894ℎ

1ℎ2ℎ2

3radic2120587

151199032(Yminus22(r) minus Y2

2(r))

E4

2(r) = ℎ

1ℎ2

2ℎ3radic2120587

151199032(Yminus12(r) minus Y1

2(r))

E5

2(r) = 119894ℎ

2

1ℎ2ℎ3radic2120587

151199032(Yminus12(r) + Y1

2(r))

E1

3(r) = ℎ

2ℎ3L1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L1

Λ1minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1radic120587

35(

1

Λ1minus 11988621

minus1

Λ1minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E2

3(r) = ℎ

2ℎ3L1015840

1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L10158401

Λ10158401minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1015840

1radic120587

35(

1

Λ10158401minus 11988621

minus1

Λ10158401minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E3

3(r) = 119894ℎ

1ℎ3L2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L2

Λ2minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L2radic120587

35(

1

Λ2minus 11988621

minus1

Λ2minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E4

3(r) = 119894ℎ

1ℎ3L1015840

2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L10158402

Λ10158402minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L1015840

2radic120587

35(

1

Λ10158402minus 11988621

minus1

Λ10158402minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E5

3(r) = 2ℎ

1ℎ2L3radic120587

3119903Y01(r) + 2 ℎ1ℎ2L3

Λ3minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L3radic2120587

105(

1

Λ3minus 11988621

minus1

Λ3minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E6

3(r) = 2ℎ

1ℎ2L1015840

3radic120587

3119903Y01(r) + 2

ℎ1ℎ2L10158403

Λ10158403minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L1015840

3radic2120587

105(

1

Λ10158403minus 11988621

minus1

Λ10158403minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E7

3(r) = 119894ℎ

2

1ℎ2

2ℎ2

3radic2120587

1051199033(Yminus23(r) minus Y2

3(r))

(21)

The above relations are not hard to prove considering thatthe first term of the product present in the integrand of (18)displays the general form 119909

120581

111990911989721199091199043where 120581+119897+119904 = 119899 implying

that only terms of the same or lower degree 119899 survive Inaddition switching to spherical coordinates and bearing inmind that the point (119909

1 1199092 1199093) resides interior of the ellipsoid

(2) give the desired resultsOn the other hand in order to evaluate the exterior

ellipsoidal harmonics provided via (14) and (15) one needsonly to express the quadratic terms (E119898

119899(119905))2 as a function of

Legendre polynomials This is easily done furnishing

(E10(119905))2

= P0(119905) = 1

(E11(119905))2

=2

3P2(119905) +

1

3

Advances in Mathematical Physics 5

(E21(119905))2

=2

3P2(119905) +

1

3minus ℎ2

3

(E31(119905))2

=2

3P2(119905) +

1

3minus ℎ2

2

(E12(119905))2

=8

35P4(119905) + 4(

1

7+Λ minus 11988621

3)P2(119905)

+ (1

5+2

3(Λ minus 119886

2

1) + (Λ minus 119886

2

1)2

)

(E22(119905))2

=8

35P4(119905) + 4(

1

7+Λ1015840 minus 1198862

1

3)P2(119905)

+ (1

5+2

3(Λ1015840minus 1198862

1) + (Λ

1015840minus 1198862

1)2

)

(E32(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

3

3)P2(119905) + (

1

5minusℎ2

3

3)

(E42(119905))2

=8

35P4(119905) + 2(

2

7minusℎ22

3)P2(119905) + (

1

5minusℎ22

3)

(E52(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

2+ ℎ23

3)P2(119905)

+ (1

5minusℎ2

2+ ℎ23

3+ ℎ2

2ℎ2

3)

(E13(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)

(E23(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1015840

1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)

(E33(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1) ((Λ

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1)

times ((Λ2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ2minus 1198862

1)2

]

(E43(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1) ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1)

times ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ1015840

2minus 1198862

1)2

]

(E53(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1) ((Λ

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1)

times ((Λ3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ3minus 1198862

1)2

]

(E63(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1) ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1)

times ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ1015840

3minus 1198862

1)2

]

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

Nonetheless distinct ellipsoidal harmonics can be repre-sented in terms of a finite set of spherical harmonics of degreeequal or less in view of the initial ellipsoidal harmonic

The present communication aims towards this directionThe section following is devoted to a brief introduction ofthe peculiarities of the ellipsoidal system Lame functionsand ellipsoidal harmonics (missing details can be found in[1]) We then continue elaborating explicit relations connect-ing ellipsoidal harmonics with the spherical ones FinallySection 3 introducing two real-world problems showcasesthe efficiency of the derived formulas

2 Mathematical Background

21 The Confocal Ellipsoidal Coordinate System The ellip-soidal coordinates (120588 120583 ]) of each point (119909

1 1199092 1199093) are

1199091=120588120583]ℎ2ℎ3

1199092=radic1205882 minus ℎ2

3radic1205832 minus ℎ2

3radicℎ23minus ]2

ℎ1ℎ3

1199093=radic1205882 minus ℎ2

2radicℎ22minus 1205832radicℎ2

2minus ]2

ℎ1ℎ2

(1)

provided that 0 le ]2 le ℎ23le 1205832 le ℎ2

2le 1205882 lt +infin

The coordinate 120588 which determines a family of confocalellipsoids is comparable to the radial variable 119903 in sphericalcoordinates On the other hand the coordinates 120583 and ]specify a family of confocal hyperboloids of one and twosheets respectively and correlate to the angular variables 120579and 120601

In the ellipsoidal coordinate system each direction isunique providing a particular perception of anisotropicspace Since any direction exhibits its own character theellipsoidal coordinate system requires the introduction ofa reference ellipsoid establishing the variations in angulardependence a direct analogy to the unit sphere This refer-ence ellipsoid defined by

3

sum119895=1

(119909119895

119886119895

)

2

= 1 (2)

is not one of a kind but must incorporate the physical realityat hand The constants

ℎ2

1= 1198862

2minus 1198862

3

ℎ2

2= 1198862

1minus 1198862

3

ℎ2

3= 1198862

1minus 1198862

2

(3)

are the squares of the semifocal distances and 119886120581 120581 = 1 2 3

with 0 lt 1198863lt 1198862lt 1198861lt +infin fixed parameters determining

the reference semiaxesA decisive aspect when concerned with boundary value

problems in ellipsoidal coordinates resides in the spectral

decomposition of the Laplacian Separating variables leadsto three identical ordinary differential equations known asLamersquos equation The corresponding solutions are the so-called Lame functions of the first kind E119898

119899 where 119899 designates

the set of nonnegative integers and119898 = 1 2 2119899+1 as wellas the matching second kind functions F119898

119899

Analytically the first second and third degree Lamefunctions of the first kind are presented below The variable119906 represents either variable 120588 isin [ℎ

2 +infin) 120583 isin [ℎ

3 ℎ2] or

] isin [0 ℎ3] Therein

E10(119906) = 1

E1205811(119906) = radic1003816100381610038161003816119906

2 minus 11988621+ 1198862120581

1003816100381610038161003816 120581 = 1 2 3

E12(119906) = 119906

2+ Λ minus 119886

2

1

E22(119906) = 119906

2+ Λ1015840minus 1198862

1

E120581+1198972

(119906) = radic10038161003816100381610038161199062 minus 11988621+ 1198862120581

1003816100381610038161003816radic10038161003816100381610038161199062 minus 11988621+ 1198862119897

1003816100381610038161003816

120581 119897 = 1 2 3 120581 = 119897

E2120581minus13

(119906) = radic10038161003816100381610038161199062 minus 11988621+ 1198862120581

1003816100381610038161003816 (1199062+ Λ120581minus 1198862

1)

120581 = 1 2 3

E21205813(119906) = radic1003816100381610038161003816119906

2 minus 11988621+ 1198862120581

1003816100381610038161003816 (1199062+ Λ1015840

120581minus 1198862

1)

120581 = 1 2 3

E73(119906) = 119906radic

10038161003816100381610038161199062 minus ℎ23

1003816100381610038161003816radic10038161003816100381610038161199062 minus ℎ22

1003816100381610038161003816

(4)

where the constants Λ and Λ1015840 as well as Λ120581and Λ1015840

120581 120581 =

1 2 3 are given as

Λ = 1198862

1minusℎ23+ ℎ22

3+radicℎ41+ ℎ22ℎ23

3

(5)

Λ1015840= 1198862

1minusℎ23+ ℎ22

3minusradicℎ41+ ℎ22ℎ23

3

(6)

Λ120581= 1198862

1minusℎ22+ (1 + 2120575

1205811) ℎ23+ ℎ2120581

5

+radicℎ41+ ℎ22ℎ23+ 3ℎ2120581

5 120581 = 1 2 3

(7)

Λ1015840

120581= 1198862

1minusℎ22+ (1 + 2120575

1205811) ℎ23+ ℎ2120581

5

minusradicℎ41+ ℎ22ℎ23+ 3ℎ2120581

5 120581 = 1 2 3

(8)

where 120575 represents the Kronecker delta The above constantssatisfy the following relations3

sum119895=1

1

Λ minus 1198862119895

= 0

3

sum119895=1

1 + 2120575120581119895

Λ120581minus 1198862119895

= 0 120581 = 1 2 3 (9)

respectively

Advances in Mathematical Physics 3

The corresponding Lame functions of the second kind are

F119898119899(119906) = E119898

119899(119906) I119898119899(119906) (10)

for every 119899 = 0 1 2 and119898 = 1 2 2119899 + 1 where

I119898119899(119906) = int

119906

1199060

d119905

(E119898119899(119905))2radic10038161003816100381610038161199052 minus ℎ23

1003816100381610038161003816radic10038161003816100381610038161199052 minus ℎ22

1003816100381610038161003816(11)

is an elliptic integralIn view of problems where the boundary consists of a

triaxial confocal ellipsoid the product of two Lame functionsbelonging to the same class defines the surface ellipsoidalharmonics S119898

119899 that is

S119898119899(120583 ]) = E119898

119899(120583)E119898119899(]) (12)

whereas

E119898

119899(120588 120583 ]) = E119898

119899(120588) S119898119899(120583 ]) (13)

designate the interior ellipsoidal harmonicsOn the other hand the exterior ellipsoidal harmonics are

specified as

F119898

119899(120588 120583 ]) = (2119899 + 1)E119898

119899(120588 120583 ]) I119898

119899(120588) (14)

where

I119898119899(120588) = int

+infin

120588

d119905

(E119898119899(119905))2radic1199052 minus ℎ2

3radic1199052 minus ℎ2

2

120588 ge ℎ2 (15)

22 Connecting Ellipsoidal and Spherical Harmonics Wealready mentioned in the introduction the paucity of generalformulas associating ellipsoidal harmonics with the samedegree or less spherical harmonics Another way to compre-hend this is the following As the triaxial ellipsoid deterioratesto a sphere the ellipsoidal harmonics reduce to the so-called spheroconal harmonics which are a form of sphericalharmonics The spheroconal system which incorporatesthe radial coordinate 119903 of the spherical system with thecoordinates of the ellipsoidal system that specify orientationover any ellipsoidal surface (120583 ]) is established on the sameparameters 119886

120581 120581 = 1 2 3 thus preserving its ellipsoidal

characteristic Nevertheless although it seems that a generalframework cannot be established it is possible to representdistinct ellipsoidal harmonics with reference to finite termsof spherical harmonics (see Figure 1 for an illustration)

Forasmuch as the spherical harmonics form a completeset any continuous function can be expanded in a series ofY119898119899(r) Therefore

E119902

119901(r) =

infin

sum119899=0

119899

sum119898=minus119899

119860119898

119899119903119899Y119898119899(r) 119901 ge 0 119902 = 1 2 2119901 + 1

(16)

and the coefficients 119860119898119899

depend solely on the referencesemiaxes 119886

1 1198862 1198863

Ellipsoidal harmonics Spherical harmonics

Cartesian coordinates

Figure 1 In order to obtain representations of ellipsoidal harmonicsin terms of spherical harmonics and vice versa one has to go throughCartesian coordinates (solid lines) A direct connection appears notto be feasible (dashed line)

Provided that on the unit sphere 1198782

∮1198782

Y119898119899(r)Y119898

1015840

1198991015840 (r) dΩ (r) = 120575

11989911989910158401205751198981198981015840 (17)

the coefficients of (16) are computed as

119860119898

119899=1

119903119899∮1198782

E119902

119901(r)Y119898119899(r) dΩ (r) (18)

Equations (16) and (18) provide the backbone of the presentedanalysis

Employing the Cartesian form of the ellipsoidal harmon-ics for degree up to three namely

E1

0(r) = 1

E120581

1(r) = ℎ

1ℎ2ℎ3

ℎ120581

119909120581 120581 = 1 2 3

E1

2(r) = L(

3

sum119895=1

1199092

119895

Λ minus 1198862119895

+ 1)

E2

2(r) = L

1015840(

3

sum119895=1

1199092

119895

Λ1015840 minus 1198862119895

+ 1)

E6minus120581

2(r) = ℎ

1ℎ2ℎ3ℎ120581

119909111990921199093

119909120581

120581 = 1 2 3

E2120581minus1

2(r) = ℎ

1ℎ2ℎ3L120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ120581minus 1198862119895

+ 1) 120581 = 1 2 3

E2120581

2(r) = ℎ

1ℎ2ℎ3L1015840

120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ1015840120581minus 1198862119895

+ 1) 120581 = 1 2 3

E7

3(r) = ℎ

2

1ℎ2

2ℎ2

3119909111990921199093

(19)

provided that r = (119909 119910 119911) and

L = (Λ minus 1198862

1) (Λ minus 119886

2

2) (Λ minus 119886

2

3)

L1015840= (Λ1015840minus 1198862

1) (Λ1015840minus 1198862

2) (Λ1015840minus 1198862

3)

L120581= (Λ120581minus 1198862

1) (Λ120581minus 1198862

2) (Λ120581minus 1198862

3) 120581 = 1 2 3

L1015840

120581= (Λ1015840

120581minus 1198862

1) (Λ1015840

120581minus 1198862

2) (Λ1015840

120581minus 1198862

3) 120581 = 1 2 3

(20)

4 Advances in Mathematical Physics

where the constants ΛΛ1015840 Λ120581 Λ1015840120581 120581 = 1 2 3 are given by

(5)ndash(8) respectively the following representations hold

E1

0(r) = radic4120587119903

0Y00(r)

E1

1(r) = ℎ

2ℎ3radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

E2

1(r) = 119894ℎ

1ℎ3radic2120587

3119903 (Yminus11(r) + Y1

1(r))

E3

1(r) = 2ℎ

1ℎ2radic120587

3119903Y01(r)

E1

2(r) = L + radic

4120587

5

L

Λ minus 11988623

1199032Y02(r)

+ radic2120587

15L(

1

Λ minus 11988621

minus1

Λ minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E2

2(r) = L

1015840+ radic

4120587

5

L1015840

Λ1015840 minus 11988623

1199032Y02(r)

+ radic2120587

15L1015840(

1

Λ1015840 minus 11988621

minus1

Λ1015840 minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E3

2(r) = 119894ℎ

1ℎ2ℎ2

3radic2120587

151199032(Yminus22(r) minus Y2

2(r))

E4

2(r) = ℎ

1ℎ2

2ℎ3radic2120587

151199032(Yminus12(r) minus Y1

2(r))

E5

2(r) = 119894ℎ

2

1ℎ2ℎ3radic2120587

151199032(Yminus12(r) + Y1

2(r))

E1

3(r) = ℎ

2ℎ3L1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L1

Λ1minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1radic120587

35(

1

Λ1minus 11988621

minus1

Λ1minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E2

3(r) = ℎ

2ℎ3L1015840

1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L10158401

Λ10158401minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1015840

1radic120587

35(

1

Λ10158401minus 11988621

minus1

Λ10158401minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E3

3(r) = 119894ℎ

1ℎ3L2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L2

Λ2minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L2radic120587

35(

1

Λ2minus 11988621

minus1

Λ2minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E4

3(r) = 119894ℎ

1ℎ3L1015840

2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L10158402

Λ10158402minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L1015840

2radic120587

35(

1

Λ10158402minus 11988621

minus1

Λ10158402minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E5

3(r) = 2ℎ

1ℎ2L3radic120587

3119903Y01(r) + 2 ℎ1ℎ2L3

Λ3minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L3radic2120587

105(

1

Λ3minus 11988621

minus1

Λ3minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E6

3(r) = 2ℎ

1ℎ2L1015840

3radic120587

3119903Y01(r) + 2

ℎ1ℎ2L10158403

Λ10158403minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L1015840

3radic2120587

105(

1

Λ10158403minus 11988621

minus1

Λ10158403minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E7

3(r) = 119894ℎ

2

1ℎ2

2ℎ2

3radic2120587

1051199033(Yminus23(r) minus Y2

3(r))

(21)

The above relations are not hard to prove considering thatthe first term of the product present in the integrand of (18)displays the general form 119909

120581

111990911989721199091199043where 120581+119897+119904 = 119899 implying

that only terms of the same or lower degree 119899 survive Inaddition switching to spherical coordinates and bearing inmind that the point (119909

1 1199092 1199093) resides interior of the ellipsoid

(2) give the desired resultsOn the other hand in order to evaluate the exterior

ellipsoidal harmonics provided via (14) and (15) one needsonly to express the quadratic terms (E119898

119899(119905))2 as a function of

Legendre polynomials This is easily done furnishing

(E10(119905))2

= P0(119905) = 1

(E11(119905))2

=2

3P2(119905) +

1

3

Advances in Mathematical Physics 5

(E21(119905))2

=2

3P2(119905) +

1

3minus ℎ2

3

(E31(119905))2

=2

3P2(119905) +

1

3minus ℎ2

2

(E12(119905))2

=8

35P4(119905) + 4(

1

7+Λ minus 11988621

3)P2(119905)

+ (1

5+2

3(Λ minus 119886

2

1) + (Λ minus 119886

2

1)2

)

(E22(119905))2

=8

35P4(119905) + 4(

1

7+Λ1015840 minus 1198862

1

3)P2(119905)

+ (1

5+2

3(Λ1015840minus 1198862

1) + (Λ

1015840minus 1198862

1)2

)

(E32(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

3

3)P2(119905) + (

1

5minusℎ2

3

3)

(E42(119905))2

=8

35P4(119905) + 2(

2

7minusℎ22

3)P2(119905) + (

1

5minusℎ22

3)

(E52(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

2+ ℎ23

3)P2(119905)

+ (1

5minusℎ2

2+ ℎ23

3+ ℎ2

2ℎ2

3)

(E13(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)

(E23(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1015840

1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)

(E33(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1) ((Λ

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1)

times ((Λ2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ2minus 1198862

1)2

]

(E43(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1) ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1)

times ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ1015840

2minus 1198862

1)2

]

(E53(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1) ((Λ

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1)

times ((Λ3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ3minus 1198862

1)2

]

(E63(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1) ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1)

times ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ1015840

3minus 1198862

1)2

]

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

The corresponding Lame functions of the second kind are

F119898119899(119906) = E119898

119899(119906) I119898119899(119906) (10)

for every 119899 = 0 1 2 and119898 = 1 2 2119899 + 1 where

I119898119899(119906) = int

119906

1199060

d119905

(E119898119899(119905))2radic10038161003816100381610038161199052 minus ℎ23

1003816100381610038161003816radic10038161003816100381610038161199052 minus ℎ22

1003816100381610038161003816(11)

is an elliptic integralIn view of problems where the boundary consists of a

triaxial confocal ellipsoid the product of two Lame functionsbelonging to the same class defines the surface ellipsoidalharmonics S119898

119899 that is

S119898119899(120583 ]) = E119898

119899(120583)E119898119899(]) (12)

whereas

E119898

119899(120588 120583 ]) = E119898

119899(120588) S119898119899(120583 ]) (13)

designate the interior ellipsoidal harmonicsOn the other hand the exterior ellipsoidal harmonics are

specified as

F119898

119899(120588 120583 ]) = (2119899 + 1)E119898

119899(120588 120583 ]) I119898

119899(120588) (14)

where

I119898119899(120588) = int

+infin

120588

d119905

(E119898119899(119905))2radic1199052 minus ℎ2

3radic1199052 minus ℎ2

2

120588 ge ℎ2 (15)

22 Connecting Ellipsoidal and Spherical Harmonics Wealready mentioned in the introduction the paucity of generalformulas associating ellipsoidal harmonics with the samedegree or less spherical harmonics Another way to compre-hend this is the following As the triaxial ellipsoid deterioratesto a sphere the ellipsoidal harmonics reduce to the so-called spheroconal harmonics which are a form of sphericalharmonics The spheroconal system which incorporatesthe radial coordinate 119903 of the spherical system with thecoordinates of the ellipsoidal system that specify orientationover any ellipsoidal surface (120583 ]) is established on the sameparameters 119886

120581 120581 = 1 2 3 thus preserving its ellipsoidal

characteristic Nevertheless although it seems that a generalframework cannot be established it is possible to representdistinct ellipsoidal harmonics with reference to finite termsof spherical harmonics (see Figure 1 for an illustration)

Forasmuch as the spherical harmonics form a completeset any continuous function can be expanded in a series ofY119898119899(r) Therefore

E119902

119901(r) =

infin

sum119899=0

119899

sum119898=minus119899

119860119898

119899119903119899Y119898119899(r) 119901 ge 0 119902 = 1 2 2119901 + 1

(16)

and the coefficients 119860119898119899

depend solely on the referencesemiaxes 119886

1 1198862 1198863

Ellipsoidal harmonics Spherical harmonics

Cartesian coordinates

Figure 1 In order to obtain representations of ellipsoidal harmonicsin terms of spherical harmonics and vice versa one has to go throughCartesian coordinates (solid lines) A direct connection appears notto be feasible (dashed line)

Provided that on the unit sphere 1198782

∮1198782

Y119898119899(r)Y119898

1015840

1198991015840 (r) dΩ (r) = 120575

11989911989910158401205751198981198981015840 (17)

the coefficients of (16) are computed as

119860119898

119899=1

119903119899∮1198782

E119902

119901(r)Y119898119899(r) dΩ (r) (18)

Equations (16) and (18) provide the backbone of the presentedanalysis

Employing the Cartesian form of the ellipsoidal harmon-ics for degree up to three namely

E1

0(r) = 1

E120581

1(r) = ℎ

1ℎ2ℎ3

ℎ120581

119909120581 120581 = 1 2 3

E1

2(r) = L(

3

sum119895=1

1199092

119895

Λ minus 1198862119895

+ 1)

E2

2(r) = L

1015840(

3

sum119895=1

1199092

119895

Λ1015840 minus 1198862119895

+ 1)

E6minus120581

2(r) = ℎ

1ℎ2ℎ3ℎ120581

119909111990921199093

119909120581

120581 = 1 2 3

E2120581minus1

2(r) = ℎ

1ℎ2ℎ3L120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ120581minus 1198862119895

+ 1) 120581 = 1 2 3

E2120581

2(r) = ℎ

1ℎ2ℎ3L1015840

120581

119909120581

ℎ120581

(

3

sum119895=1

1199092119895

Λ1015840120581minus 1198862119895

+ 1) 120581 = 1 2 3

E7

3(r) = ℎ

2

1ℎ2

2ℎ2

3119909111990921199093

(19)

provided that r = (119909 119910 119911) and

L = (Λ minus 1198862

1) (Λ minus 119886

2

2) (Λ minus 119886

2

3)

L1015840= (Λ1015840minus 1198862

1) (Λ1015840minus 1198862

2) (Λ1015840minus 1198862

3)

L120581= (Λ120581minus 1198862

1) (Λ120581minus 1198862

2) (Λ120581minus 1198862

3) 120581 = 1 2 3

L1015840

120581= (Λ1015840

120581minus 1198862

1) (Λ1015840

120581minus 1198862

2) (Λ1015840

120581minus 1198862

3) 120581 = 1 2 3

(20)

4 Advances in Mathematical Physics

where the constants ΛΛ1015840 Λ120581 Λ1015840120581 120581 = 1 2 3 are given by

(5)ndash(8) respectively the following representations hold

E1

0(r) = radic4120587119903

0Y00(r)

E1

1(r) = ℎ

2ℎ3radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

E2

1(r) = 119894ℎ

1ℎ3radic2120587

3119903 (Yminus11(r) + Y1

1(r))

E3

1(r) = 2ℎ

1ℎ2radic120587

3119903Y01(r)

E1

2(r) = L + radic

4120587

5

L

Λ minus 11988623

1199032Y02(r)

+ radic2120587

15L(

1

Λ minus 11988621

minus1

Λ minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E2

2(r) = L

1015840+ radic

4120587

5

L1015840

Λ1015840 minus 11988623

1199032Y02(r)

+ radic2120587

15L1015840(

1

Λ1015840 minus 11988621

minus1

Λ1015840 minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E3

2(r) = 119894ℎ

1ℎ2ℎ2

3radic2120587

151199032(Yminus22(r) minus Y2

2(r))

E4

2(r) = ℎ

1ℎ2

2ℎ3radic2120587

151199032(Yminus12(r) minus Y1

2(r))

E5

2(r) = 119894ℎ

2

1ℎ2ℎ3radic2120587

151199032(Yminus12(r) + Y1

2(r))

E1

3(r) = ℎ

2ℎ3L1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L1

Λ1minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1radic120587

35(

1

Λ1minus 11988621

minus1

Λ1minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E2

3(r) = ℎ

2ℎ3L1015840

1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L10158401

Λ10158401minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1015840

1radic120587

35(

1

Λ10158401minus 11988621

minus1

Λ10158401minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E3

3(r) = 119894ℎ

1ℎ3L2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L2

Λ2minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L2radic120587

35(

1

Λ2minus 11988621

minus1

Λ2minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E4

3(r) = 119894ℎ

1ℎ3L1015840

2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L10158402

Λ10158402minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L1015840

2radic120587

35(

1

Λ10158402minus 11988621

minus1

Λ10158402minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E5

3(r) = 2ℎ

1ℎ2L3radic120587

3119903Y01(r) + 2 ℎ1ℎ2L3

Λ3minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L3radic2120587

105(

1

Λ3minus 11988621

minus1

Λ3minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E6

3(r) = 2ℎ

1ℎ2L1015840

3radic120587

3119903Y01(r) + 2

ℎ1ℎ2L10158403

Λ10158403minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L1015840

3radic2120587

105(

1

Λ10158403minus 11988621

minus1

Λ10158403minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E7

3(r) = 119894ℎ

2

1ℎ2

2ℎ2

3radic2120587

1051199033(Yminus23(r) minus Y2

3(r))

(21)

The above relations are not hard to prove considering thatthe first term of the product present in the integrand of (18)displays the general form 119909

120581

111990911989721199091199043where 120581+119897+119904 = 119899 implying

that only terms of the same or lower degree 119899 survive Inaddition switching to spherical coordinates and bearing inmind that the point (119909

1 1199092 1199093) resides interior of the ellipsoid

(2) give the desired resultsOn the other hand in order to evaluate the exterior

ellipsoidal harmonics provided via (14) and (15) one needsonly to express the quadratic terms (E119898

119899(119905))2 as a function of

Legendre polynomials This is easily done furnishing

(E10(119905))2

= P0(119905) = 1

(E11(119905))2

=2

3P2(119905) +

1

3

Advances in Mathematical Physics 5

(E21(119905))2

=2

3P2(119905) +

1

3minus ℎ2

3

(E31(119905))2

=2

3P2(119905) +

1

3minus ℎ2

2

(E12(119905))2

=8

35P4(119905) + 4(

1

7+Λ minus 11988621

3)P2(119905)

+ (1

5+2

3(Λ minus 119886

2

1) + (Λ minus 119886

2

1)2

)

(E22(119905))2

=8

35P4(119905) + 4(

1

7+Λ1015840 minus 1198862

1

3)P2(119905)

+ (1

5+2

3(Λ1015840minus 1198862

1) + (Λ

1015840minus 1198862

1)2

)

(E32(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

3

3)P2(119905) + (

1

5minusℎ2

3

3)

(E42(119905))2

=8

35P4(119905) + 2(

2

7minusℎ22

3)P2(119905) + (

1

5minusℎ22

3)

(E52(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

2+ ℎ23

3)P2(119905)

+ (1

5minusℎ2

2+ ℎ23

3+ ℎ2

2ℎ2

3)

(E13(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)

(E23(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1015840

1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)

(E33(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1) ((Λ

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1)

times ((Λ2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ2minus 1198862

1)2

]

(E43(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1) ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1)

times ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ1015840

2minus 1198862

1)2

]

(E53(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1) ((Λ

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1)

times ((Λ3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ3minus 1198862

1)2

]

(E63(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1) ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1)

times ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ1015840

3minus 1198862

1)2

]

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

where the constants ΛΛ1015840 Λ120581 Λ1015840120581 120581 = 1 2 3 are given by

(5)ndash(8) respectively the following representations hold

E1

0(r) = radic4120587119903

0Y00(r)

E1

1(r) = ℎ

2ℎ3radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

E2

1(r) = 119894ℎ

1ℎ3radic2120587

3119903 (Yminus11(r) + Y1

1(r))

E3

1(r) = 2ℎ

1ℎ2radic120587

3119903Y01(r)

E1

2(r) = L + radic

4120587

5

L

Λ minus 11988623

1199032Y02(r)

+ radic2120587

15L(

1

Λ minus 11988621

minus1

Λ minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E2

2(r) = L

1015840+ radic

4120587

5

L1015840

Λ1015840 minus 11988623

1199032Y02(r)

+ radic2120587

15L1015840(

1

Λ1015840 minus 11988621

minus1

Λ1015840 minus 11988622

)

times 1199032(Yminus22(r) + Y2

2(r))

E3

2(r) = 119894ℎ

1ℎ2ℎ2

3radic2120587

151199032(Yminus22(r) minus Y2

2(r))

E4

2(r) = ℎ

1ℎ2

2ℎ3radic2120587

151199032(Yminus12(r) minus Y1

2(r))

E5

2(r) = 119894ℎ

2

1ℎ2ℎ3radic2120587

151199032(Yminus12(r) + Y1

2(r))

E1

3(r) = ℎ

2ℎ3L1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L1

Λ1minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1radic120587

35(

1

Λ1minus 11988621

minus1

Λ1minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E2

3(r) = ℎ

2ℎ3L1015840

1radic2120587

3119903 (Yminus11(r) minus Y1

1(r))

+ℎ2ℎ3L10158401

Λ10158401minus 11988623

radic120587

211199033(Yminus13(r) minus Y1

3(r))

+ ℎ2ℎ3L1015840

1radic120587

35(

1

Λ10158401minus 11988621

minus1

Λ10158401minus 11988622

)

times 1199033(Yminus33(r) minus Y3

3(r))

E3

3(r) = 119894ℎ

1ℎ3L2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L2

Λ2minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L2radic120587

35(

1

Λ2minus 11988621

minus1

Λ2minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E4

3(r) = 119894ℎ

1ℎ3L1015840

2radic2120587

3119903 (Yminus11(r) + Y1

1(r))

+ 119894ℎ1ℎ3L10158402

Λ10158402minus 11988623

radic120587

211199033(Yminus13(r) + Y1

3(r))

+ 119894ℎ1ℎ3L1015840

2radic120587

35(

1

Λ10158402minus 11988621

minus1

Λ10158402minus 11988622

)

times 1199033(Yminus33(r) + Y3

3(r))

E5

3(r) = 2ℎ

1ℎ2L3radic120587

3119903Y01(r) + 2 ℎ1ℎ2L3

Λ3minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L3radic2120587

105(

1

Λ3minus 11988621

minus1

Λ3minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E6

3(r) = 2ℎ

1ℎ2L1015840

3radic120587

3119903Y01(r) + 2

ℎ1ℎ2L10158403

Λ10158403minus 11988623

radic120587

71199033Y03(r)

+ ℎ1ℎ2L1015840

3radic2120587

105(

1

Λ10158403minus 11988621

minus1

Λ10158403minus 11988622

)

times 1199033(Yminus23(r) + Y2

3(r))

E7

3(r) = 119894ℎ

2

1ℎ2

2ℎ2

3radic2120587

1051199033(Yminus23(r) minus Y2

3(r))

(21)

The above relations are not hard to prove considering thatthe first term of the product present in the integrand of (18)displays the general form 119909

120581

111990911989721199091199043where 120581+119897+119904 = 119899 implying

that only terms of the same or lower degree 119899 survive Inaddition switching to spherical coordinates and bearing inmind that the point (119909

1 1199092 1199093) resides interior of the ellipsoid

(2) give the desired resultsOn the other hand in order to evaluate the exterior

ellipsoidal harmonics provided via (14) and (15) one needsonly to express the quadratic terms (E119898

119899(119905))2 as a function of

Legendre polynomials This is easily done furnishing

(E10(119905))2

= P0(119905) = 1

(E11(119905))2

=2

3P2(119905) +

1

3

Advances in Mathematical Physics 5

(E21(119905))2

=2

3P2(119905) +

1

3minus ℎ2

3

(E31(119905))2

=2

3P2(119905) +

1

3minus ℎ2

2

(E12(119905))2

=8

35P4(119905) + 4(

1

7+Λ minus 11988621

3)P2(119905)

+ (1

5+2

3(Λ minus 119886

2

1) + (Λ minus 119886

2

1)2

)

(E22(119905))2

=8

35P4(119905) + 4(

1

7+Λ1015840 minus 1198862

1

3)P2(119905)

+ (1

5+2

3(Λ1015840minus 1198862

1) + (Λ

1015840minus 1198862

1)2

)

(E32(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

3

3)P2(119905) + (

1

5minusℎ2

3

3)

(E42(119905))2

=8

35P4(119905) + 2(

2

7minusℎ22

3)P2(119905) + (

1

5minusℎ22

3)

(E52(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

2+ ℎ23

3)P2(119905)

+ (1

5minusℎ2

2+ ℎ23

3+ ℎ2

2ℎ2

3)

(E13(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)

(E23(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1015840

1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)

(E33(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1) ((Λ

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1)

times ((Λ2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ2minus 1198862

1)2

]

(E43(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1) ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1)

times ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ1015840

2minus 1198862

1)2

]

(E53(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1) ((Λ

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1)

times ((Λ3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ3minus 1198862

1)2

]

(E63(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1) ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1)

times ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ1015840

3minus 1198862

1)2

]

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

(E21(119905))2

=2

3P2(119905) +

1

3minus ℎ2

3

(E31(119905))2

=2

3P2(119905) +

1

3minus ℎ2

2

(E12(119905))2

=8

35P4(119905) + 4(

1

7+Λ minus 11988621

3)P2(119905)

+ (1

5+2

3(Λ minus 119886

2

1) + (Λ minus 119886

2

1)2

)

(E22(119905))2

=8

35P4(119905) + 4(

1

7+Λ1015840 minus 1198862

1

3)P2(119905)

+ (1

5+2

3(Λ1015840minus 1198862

1) + (Λ

1015840minus 1198862

1)2

)

(E32(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

3

3)P2(119905) + (

1

5minusℎ2

3

3)

(E42(119905))2

=8

35P4(119905) + 2(

2

7minusℎ22

3)P2(119905) + (

1

5minusℎ22

3)

(E52(119905))2

=8

35P4(119905) + 2(

2

7minusℎ2

2+ ℎ23

3)P2(119905)

+ (1

5minusℎ2

2+ ℎ23

3+ ℎ2

2ℎ2

3)

(E13(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1minus 1198862

1) +

1

3(Λ1minus 1198862

1)2

)

(E23(119905))2

=16

231P6(119905) +

8

7(3

11+2

5(Λ1015840

1minus 1198862

1))P4(119905)

+ 2 (5

21+4

7(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)P2(119905)

+ (1

7+2

5(Λ1015840

1minus 1198862

1) +

1

3(Λ1015840

1minus 1198862

1)2

)

(E33(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1) ((Λ

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ2minus 1198862

1)

times ((Λ2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ2minus 1198862

1)2

]

(E43(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1) ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

2minus 1198862

1) minus ℎ2

3)

+1

3(Λ1015840

2minus 1198862

1)

times ((Λ1015840

2minus 1198862

1) minus 2ℎ

2

3) minus ℎ2

3(Λ1015840

2minus 1198862

1)2

]

(E53(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1) ((Λ

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ3minus 1198862

1)

times ((Λ3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ3minus 1198862

1)2

]

(E63(119905))2

=16

231P6(119905)

+8

7[3

11+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)]P4(119905)

+ 2 [5

21+2

7(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1) ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2)]P2(119905)

+ [1

7+1

5(2 (Λ1015840

3minus 1198862

1) minus ℎ2

2)

+1

3(Λ1015840

3minus 1198862

1)

times ((Λ1015840

3minus 1198862

1) minus 2ℎ

2

2) minus ℎ2

2(Λ1015840

3minus 1198862

1)2

]

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

(E73(119905))2

=16

231P6(119905)

+8

7[3

11minus1

5(ℎ2

2+ ℎ2

3)]P4(119905)

+ 2 [5

21minus2

7(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]P2(119905)

+ [1

7minus1

5(ℎ2

2+ ℎ2

3) +

1

3(ℎ2

2ℎ2

3)]

(22)

3 Applications

In order to demonstrate the practicality of the precedingrelations two physical applications are illustrated The firstproblem comprises the field of low-Reynolds number hydro-dynamics while the second one emerges from the area of low-frequency electromagnetic scattering

31 Particle-in-Cell Models for Stokes Flow in EllipsoidalGeometry A dimensionless criterion which determines therelative importance of inertial and viscous effects in fluiddynamics is the Reynolds number (Re) The pseudosteadyand nonaxisymmetric creeping flow (Re ≪ 1) of an incom-pressible viscous fluid with dynamic viscosity 120583

119891and mass

density 120588119891is described by the well-known Stokes equations

Let us consider any smooth and bounded or unboundedthree-dimensional domain119881(R3) depending on the physicalproblem in terms of the position vector r = (119909

1 1199092 1199093)

Therein Stokes flow governing equations connect the bihar-monic vector velocity k with the harmonic scalar totalpressure field 119875 via

120583119891Δk = nabla119875 nabla sdot k = 0 (23)

for every r isin 119881(R3)The left-hand side of (23) states that in creeping flow the

viscous force compensates the force caused by the pressuregradient while the right-hand side of it secures the incom-pressibility of the fluid The fluid total pressure is related tothe thermodynamic pressure 119901 through

119875 = 119901 + 120588119891119892ℎ (24)

where the contribution of the term 120588119891119892ℎ with 119892 being the

acceleration of gravity refers to the gravitational pressureforce corresponding to a height of reference ℎ Once thevelocity field is obtained the harmonic vorticity field 120596 isdefined as

120596 =1

2nabla times k (25)

while the stress dyadic Π is assumed to be

Π = minus119901I + 120583119891[nabla otimes k + (nabla otimes k)⊺] (26)

in terms of the unit dyadic I = sum3119895=1

x119895otimesx119895 where the symbols

otimes and ⊺ denote juxtaposition and transposition respectively

One of the important areas of applications concerns theconstruction of particle-in-cell models which are useful inthe development of simple but reliable analytical expressionsfor heat and mass transfer in swarms of particles in thecase of concentrated suspensions In applied type analysisit is not usually necessary to have detailed solution of theflow field over the entire swarm of particles taking intoaccount the exact positions of the particles since suchsolutions are cumbersome to use Thus the technique ofcell models is adopted where the mathematical treatmentof each problem is based on the assumption that a three-dimensional assemblage may be considered to consist of anumber of identical unit cells Each of these cells contains aparticle surrounded by a fluid envelope containing a volumeof fluid sufficient to make the fractional void volume in thecell identical to that in the entire assemblage

With the aim of mathematical modeling of the particularphysical problem we inherit chosen among others cited inoriginal paper [5] from where the main results were drawnthe Happel cell model [6] in which both the particle andthe outer envelope enjoy spherical symmetry In view ofthe Happel-type model [5 6] we consider a fluid-particlesystem consisting of any finite number of rigid particles ofarbitrary shape Introducing the particle-in-cell model weexamine the Stokes flow of one of the assemblages of particlesneglecting the interaction with other particles or with thebounded walls of a container Let 119878

119886denote the surface of

the particle of the swarm which is solid is moving witha known constant translational velocity U = (119880

1 1198802 1198803)

in an arbitrary direction and is rotating also arbitrarilywith a defined constant angular velocity Ω = (Ω

1 Ω2 Ω3)

It lives within an otherwise quiescent fluid layer which isconfined by the outer surface denoted by 119878

119887 Following the

formulation of Happel [6] extended to three-dimensionalflows the necessary nonslip flow conditions

k = U +Ω times r r isin 119878119886 (27)

are imposed on the surface of the particle while the velocitycomponent field normal to 119878

119887and the tangential stresses are

assumed to vanish on 119878119887 that is

n sdot k = 0 r isin 119878119887 (28)

n sdot Π sdot (I minus n otimes n) = 0 r isin 119878119887 (29)

where n is the outer unit normal vector Equations (23)ndash(29)define a well-posed Happel-type boundary value problem for3D domains 119881(R3) bounded in our case by two arbitrarysurfaces 119878

119886and 119878119887

Papkovich (1932) and Neuber (1934) proposed a differ-ential representation of the flow fields in terms of harmonicfunctions [7 8] which is derivable from the well-knownNaghdi-Hsu solution [9] and it is applicable to axisymmetricbut also to nonaxisymmetric domains as in our caseWeprofitby the major advantage of the Papkovich-Neuber differentialrepresentation which can be used to obtain solutions ofcreeping flow in cell models where the shape of the particlesis genuinely three-dimensional Let us notice that the loss of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

symmetry is caused by the imposed rotation of the particlesThus in terms of two harmonic potentialsΦ and Ψ

ΔΦ = 0 ΔΨ = 0 (30)

the Papkovich-Neuber general solution represents the veloc-ity and the total pressure fields appearing in (23) via the actionof differential operators onΦ and Ψ that is

k = Φ minus1

2nabla (r sdotΦ + Ψ) (31)

119875 = 1198750minus 120583119891nabla sdotΦ (32)

whereas1198750is a constant pressure of reference usually assigned

at a convenient point It can be easily confirmed that (31) and(32) satisfy Stokes equations (23)

Ellipsoidal geometry provides us with the most widelyused framework for representing small particles of arbi-trary shape embedded within a fluid that flows accordingto Stokes law This nonaxisymmetric flow is governed bythe genuine 3D ellipsoidal geometry which embodies thecomplete anisotropy of the three-dimensional space Theabove Happel-type problem (23)ndash(26) accompanied by theappropriate boundary conditions (27)ndash(29) is solved withthe aim of the Papkovich-Neuber differential representation(30)ndash(32) using ellipsoidal coordinates The fields are pro-vided in a closed form fashion as full series expansions ofellipsoidal harmonic eigenfunctions The velocity to the firstdegree which represents the leading term of the series issufficient formost engineering applications and also providesus with the corresponding full 3D solution for the sphere[10] after a proper reduction The whole analysis is based onthe Lame functions and the theory of ellipsoidal harmonicsIn fact only harmonics of degree less than or equal to twoare needed to obtain the velocity field of the first degreeAnalytical expressions for the leading terms of the totalpressure the angular velocity and the stress tensor fields areprovided in [5] as well

Introducing the two boundary ellipsoidal surfaces withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119886at 119878119886for the inner on the particle and 120588 = 120588

119887at 119878119887

for the outer boundary of the fluid envelope then 119881(R3) =(120588 120583 ]) 120588 isin (120588

119886 120588119887) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3] confines the

actual domain of the flow study In view of the orthonormalunit vector in ellipsoidal coordinates and in terms ofthe connection formulae between ellipsoidal and sphericalharmonics the mixed Cartesian-ellipsoidal form of the mainresults obtained in [5] admits more tractable expressions nowvia the spherical harmonics where any numerical treatmentis much more feasible

After some extended algebra the velocity field renders

k = U +Ω times r + Z (120588) +3

sum120581=1

H120581(120588)E120581

1(r)

+

2radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)E119895

1(r) +

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2(r)]]]

]

(33)

provided that

Z (120588)

= minusE73(120588119887)U

sdot

3

sum119895=1

[(I10(120588) minus I1

0(120588119886)) + (E119895

1(120588119886))2

(I1198951(120588) minus I119895

1(120588119886))]

timesx119895otimes x119895

119873119895

(34)

H120581(120588)

=3

2

3

sum119895=1

119895 =120581

ℎ119895119890120581

119895(I1205811(120588) minus I120581

1(120588119886))

minus ℎ120581119890119895

120581(I1198951(120588) minus I119895

1(120588119886))

+ [ℎ119895119890120581

119895(E1198951(120588119886))2

+ ℎ120581119890119895

120581(E1205811(120588119886))2

]

times (I119895+1205812

(120588) minus I119895+1205812

(120588119886))

x119895

ℎ119895

120581 = 1 2 3

(35)

whereas

Θ120581(120588) = minus

2ℎ120581E73(120588119887)

ℎ1ℎ2ℎ3

U sdot x120581

119873120581

[1 minus (E1205811(120588119886)

E1205811(120588)

)

2

]

120581 = 1 2 3

(36)

Φ119897

120581(120588) =

3ℎ120581

ℎ1ℎ2ℎ3

119890119897

120581[1 minus (

E1205811(120588119886)

E1205811(120588)

)

2

]1

(E1198971(120588))2

120581 119897 = 1 2 3 120581 = 119897

(37)

On the other hand the constants119873120581 120581 = 1 2 3 and 119890119897

120581 120581 119897 =

1 2 3 120581 = 119897 present in (34) (36) and (35) (37) respectivelycan be found analytically in [5]

Once the above constants are calculated the velocity fieldis obtained in terms of the applied fieldsU andΩ via (33)Thetotal pressure field assumes the expression

119875 = 1198750+ 120583119891

E73(120588)

(1205882 minus 1205832) (1205882 minus ]2)

times[[[

[

3

sum119895=1

Θ119895(120588)

(1205882 minus 1205882119886)E119895

1+

3

ℎ1ℎ2ℎ3

3

sum119894119895=1

119894 =119895

ℎ119894119890119895

119894

(E119894+1198952(120588))2E119894+119895

2

]]]

]

(38)

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

Matching results (25) and (33) we arrive at

120596 = Ω +ℎ1ℎ2ℎ3

2

3

sum119895=1

x119895timesH119895(120588)

ℎ119895

+radic1205882 minus ℎ2

3radic1205882 minus ℎ2

2

2radic1205882 minus 1205832radic1205882 minus ]2

times [

[

dZ (120588)d120588

+

3

sum119895=1

dH119895(120588)

d120588E119895

1]

]

minusℎ1ℎ2ℎ3

4radic1205882 minus 1205832radic1205882 minus ]2

times[[[

[

3

sum119895=1

Θ119895(120588)

ℎ119895

x119895+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)(

E1198941

ℎ119895

x119895+E119895

1

ℎ119894

x119894)]]]

]

(39)

for the vorticity fieldOn the other hand substitution of (38) and (33) with

respect to (24) into (26) reveals that the stress tensor yields

Π = (120588119891119892ℎ minus 119875) I

minus 120583119891

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2

times ( otimesdZ (120588)d120588

+dZ (120588)d120588

otimes )

+

3

sum119895=1

[[

[

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2E119895

1

times ( otimesdH119895(120588)

d120588+dH119895(120588)

d120588otimes )

+ℎ1ℎ2ℎ3

ℎ119895

times (x119895otimesH119895(120588) +H

119895(120588) otimes x

119895)]]

]

minus120583119891

radic1205882 minus 1205832radic1205882 minus ]2

times

radic1205882 minus ℎ23radic1205882 minus ℎ2

2

radic1205882 minus 1205832radic1205882 minus ]2 otimes

times[[[

[

3

sum119895=1

dΘ119895(120588)

d120588E119895

1+

3

sum119894119895=1

119894 =119895

dΦ119895119894(120588)

d120588E119894+119895

2

]]]

]

+ℎ1ℎ2ℎ3

2

times [

[

3

sum119895=1

Θ119895(120588)

ℎ119895

( otimes x119895+ x119895otimes )

+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)

times (E1198941

ℎ119895

( otimes x119895+ x119895otimes )

+E119895

1

ℎ119894

( otimes x119894+ x119894otimes ))]

]

minus120583119891

2S[[[

[

3

sum119895=1

Θ119895(120588)E119895

1+

3

sum119894119895=1

119894 =119895

Φ119895

119894(120588)E119894+119895

2

]]]

]

(40)

where the dyadic S is given as a function of the metriccoefficients in ellipsoidal geometry in the form

S = 2

radic1205882 minus 1205832radic1205882 minus ]2

times 120588

ℎ120588

[minus(1

1205882 minus 1205832+

1

1205882 minus ]2) otimes

+ otimes

1205882 minus 1205832+

otimes

1205882 minus ]2]

+120583

ℎ120583(1205882 minus 1205832)

( otimes + otimes )

+]

ℎ] (1205882 minus ]2)

( otimes + otimes )

(41)

With the aim of the much helpful connection formulasderived in this work we can interpret the ellipsoidal har-monics E120581

1and E6minus120581

2 120581 = 1 2 3 in terms of the spherical

harmonics

32 Low-Frequency Electromagnetic Scattering by Impen-etrable Ellipsoidal Bodies Practical applications in severalphysical areas relative to electromagnetism (eg geophysics)are often concerned with the problem of identifying andretrieving metallic anomalies and impenetrable obstacleswhich are buried under the surface of the conductive EarthThe detailed way where these problems are dealt with can be

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

found analytically in relevant reference [11] of this particularapplication

In that sense we consider a solid body of arbitraryshape with impenetrable surface 119878 The perfectly electricallyconducting target is embedded in conductive homogeneousisotropic and nonmagnetic medium of conductivity 120590 and ofpermeability 120583 (usually approximated by the permeability offree space 120583

0) where the complex-valued wave number at the

operation frequency

119896 = radic119894120596120583120590 = radic120596120583120590

2(1 + 119894) (42)

is provided in terms of imaginary unit 119894 and low circularfrequency 120596 the dielectric permittivity 120576 being 120576 ≪ 120590120596The external three-dimensional space is 119882(R3) consideredto be smooth and bounded or not bounded depending on thephysical problem Without loss of generality the harmonictime-dependence exp(minus119894120596119905) of all field quantities is impliedthus they are written in terms of the position vector r =

(1199091 1199092 1199093) The metallic object is illuminated by a known

magnetic dipole source m located at a prescribed positionr0and arbitrarily orientated which is defined via

m =

3

sum119895=1

119898119895x119895 (43)

It is operated at the acceptable low-frequency regime andproduces three-dimensional incident waves in an arbitrarydirection the so-called electromagnetic incident fields Hin

and Ein which are radiated by the magnetic dipole (43) andare scattered by the solid target creating the scattered fieldsH119904 and E119904 Therein

H119905 = Hin+H119904 E119905 = Ein

+ E119904 (44)

for every r isin 119882(R3) minus r0 are the total magnetic and electric

fields given by superposition of corresponding incidentand scattered fields where the singular point r

0has been

excludedIn order to put those tools together within the framework

of the low-frequency diffusive scattering theory method[12 13] we expand the incident (in) the scattered (119904)and consequently the total (119905) electromagnetic fields in aRayleigh-like manner of positive integral powers of 119894119896 where119896 is given by (42) such as

H119909 =infin

sumℓ=0

H119909ℓ(119894119896)ℓ E119909 =

infin

sumℓ=0

E119909ℓ(119894119896)ℓ 119909 = in 119904 119905 (45)

So Maxwellrsquos equations

nabla times E119909 = 119894120596120583H119909

nabla timesH119909 = (minus119894120596120576 + 120590)E119909120576≪120590120596

cong 120590E119909 119909 = in 119904 119905(46)

are reduced into low-frequency forms

120590nabla times E119909ℓ= minusH119909ℓminus2

for ℓ ge 2

nabla timesH119909ℓ= 120590E119909ℓ

for ℓ ge 0 119909 = in 119904 119905(47)

where either in (46) or in (47) magnetic and electric fieldsare divergence-free for any r isin 119882(R3) minus r

0 that is

nabla sdotH119909 = nabla sdot E119909 = 0

nabla sdotH119909ℓ= nabla sdot E119909

ℓ= 0 for ℓ ge 0 119909 = in 119904 119905

(48)

The gradient operator nabla involved in the above relationsoperates on r

Letting R = |r minus r0| the electromagnetic incident fields

generated by the magnetic dipolem take the expressions [12]

Hin=

1

4120587[(1198962+119894119896

119877minus

1

1198772)m

minus(1198962+3119894119896

119877minus

3

1198772)r otimes r sdotm1198772

]119890119894119896119877

119877

Ein= [

120596120583119896

4120587(1 +

119894

119896119877)m times r119877

]119890119894119896119877

119877

(49)

where the symbol otimes denotes juxtaposition Extended alge-braic calculations upon (49) lead to the low-frequency forms(45) for the incident fields (119909 = in) where by implying thehypothesis of low frequency we conclude that the first fourterms of the expansions are adequate (see [11] for details)

Therefore our analysis has confined these importantterms of the expansions for the scattered fields as well Thoseare the static term (Rayleigh approximation) for ℓ = 0 andthe dynamic ones for ℓ = 1 2 3 while the terms for ℓ ge 4 arebeing expected to be very small and consequently they areneglected Hence the scattered magnetic field

H119904 = H1199040+H1199042(119894119896)2+H1199043(119894119896)3+ O ((119894119896)

4) (50)

and the scattered electric field

E119904 = E1199042(119894119896)2+ O ((119894119896)

4) (51)

inherit similar forms to those of the incident fields [11]whereH119904

0H1199042H1199043 and E119904

2are to be evaluated Substituting the

wave number 119896 of the surrounding medium from (42) intorelations (50) and (51) trivial analysis yields

H119904 = H1199040+ (120596120583120590)radic

120596120583120590

2H1199043

+ 119894 (120596120583120590) [radic120596120583120590

2H1199043minusH1199042] + O ((119894119896)

4)

(52)

E119904 = minus119894 (120596120583120590)E1199042+ O ((119894119896)

4) (53)

respectivelyThe electric field (53) is purely imaginary-valuedneeding only E119904

2 while the magnetic field is complex-valued

noticing that the electromagnetic fields H1199042and H119904

0are

adequate for the full solution since the contribution ofH1199043 as

the outcome of the corresponding constant field stands for avery small correction to both real and imaginary parts of thescattered magnetic field (52) On the other hand H119904

1= E1199040=

E1199041= E1199043= 0 in absence of the corresponding incident fields

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

Straightforward calculations on Maxwellrsquos equations (47)for 119909 = 119904 and elaborate use of identitynablatimesnablatimesf = nabla(nablasdotf)minusΔf f being any vector result in mixed boundary value problemswhich are possibly coupled to one anotherThose are given by

ΔH1199040= 0 997904rArr H119904

0= nablaΦ

119904

0 since nabla sdotH119904

0= 0 nabla timesH119904

0= 0(54)

ΔH1199042= H1199040997904rArr H119904

2= Φ119904

2+1

2(rΦ1199040)

120590E1199042= nabla timesH119904

2 since nabla sdotH119904

2= nabla sdot E119904

2= 0

(55)

ΔH1199043= 0 997904rArr H119904

3= nablaΦ

119904

3 since nabla sdotH119904

3= 0 (56)

which are written in terms of the harmonic potentialsΦ1199040Φ1199042

and Φ1199043that satisfy

ΔΦ119904

0= ΔΦ

119904

3= 0 ΔΦ

119904

2= 0 (57)

It is worth mentioning that for ℓ = 0 3 standard Laplaceequations must be solved for H119904

0and H119904

3fields whilst the

inhomogeneous vector Laplace equation (55) coupled withthe solution of (54) is a Poisson partial differential equationprovided that the zero-order scattered field H119904

0is obtained

the second-order scattered fieldH1199042can bewritten as a general

vector harmonic function Φ1199042plus a particular solution

(12)(rΦ1199040) As for the scattered electric field E119904

2for ℓ = 2

it is given by the curl of the corresponding magnetic field via(55)

The set of problems (54)ndash(57) has to be solved by using theproper perfectly electrically conducting boundary conditionson the surface 119878 of the perfectly electrically conducting bodyThey concern the total fields (44) at each order ℓ where usingoutward unit normal vector the normal component of thetotal magnetic field and the tangential component of the totalelectric field are canceling Hence combining (44) and (45)we readily obtain

n sdot (Hinℓ+H119904ℓ) = 0 for ℓ = 0 2 3

n times (Ein2+ E1199042) = 0 r isin 119878

(58)

where the Silver-Muller radiation conditions [12] at infinitymust automatically be satisfied

The most arbitrary shape which is consistent with thewell-known orthogonal curvilinear systems we know andreflects the random kind of objects that can be foundunderground is the ellipsoidal coordinate systemThereforewe can readily apply the mathematical tools of the ellip-soidal nonaxisymmetric geometry to our scattering modeldescribed earlier and after cumbersome yet rigorous cal-culations in terms of ellipsoidal harmonic eigenfunctions[1] we may provide the scattered fields as infinite seriesexpansionsThisworkwas accomplished in relevant reference[11] where the fields were presented in terms of the theoryof ellipsoidal harmonics and a proper reduction process tothe spherical model [14] was demonstrated At this pointwe must mention that in electromagnetic problems such as

ours where the source is far away from the object the low-frequency approximation is attainable and therein the use ofthe first degrees of the ellipsoidal harmonic eigenfunctionsin our case let us say 119899 = 0 1 2 is more than enough toreach convergence Thus the results collected in [11] use thistype of harmonics which was the key to obtaining closed-type forms for the ℓ-order (ℓ = 0 1 2 3) low-frequencyelectromagnetic fields (52) and (53) in terms of ellipsoidalharmonics

Introducing the boundary ellipsoidal metal surface withrespect to the radial ellipsoidal variable 120588 isin [ℎ

2 +infin) as

120588 = 120588119904= 1198861at 119878 119886

1being the major axis of the ellipsoid under

consideration then the region of electromagnetic scatteringis119882(R3) = (120588 120583 ]) 120588 isin (119886

1 +infin) 120583 isin [ℎ

3 ℎ2] ] isin [0 ℎ

3]

In view of the orthonormal unit vector in ellipsoidalcoordinates and in terms of the connection formulae betweenellipsoidal and spherical harmonics the mixed Cartesian-ellipsoidal form of the main results obtained in [11] providesus with more tractable expressions via the known sphericalharmonics where any numerical interpretation is mucheasier

Collecting everything together the scattered magneticfield is

H119904 =2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times

[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (119887119898

119899+ (120596120583120590)radic

120596120583120590

2119886119898

119899) + 119894 (120596120583120590)

times(radic120596120583120590

2119886119898

119899

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

minus (c119898119899+119887119898119899

2r) I119898119899(120588)E119898

119899)

+ O ((119894119896)4)

(59)

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

while for the scattered electric field we have

E119904 = minus119894120596120583

times

2

sum119899=0

2119899+1

sum119898=1

(2119899 + 1)

times[[

[

I119898119899(120588) nablaE

119898

119899

minus

[E119898119899(120588)]2radic1205882 minus 1205832radic1205882 minus ]2

E119898

119899

]]

]

times (c119898119899+119887119898119899

2r)

+ O ((119894119896)4)

(60)

The coefficients 119886119898119899 119887119898119899 and c119898

119899are found explicitly in [11]

Moreover the gradient present in (59) and (60) is easilyevaluated since

nabla (119903119899Y119898119899(r))

= 119903119899[119899

Y119898119899(r)119903

r

+1

sin2120579(119899119895119898

119899+1Y119898119899+1

(r) minus (119899 + 1) 119895119898119899Y119898119899minus1

(r))

+ 119894119898

119903 sin 120579Y119898119899(r) 120601]

(61)

where

119895119898

119899= radic

1198992 minus 1198982

41198992 minus 1 (62)

The ellipsoidal harmonicsE119898119899comprise the amenable derived

spherical harmonics

4 Conclusions

We introduce explicit formulas relating ellipsoidal harmonicsin terms of spherical harmonics up to the third degreenamely sixteen harmonics in total valid in the interiorand exterior of a confocal triaxial ellipsoid For a numberof physical problems depending on the prescribed condi-tions the existence of source terms et cetera these sixteenellipsoidal harmonics are adequate to ensure convergence ofthe related expansion to an extent reaching almost over 95per cent For these kinds of problems (two applications areadvertised in Section 3) the expressions supplied can be used

directly without prior knowledge of the theory of ellipsoidalharmonics

For the sake of completeness we indicate that an essen-tially identical process can be commenced correlating spher-ical harmonics with the associated ellipsoidal harmonicswhich however in view of practical problems is meaning-less

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Michael Doschoris gratefully acknowledges the contributionof the ldquoARISTEIArdquo Action of the ldquoOPERATIONAL PRO-GRAMME EDUCATION AND LIFELONG LEARNINGrdquocofunded by the European Social Fund (ESF) and NationalResources

References

[1] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press 2012

[2] G Dassios and K Satrazemi ldquoLame functions and Ellipsoidalharmonics up to degree sevenrdquo Submitted to InternationalJournal of Special Functions and Applications

[3] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[4] J P Bardhan and M G Knepley ldquoComputational scienceand re-discovery open-source implementation of ellipsoidalharmonics for problems in potential theoryrdquo ComputationalScience and Discovery vol 5 no 1 Article ID 014006 2012

[5] P Vafeas and G Dassios ldquoStokes flow in ellipsoidal geometryrdquoJournal ofMathematical Physics vol 47 no 9 Article ID 093102pp 1ndash38 2006

[6] J Happel ldquoViscous flow in multiparticle systems slow motionof fluids relative to beds of spherical particlesrdquo AIChE Journalvol 4 pp 197ndash201 1958

[7] X S Xu and M Z Wang ldquoGeneral complete solutions ofthe equations of spatial and axisymmetric Stokes flowrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol44 no 4 pp 537ndash548 1991

[8] H Neuber ldquoEin neuer Ansatz zur Losung raumlicher Problemeder Elastizitatstheorierdquo Journal of Applied Mathematics andMechanics vol 14 pp 203ndash212 1934

[9] P M Naghdi and C S Hsu ldquoOn the representation of displace-ments in linear elasticity in terms of three stress functionsrdquoJournal ofMathematics andMechanics vol 10 pp 233ndash245 1961

[10] G Dassios and P Vafeas ldquoThe 3D Happel model for completeisotropic Stokes flowrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 46 pp 2429ndash2441 2004

[11] G Perrusson P Vafeas and D Lesselier ldquoLow-frequencydipolar excitation of a perfect ellipsoidal conductorrdquo Quarterlyof Applied Mathematics vol 68 no 3 pp 513ndash536 2010

[12] G Dassios and R Kleinman Low Frequency Scattering OxfordUniversity Press Oxford UK 2000

[13] C Athanasiadis ldquoThe multi-layered ellipsoid with a soft corein the presence of a low-frequency acoustic waverdquo Quarterly

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Advances in Mathematical Physics

Journal of Mechanics and AppliedMathematics vol 47 no 3 pp441ndash459 1994

[14] P Vafeas G Perrusson and D Lesselier ldquoLow-frequencysolution for a perfectly conducting sphere in a conductivemedium with dipolar excitationrdquo Progress in ElectromagneticsResearch vol 49 pp 87ndash111 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of