H.K. Moffatt- On the behaviour of a suspension of conducting particles subjected to a time-periodic...

8/3/2019 H.K. Moffatt- On the behaviour of a suspension of conducting particles subjected to a time-periodic magnetic field

1/21

J.Fluid Mech. (1990),ol. 218, p p . 509-529Printed in Great Brita in

509

On the behaviour of a suspension of conductingparticles subjected to a time-periodic magnetic fieldBy H. K. MOFFATT

Department of Applied Mathematics & Theoretical Physics, University of Cambridge,Silver Street, Cambridge CB3 S E W , UK(Received 23 January 1990)

The force F ( x )and torque G(x) acting on a conducting body or particle suspendedat position x in a magnetic field Re (&x) e-iwt)are determined to leading order in theratio of the scale a of the particle to the scale L of the field. When the particle isspherical, F decomposes naturally into an irrotational 'lift ' force FL nd a solenoidal'drag' force P elated to G by

F "+ $V A G = 0.This relationship is important when the action of the field on a suspension of suchspheres is considered, because it implies that the net effective force per unit volumeacting to generate bulk flow is zero in any region where the concentration c isuniform. However, non-uniformity is generated by the force ingredient FL nd bulkflow is then generated through interaction of G and V c . These effects aredemonstrated for two examples involving rotat ing and travelling fields. Interactionsof a finite number N of spheres are also considered, and in particular it is shown thatwhen the field is a uniform rotating one, the governing dynamical system is atleading order Hamiltonian with four independent integral invariants. When N 2 4,the system in general exhibits chaos.

1. IntroductionWhen an electrically conducting particle is suspended in an a.c. (time-periodic)

magnetic field, electric currents are induced in the particle ; these interact with theapplied field to give a Lorentz force distribution, which has a mean (i.e. time-averaged) part , and a fluctuating part . There is then in general a total mean force Fand torque (or couple) G acting on the particle.

Although this type of induction problem is classical in character (Lamb 1883),there are certain features that do not seem to be widely understood. The featuresthat we shall focus on in this paper are well illustrated by three differentconfigurations, sketched in figure 1 (a-c), which are believed to be prototypical. Thefirst (figure 1 a ) is the rotating field configuration, in which a uniform field is causedto rota te with angular velocity w by the use of three-phase windings in the externalcoils carrying the source currents. If a conducting sphere is suspended in this field,then i t experiences a torque G tending to rotate it in the same sense as the rotationof the field. If the sphere is suspended on a wire, it will rotate until theelectromagnetic torque is balanced by the torque transmitted through the twistedwire and will then attain a position of equilibrium. This mechanism was firstidentified by Braunbeck (1932) . If, on the other hand, it is suspended in a (non-conducting) viscous fluid, then it will rotate with angular velocity 0 determined by

17-2

www.moffatt.tc


2/21

510w

H . K.Moffatt4 F L

-IGURE. Prototype fields, and their effect on a suspended sphere: (a ) otating field;( b ) travelling field; ( c ) levitating field.

( b )

the condition that the sum of the electromagnetic and viscous torques acting on thesphere be zero.The second configuration (figure 1 b ) is that of the travelling field, in which theapplied field has a progressive wave form - ei(kx-wt)again through the use of three-phase external windings. If a conducting sphere is placed in the symmetric positionP indicated in the figure, then it experiences a drag force FD n the direction oftravel of the wave, and tending to accelerate the sphere in this direction. If the sphereis surrounded by viscous fluid, then this acceleration will be arrested when the forceFD s balanced by the viscous drag acting on the sphere.

The third configuration (figure 1 c ) is th at of the levitating field which may beproduced by single-phase source currents. There is then a levitating or lift force FLacting on a suspended sphere in the direction indicated.

In more general situations, we shall find that the body experiences a torque G ndboth force ingredients FD nd FL imultaneously. (Forexample the sphere in the non-symmetric position Q in figure 1b experiences a force FD+FL nd a torque G n thesense indicated.) These depend on the position vector x of the centre of the sphere,and we shall find that there is a simple relation between the force ingredient FD ( x )and the torque G(x),namely,

FD = -gV A G. ( 1 . 1 )It is worth noting th at in both the problems of figures 1 ( a ) nd 1 ( b ) , t is the relativemotion between sphere and field which leads to induced currents and so to the

Lorentz force distribution. Thus, in the travelling field case (figure 1 b ) , in a frame ofreference moving with the field (i.e. with velocity V = ( w / k ,O , O ) ) , the field is a t rest,and the sphere moves with velocity - V .The mean drag force FD cting on the sphereis of course unaffected by this change of viewpoint (provided V / c < 1 and relativisticeffects are negligible).


3/21


4/21

512 H . K . Moffattwith G through the term -$ A Vc to produce a bulk flow. This flow is determinedin $6 or the case of a rotating multipole field, and for the travelling field. The variousthreads are drawn together in the concluding $ 7 .

2. Action of a time-periodic magnetic field o n a conducting particleConsider then a time-periodic magnetic field of the form

B , ( x , t ) = B F ) (x ) c o s w t + B b i ) ( x ) s i n w t Re ( B , ( X ) ~ - ~ ~ ~ ) , 2 .1 )where Bo = BF)+ Bf). In what follows, the real par t of expressions like B0 -iwt willbe understood. In the seecial situation in which B f ) ( x ) nd Br) (x ) re everywhereparallel, we may write Bo = BbT)ei#(x),o that

B , ( x , ) = BbT)(x)os ( $ ( x ) w t ) , (2 .2 )and the field is a single-phase field. Generally, however, Bf)nd B r ) are non-parallel,and we have a multiphase field. Note tha t, with * representing a complex conjugate,

iB, A 8 = 2 ~ r ) ~ ( i ), (2 .3 )and this is real and non-zero for a multiphase field.

We suppose that the field Bo ( x , ) is produced by a system of currents in coils orconductors tha t are external to a domain 9 n which the background medium is non-conducting. Hence Bo(x)s a potential field in 9,.e.

B , ( x )= vYo, WO 0, (2 .4 )where Y 0 ( x ) s a complex-valued scalar field. The lengthscale L characteristic ofvariation of B0 (defined by L-l - /VBoIl/llBoll here 1 1 . . . I is some suitably definednorm) is determined by the scale and remoteness of the source currents.

Suppose now that we introduce a particle of linear scale a small compared with L ,and uniform conductivity v , nto this field. Let S = (2 /p0vw)i where po= 4n x lO- '(S.I. units), and let

h = cc/S = ($2p0 w ) i , (2 .5 )a dimensionless parameter which plays an important part in what follows. Whenh 9 1, S is the familiar skin depth to which the magnetic field penetrates. Whenh 5 1, the field penetrates throughout the conductor.Let j ( x , t )= j ( ~ ) e - ~ ~ ~e the current induced in the particle. Then the dipole

moment of the associated magnetic field is p ( t )= where

B being the volume occupied by the particle. Since the induction problem is linear,there must exist a linear relationship between p and & x ) . Taking origin at thecentre of volume of the particle, this linear relationship may be expressed in termsof the value of B, and its derivatives at x = 0 :


5/21

A suspens ion of particles in a periodic magnetic field 513m[B,(x , ) +Sx .VB , (x , t ) ]"\

FIGURE. Elementary representation of a dipole in terms of magnetic poles &m at separation6x. The resultant force F and torque G are given by (2 .10) .

where ail ,P i l k , . . .are dimensionless tensor coefficients dependent on h and on theshape of the particle. This is effectively a power series in the small parameter a / L ,since the nth derivatives of Bo are of order L-"IB0l.

If the particle is spherical, then there is a remarkable simplificationt of the series( 2 . 7 ) for then the tensors ail,Pilk,. . .must be isotropic, i.e.aij = adij, Pi jk = 0, yijkl = y(l'8ij 8k l + y'2'8ik 8jl+ y'3'8{, j k > . . (2 .8)

Hence, firstly, all the odd-derivative terms of ( 2 . 7 )vanish. Secondly, since V * B, = 0and (from ( 2 . 4 ) )V zB o= 0 , each contribution to the third term of ( 2 . 7 )vanishes, anda fortiori , all higher-order even-derivative terms vanish also ! Hence we are left with

r; = 4npu,1aa3B0, ( 2 . 9 )where a(h) s a complex-valued scalar, which may be described as the 'inductioncoefficient'. It is important to emphasize here that 8, in ( 2 . 9 ) is evaluated at thecentre of symmetry of the particle, x = 0.

Consider now the mean force F and torque G acting on the particle. Regarding thedipolep as a pair of magnetic poles f at vector separation 6x (figure 2 ) ,elementaryconsiderations give F = ( p - V B , ) = ;Re ( F e V&) 1 (2 .10)andwhere (. . .) denotes the time-average. A formal proof is given in Appendix A thatthese are indeed the correct expressions for F and G a t leading order in a / L .Appendix A also indicates how corrections to these expressions at order ( a / L ) 2(associated with the induced quadrupole) may be obtained.

G = p A B , ) =;Re(@ A & ) j '

Substituting ( 2 .9 ) for the case of a sphere, we have immediatelyF = 27ca3,uu,'Re (a8, VB,*),G = 2na3,u;l Re (aB, &).

( 2 . 1 1 ~ )(2 .11b)

Using ( 2 . 3 ) , he expression for G may be written alternativelyG = 4na3pu,l a(i)B(r)A B(i), (2 .12)

t It may be wo r th noting tha t the same simplification occurs if the particle has cubic symnietr!.or the symmetry of any other regular solid.


6/21

514 H . K . Moffattwhere a = a @ ) id'). Moreover, using (2.4) t is easy to show that F = FL+FDwhere

(2.13)and FD = - na3p;1 a(i)v ( ~ ( r )A B(i)). (2.14)Here, and subsequently, FL ,FD and G can be regarded as functions of the positionx a t which the centre of the sphere is placed, Note that FL ( x )s an irrotational forcefield, while FD ( x )s solenoidal.

h'ote also that a non-zero torque G is associated with the imaginary pa rt of a (i.e.with phase-lag between B , and ,U) and with non-parallel real and imaginary parts ofB0(characteristic of a multiphase field). Such a field locally rotates with fluctuatingamplitude and period 2n/w in the plane of B f ) and B t ) .The field FD is evidentlyrelated to G by

FD+ iV A G = 0. (2.15)

FL= 2na3pGia ( I . ) (B f )V B@ )+B(i).VB ( i ) )= 7~(7,3~u, l (r)VlBo12,

This simple result holds however only under the isotropic condition ail = asii.Consider now the three prototype configurations of figure 1 (a-c) .

( a ) Rotating jield (figure 1a )Let Bc) = Bo(l, ,O ) , Bhi)= B,(O, 1 , O ) in Cartesian coordinates. Then

B f ) A B f ) = Bi(O,O, ) , (2.16)and so from (2.12)

G = 47~p~,~a~B~cz( ')(O,O,) . (2.17)We expect that the torque should be in the same sense as the field rotation, i.e.G , > 0 , and so a ( ' ) ( A )hould be positive for all A . This is confirmed below. Note that,since Bo is uniform, F = 0 for this configuration.

Let( b ) Travelling jield (figure 1 b )

g o ( x )= Aeikxsinhky (k> 0, IyI < b ) (2.18)so thatHence from (2 .13) and (2.14), we find easilyB o ( x )= V g 0= Ak(i sinh Icy,cosh Icy, 0)eikx. (2.19)

FL= 2np;l a31AI2k3a(') inh 2ky t l / , (2.20)and P = 2n,u~,' 31AI2 301(')cosh 2kytx. (2.21)

G = - n p ~ , ~31AI2 2aCi)sinh 2ky t z . (2.22)Thus, for the sphere P in figure 1 b ) on the plane of symmetry (y = 0) we haveFL= 0, G = 0, FD = 2xc~~,u;llA1~3a ( i ) t x . ote that here, the in-phase component ofthe dipole moment gives a contribution to the force which averages to zero. Forthe sphere Q, however, FL ,FD and G are all non-zero. Again, since, we expect thedrag force FD to be in the direction of travel of the field, i.e. the positive x-direction,we expect that & ) ( A ) > 0 for all A.As noted above, the result FD+iV A G = 0 does not hold if ail is non-isotropic. Inthis more general situation, we define FD as that part of the force involving the

Moreover, from (2.12),


7/21

A suspension of particles in a periodic magnetic $eld 515imaginary part a$)of ail.Then for example, if ail = diag (a , , y ) , for the travellingfield configuration we find

(FD;I7 A G), = T C U ~ ~ U , ' ~ A / ~3(a(i)-/3(i)) . (2.23)

Let( c ) Levitatin g field (figure 1c )

Yo= A cos kxePky ( k > 0, y > 0) (2.24)so that Bo = V Y o = -Ak(sinkx,~oskx,O)e-~~. (2.25)Then obviously 8, A B,*= 0, so that G = 0 no matter where the sphere is placed, andso FD = 0 also. Here therefore it is the out-of-phase component of the dipole thatgives a contribution to the force that averages to zero. However, from (2.13) we findan in-phase contribution :

FL = - np;' a3a(r)lA12 3 e-2kYye"' (2.26)We expect that this force will be upwards since the mean magnetic pressure is greateron the lower surface of the sphere (where the mean field strength is greater). Thismeans that & ) ( A ) should be negative for all A. This also is confirmed below. A sphereof mass m will then be levitated provided

2np;' U ~ J ~ ( ~ ) JAI2 3 > mg (2.27)and will then rise to an equilibrium height Y given by

(2.28)It remains to find the function a(h )= ~ ~ ( ~ ) ( h ) + i c t ( ~ ) ( h ) .he details of this

calculation are given in Appendix B, and the result in the case of a spherical particleis

(2.29)(sinh 2h- in 2h)a"'(h) = -++4h(cosh 2h - OS 2h)3(sinh 2h + sin 2h)a'i'(h) = -- +4h2 4h(cosh2h - OS 2h) (2.30)

These functions are plotted in figure 3, and the expectations a(')(h)< 0,all h are indeed confirmed.Note also the asymptotic behaviour :

> 0 forx .178 when h z 2.4.

(i) h 4 1 : a'" rv --45h4, a(i)- hh2, (2.31)has a maximum value(ii) h % 1: (2.32)

the last results being exact to within exponentially small terms. In the limit h = CO(corresponding to a perfectly conducting sphere), a = -; nd the field is totallyexpelled from the sphere; in this limit, both G and FD re zero, but the lift force FLis generally non-zero.Consider some orders of magnitude for the case of spheres of aluminium for which7 = ( p 0 r ) - lx 0.16 m2s-l. To achieve maximum drag or torque ( hx .4), we then


8/21

5160.5

A suspension of particles in a periodic magnetic field

a")

I0 2 4 6 8 10h

FIGURE . The real and imaginary parts of the induction function a(h) &)(A) +ia(')(h)or asphere.require a2wx 1.8m2s-l . For a frequency of the order of megahertz (w - 106s - l ) , thisis achieved with spheres of radius a x 1 .3 mm. If such spheres are suspended in anon-conducting liquid, they can be levitated by a field B of order 10-1 T( 103G ).Incidentally this would be an unconventional and rather dramatic method of heatingthe liquid! These orders of magnitude appear quite feasible, so that the effectsdescribed in this and subsequent sections should certainly be amenable toexperimental verification. The effects must inevitably become much weaker as theparticle size decreases if only because of the practical upper limits on the frequencyand strength of magnetic fields that may be produced in the laboratory.

3. Interaction of two spheres suspended in a viscous fluid and subjected toa rotating fieldWe have seen that a single sphere subjected to a uniform field of strength B,,

rotating in the (x, )-plane, experiences a torqueG = 4 ~ a ~ p ; li d i ) g Z . (3 .1 )

Suppose now that the sphere is surrounded by incompressible fluid of density p andkinematic viscosity v ; ha t, under the action of the torque G , it rotates with constantangular velocity 52; and that the Reynolds number is small, i.e.Re = Q a 2 / vQ 1, (3 .2 )so that all inertial effects are negligible. Then the velocity field in the fluid (see, forexample, Landau & Lifshitz 1959, p. 68) is

(3 .3 )( x )= (aA x ) ( ~ / r ) ~ ,x being measured from the centre of the sphere, and the viscous torque on the sphereis

G, = - Rpva352. (3 .4 )In equilibrium, G + G, = 0, i.e.

0 = (Bia( i ) /2p0 v ) c ? ~ , (3 .5 )and the condition (3 .2 ) is satisfied provided

VZ, a(i)a2/2v2Q 1 (3 .6 )


9/21

A suspension of particles in a eriodic m agnetic field 517

(4FIQURE. a)Mutually induced rotation of two equal spheres subjected to a rotating magneticfield. Each sphere spins with angular velocity R (given by (3 .5 ) ) nd moves in the couplet field (3 .3)of the other sphere. ( b )Meridional projections of sphere trajectories, as given by (3.16).where V, = B o / ( , u o p ) ~s the Alfvkn speed associated with B,. Kote tha t the angularmomentum of the sphere is instantaneously determined in this approximation.

This derivation is only valid provided the deduced angular velocity a is small inmagnitude compared with the angular velocity w of the field, since otherwise the factth at the particle is rotating would have to be taken into account? in the inductionproblem. This requires that

M 2 4 h2/cr(h), ( 3 . 7 )where M = V, ,a / (yv)~ s the Hartmann number.

Suppose now that we have two equal spheres with centres at x l ( t )and x 2 ( t ) ,whereIx,- x,I >> a . Then the dominant interactive effect is given by the tendency of eachsphere to move in the velocity field induced by rotation of the other. Consider firstthis effect alone ; the equations of motion are then

where r12= Ix2-x11.Hence the midpoint x = i(x,+x,) of the line of centres remainsfixed. Taking this point as origin and x, = x, x2= -x , r = 1x1= ir12,we have( 3 . 9 )

so that r remains constant, and the point x ( t ) rotates about an axis through X( 0 )with angular velocity $ 2 ( ~ / r ) ~figure 4 a ) .

There is a correction to this behaviour at higher order in ( a / r ) esulting from thefact that each sphere experiences a force + F due to the magnetic field gradient

t It is not in fact difficult to do this - see Moffatt (1980)- since the inducing angular velocityo need merely be replaced by o-0 .


10/21

518 H . K . Moffuttinduced by the presence of the other sphere, thus giving an additional velocity fwhere, assuming a Stokes drag,

V = (67cpva)-lF . (3 .10)The field in the neighbourhood of the sphere a t x, ( = x) is to good approximation thebasic uniform field plus the dipole perturbation associated with the sphere atx z ( = x), i.e.

From (2 .11u) , he force on sphere 1 is then(3 .11)

(3 .12)With 8,= ( B 0 / 4 2 )1 ,i ,0), as appropriate for a field of strength B, rotating in the( x ,y)-plane, this reduces to

where R = ( x , , 0 ) , = ( O , O , ) . Similarly, the force on sphere 2 is - .The additionalvelocities f are then always in the meridian plane which rotates with the spheres.With coordinates (R ,$, z ) (and r2 = R 2+ 2 ) the corrected vector equation

(3 .14)has components

dR (4z2-RZ)R d$ Qa3 dz (2Z2-3R2)X_ - - - = C , (3 .15a ,b , c )dt r7 ' dt 4r3 ' dt r7cwhere C = a51aI2VA/16v,u,,. The R and z equations have a n integral

ZRZ - const.(R 2+2): (3 .16)and the spheres therefore move on these curves on a meridian plane $ = # ( t )whichrotates according to (3 .15b)(figure 4b) . The description is of course valid only for solong as r % a.

Note that the ratio of the second term on the right of (3 .14) o the first has orderof magnitude

and is small provided

(3 .17)

(3 .18)Under this condition (satisfied provided h is not too large), the relative driftassociated with the forces f is weak. When more than two spheres are considered,the separation of any two centres changes because of perturbations associated with


11/21

A suspension of particles in a periodic magnetic field 519other spheres, and it then seems reasonable to neglect the interactive force f in afirst approximation.There are of course further higher-order effects which we have ignored in the abovetreatment. For example, the velocity - V of sphere 2 induces a velocity of orderI V(a/r)l in the neighbourhood of sphere 1 ; this leads to a further term on the rightof (3.14),but smaller in order of magnitude than those already exhibited. Equation(3.14) is at best a reasonable approximation, uniformly valid (for all A ) whena / r < 1 .4. Interaction of N spheres

Suppose now that we have N equal spheres with centres at x,(t) (n = 1 , 2 , . . N )and separations rmn = Ix, - ,I. Let r be the minimum separation ; then under thecondition (3.18), he equations of motion a t leading order are

where the prime on the summation indicates that the term for which n = rn isomitted. With f2 = ( O , O , Q ) and x, = (x,, y m , z m ) ,we have immediately from (4.1)dz,= 0, i.e. z , = Z,(const.),dt

i.e. each sphere centre moves on a plane z , = 2 I ts coordinates on this plane areeasily seen to satisfy

(4.3)aHdxm = aH dy m -dt ay dt axm where (4.4)The equations (4.3) (for m = 1 , 2 , . . N ) constitute an autonomous Hamiltoniansystem of order 2N. An immediate integral is

H(x,, yZ = const. (4.5)The similarity with the N-vortex problem of classical hydrodynamics is now clear,

only the Hamiltonian H having different form. The following discussion is guided bytha t for the N-vortex problem (Aref 1984) ; it admits trivial extension to the case ofN spheres of different radii and/or conductivities. Kote first the obvious furtherintegrals of the system (4.3), 4.4)

Hence the three-sphere problem (N = 3) is described by a sixth-order system withfour integrals, all of which are independent. It is therefore integrable, the mth spherecentre following a quasi-periodic orbit in the plane z , = 2 Note however thatinclusion of the weak forces of interaction f ,, between the spheres (as discussedin the previous section) will probably trigger chaotic behaviour even for the three-sphere system.For N = 4, the system is eighth-order, with four independent integrals, and themotion of the sphere centres may be expected to be generally chaotic in these


12/21


13/21

A suspension of particles in a periodic magnetic $eld 52 1cG ( x ) s present, angular momentum balance for a 'macro-particle ' requires that thebulk stress tensor gtjmust have an antisymmetric part given by

(5 .2 )ll?) = a ( . , j - g j i ) = &d j k c o k ,with a corresponding contribution to the Navier-Stokes equation

Hence the Navier-Stokes equation for the bulk motion u ( x , ) becomes(5.4)

If c is uniform, then the terms involving RD and 6 exactly compensate by virtueof (5 .1b ) ; moreover, since flL is irrotational, the term cFL may be absorbed in thepressure term through definition of a modified pressure

P = p - ~ c ( ~ u , p ) - l a " ' 1 ~ , 1 2 . ( 5 . 5 )Apart from this pressure adjustment and the appearance of a (generally non-uniform) antisymmetric stress g$), here is no effect a t the macroscopic level, and inparticular no macroscopic velocity field is generated. This is an astonishingconclusion, given that each conducting sphere generates a couplet and a Stokeslet byvirtue of its rotation and translation relative to the fluid. It just so happens that FDand G are related by the special condition F D+aV A G = 0 , which leads to vanishingof the rotational part of the mean driving force in (5.4) when c is uniform. Hence thenet effect of superposition of all the couplets and Stokeslets is zero.

A similar conclusion has been reached in a related, bu t more restricted, context byJansons (1983) who found that a homogeneous ferromagnetic fluid in a circularcylinder subjected to a uniform rotating magnetic field is not set in bulk rotation,although each microscopic dipole rotates with the field. The antisymmetric stressestablished in the fluid is absorbed a t the boundary which exerts a couple equal andopposite to the integrated torque acting electromagnetically on the fluid. The fluid,although a t rest (in bulk), is stressed in the same way as would be an elastic mediumsubjected to a torque in 9 ut fixed on a9. The way in which the phenomenon ofmagnetically induced rotation may be applied in the context of viscometry has beendiscussed by Brancher (1988).

6. Role of the particle force F in generating inhomogeneityrelative to the background fluid where

Each sphere in the suspension considered moves with velocity V = VL+ VDVL = (67cvpa)-lF L , VD = (67cvpa) - lFD, (6 . la ,b )

with v A V L = o , v * D = o . (6 .2)This relative velocity is large compared with any bulk velocity generated, which isa t most of order lVcl. Hence c(x, ) satisfies a conservation equation which, at leadingorder, is

ac-+ V - (Vc)= 0,at


14/21

522 H . K.Moffatt

FIQURE. Lift force on suspended spheres: (a )a rotating field provides a force F owards thecentre of rotation so that all spheres are driven to the interior of the domain 9 ; ( b )a single phasefield of the form (2 .25)produces a force field F n the y-direction, so that all spheres 'sediment 'towards the lower boundary.acor - + V . V c = - c V . V = - c V . V L . (6.4)at

Since V VL is in general non-zero, this means that c will not remain uniform evenif uniform initially. In fact, from ( 6 . 1 ~ )nd (2 .13) ,V VL = (a2cd')/6p0pv)211?012, (6.5)

so that inhomogeneity must develop if V211?olz 0.There is a further source of inhomogeneity arising from conditions at the boundary8 9 of the fluid domain, where in general V - n + 0. If V . < 0 (where n is the unitoutward normal) then all suspended spheres near the boundary move into theinterior of 9 eaving a layer nearC9where c = 0. If V - n > 0 on the other hand, thenspheres are driven onto the boundary just as in a process of gravitationalsedimentation of particles onto a solid base. Both situations can arise, as illustratedin figure 6(a,b ) .As soon as inhomogeneities of c appear in the suspension, a bulk flow will in generalappear also, as may be seen by rewriting (5.4) n the form

p-Du = -vp+ cflL-$ A Vc+pvv2u.DtThe term -$ A Vc is now directly responsible for driving the flow. Moreover, since

V A (cflL)= -FL A VC, (6.7)the term cflL may also generate vorticity like the buoyancy force in a Boussinesqfluid. The former effect is present in the prototype configurations of figures 1 a ,b ) ,and will now be analysed in detail.

6.1. Rotating multipole fieldSuppose tha t 9 s the cylindrical domain r < b , and that

y0 Arm eimewith cylindrical polar coordinates ( r , O , z ) , where rn is a positive integer. The casem = 1 gives F = 0, G = const. and cannot therefore generate a bulk flow if, as we


15/21

A suspension of particles in a periodic magnetic jield 523

w

FIGURE. Action of a rotating field (rn= 2) on an initially uniform suspension. The spheres areuniformly concentrated within a shrinking cylinder of radius b enS1, nd a vortex flow with uniform-vorticity core is established.shall suppose, the concentration c is initially uniform. We shall therefore supposethat m > 1 . The behaviour is sufficiently well illustrated by the choice m = 2 , forwhich we find from (2 .13)

FL= - r& ( 6 . 9 )where k = - 67ca3p;ldr)IAI2 ( > 0) . (6 .10)The corresponding inward particle velocity is

VL= - -re , , (6.11)where s = (67cpva)-' k. The conservation equation ( 6 . 3 ) , with effective initialcondition

CO ( r < b)0 ( r > b )c ( r , O )=

then has solutionc,, e2st ( r < b e-st)

(0 ( r > be-st),c ( r , t )=

(6.12)

(6.13)i.e. the suspendedcylinder of radius spheres are simply uniformly concentrated within a contractingR ( t )= bevst.The situation is shown in figure 7 .The torque distribution 6 for this field is

6 = Gor22z , (6.14)where Go= 12p;l cdi)lA12, (6.15)Hence -iG A Vc = + G o c o e 2 s t 8 ( r - R ( t ) ) R 2 2 0 ( 6 . 1 6 )so that we have an effective force in the 8-direction concentrated on r = R ( t ) .are negligible then, from ( 6 . 6 ) ,w(r, ) satisfies

-The driven flow has the form U = (0,w( r , ), 0) and if we suppose that inertia forces

( 2 $) = -$ ZstR28(rR ( t ) ) . ( 6 . 1 7 )


16/21

524 H . K . MoffattThe solution satisfying v ( b )= 0 and v ( 0 ) finite is

(6.18)

The circulation at r = R(t) sK = 2 n R v ( R , ) = 2nG0c0(pv)-lb(1- -st), (6.19)

tending to the constant value 2nG0cob/pv as t+ CO. Thus the asymptotic flow is thatdue to a concentrated line vortex with superposed rigid-body rotation to satisfyv ( b )= 0.6 .2 . Tra vellin g fieldYo= A eikx inh ky

Now let 9 e the channel (yI < b and let(6.20)

as in $ 2 . Then FL s given by (2 .20)so that the concentration c(y , ) satisfiesac aat ay- -(&csinh2ky),

where vo= -Lna21Al2k3a(') (rUoPv)-l ( > O ) ,and we require the solution satisfying the initial condition

(6 .21)(6 .22)

(6.23)First consider the equation

- Ksinh2kY (6 .24)Ydt---with initial condition Y(0)= Y,. The solution is given by

tanh kY(t) = e-2voktanh (ICY,) (6 .25)and, in particular, the layer of spheres initially at (or very near) the boundaryy = b moves at time t to

y = & ( t )= k-l tanh-l (eezvOkt anh kb). (6.26)The solution of (6.21) for IyI < & ( t ) may be found by the change of variableX = tanh ky, and is

(6 .27)Also c(y , t ) = 0 for IyI > T ( t ) .There is a discontinuity of c across y = K ( t ) given by

Ac = [c] = - 0 (ezvOktosh2kb - -2vokt inh2kb ) (6 .28)(see figure 8 ) .


17/21

A susp ensio n of par ticles i n a perio dic magnetic field 525

y = b

(a ) kb G 1 (b)kb p 1FIGURE . Action of a travelling field on an initially uniform suspension of spheres in the channelJyI< b. The situation is symmetric about y = 0, and only the upper half of the channel is shown.The suspension is concentrated a t time t within the slab IyI < K ( t ) . a) b < 1: a piecewise-constantvorticity distribution is established. ( b )kb % 1 a &function in concentration appears on y = K ( t ) ,and this generates an associated vortex sheet in the bulk flow.

If kb 1 , the situation is very similar to tha t considered in the previous example.In this case, c(y , ) - oe2vo'o"t or IyI < q(t) be-"okt, so that the velocityU = u ( y , )6 , is driven by the discontinuities of c across y = & Y,(t). Neglecting inertia,and with Gh = ,u;l 2na31AI2 2a('), we find

(6.29)

so that, as t -+ CO , a vorticity discontinuity[U, ] = 2Gh cokblvp (6.30)

tends to form on the plane y = 0 towards which the suspended spheres are driven,If kb P 1 , the situation is more complicated because the bulk velocity is driven bythe variation of concentration for IyI < q(t) s well as by the jumps in c acrossy = f ( t ) . The total variation of concentration however is confined to layers ofthickness k-l on y = k , ( t ) , nd the conductivity distribution may be approximated

4 Y l t ) = co[l -H(y-Y,)+(b-Y,)S(y-Y,)I , (6.31)by

where H ( x ) s the Heaviside function. A simple calculation then gives

l 0 (6.32)

so th at now a vortex sheet appears on y = q(t).f course in reality, this vortex sheetis 'smoothed' through the layer of thickness k-'.The two situations are sketched in figure 8(a,b).


18/21

526 H . K . Moffatt7. Conclusions

We have shown that a single conducting particle placed in a time-periodicmagnetic field of the form Re (B,(x)e-iwt) in general experiences a force F and atorque G , which are functions of the position x of the centre of volume of the particle.The force F has a natural decomposition F = FL+FD, here V A FL= 0, V FD = 0 ;and if the relationship between the induced dipole moment 1; and B is isotropic (asfor a spherical particle) then there is a simple relationship between FD nd G , namely

F D + i V A G = 0.If two or more particles are suspended in a viscous fluid and subjected to such afield, then each particle will move in response to the force and torque acting on it and

will at the same time be convected by the Stokes flow (a superposition of Stokesletsand couplets) induced by the other particles. Examples of such interactions are givenin $63 and 4 for the particular case of a uniform rotating field, in which case there isa compelling analogy with the classical problem of interaction of point vortices. Inparticular, the motion of four or more spheres must in general exhibit chaos.

When a suspension of particles is considered, it is shown that the condition ( 7 . 1 )implies that in any region in which the concentration c is uniform there is no net localforce to drive a bulk flow. Bulk flow can be driven only through interaction of thebody torque distribution e with the spatial gradient of c through a term -$ A Vcin the bulk equation of motion. Inhomogeneities of c are generated by the levitation force FL hrough a process analogous to sedimentation in a gravitational field. Twoexamples of flows driven by such a combination of effects are analysed in $6.An initial aim of this investigation was to determine whether a homogeneoussuspension could be set in rotation by a rotating magnetic field, as happens for ahomogeneous fluid conductor (Moffatt 1965). The results of $35, 6 show that auniform rotating field acting on a homogeneous suspension will generate no bulkflow ; but that rotation can be induced by a rotating multipole field ; and similarlythat bulk transport can be generated by a travelling field, although only with anaccompanying development of strong inhomogeneity in the suspension.

If the constituent particles of the suspension do not have spherical, or equivalent,symmetry, so that the relation between 1; and B ,, although linear, is non-isotropic,then ( 7 . 1 )no longer holds, and more subtle effects may be anticipated. For example,if the particles are needle-shaped, then as a particle rotates under the action of atorque G , the induction tensor aij in the relationship between pi and BOjwillbecometime-dependent and the force and torque on the particle will change accordingly.Determination of the motion of such a particle is then quite a complex problem ; butthis is just a preliminary to understanding the behaviour of a suspension of suchparticles.

There are certain obvious points of contact in all this with the theory offerromagnetic suspensions (Rosensweig 1985) in which microscopic dipoles respondto the direction and strength of an applied magnetic field, whether steady orunsteady. The novel features in a suspension of conducting particles are that thedipole moments are themselves induced by time-variation of the applied field andthat the relation between 1; and B, is then linear. These features lead to a class ofproblems that have some fundamental interest, and that may have some practicalapplication in relation to metallurgical processing.


19/21

A suspension of particles in a periodic magnetic field 527Appendix A. Justification of (2.10)unperturbed field Y,(x) in Taylor seriesChoosing origin ,at the centre of volume of the body, we may expand the

1!@o(X)-@o(o)= b, xi+ $i j xi XI+5 ijk xi xjxk + . . . ,the corresponding expansion for 8, (x) being

B , , ( x )= b , + ~ i j x j + ~ d , , , ~ j x k + . . . .Here, b, = B,, (O) , c,, = ( ~ ~ o i / ~ x , ) x - o ,tc.

c = cji, cii = 0,

!@=@,+!PI,

and obviously Uwith similar constraints on higher-order coefficients.The induced currents in the body perturb this external field to the form

wherei.e. a sum of dipole, quadrupole and higher-order ingredients. The field outside thebody is then 8 = 8,+8,where

The mean Lorentz force distribution in the body is

where p, qj Re (B,Bj* Sij), (A 9)the Maxwell stress tensor. Since J = 0 outside the body, the total force F and coupleG may be expressed as surface integrals over a sphere A of radius R containing thebody :

These expressions must obviously be independent of R ; hence the only part of q,contributing to (A 10)must be that part (Tf) say) proportional to r - 2 ,and the onlypart contributing to (A 11) must be that part (Tjf")proportional to r -3 .It is easy to see that the leading-order contribution to Ti,F)comes from interactionof the fields c,, xj and ,L, (a2 /&, ax,) l / r ) , nd the leading-order contribution to Tl,G)comes from interaction between B , and ,Lja2/axi x,) ( l / r ) explicitly, at leadingorder,


20/21

528 H . K . Moffattand

Hence by substitution in (A 10)and (A 11)respectively, F and G may be calculated.With repeated use of the isotropic integral

which are, as expected, identical with (2 .10 ) (using (A 3 ) ) .This is of course reassuring, but it has the added bonus that higher-order terms in( U / L ) ~ay be obtained if required. The correction to4,of order ( U / L ) ~ ,nvolves theproduct ,liJkdGk,or equivalently a term of the form

Similarly, the correction to G, at order ( u / L ) ~s

Appendix B. Determination of & ( A )for a spherical particleSince U is independent of B o ( x ) , e may assume tha t B, is uniform and real, so that

the applied field is Bocoswt. For r > a (where a is the radius of the sphere), themodified magnetic potential is thenY ( x ) b,xi 1-- ,( 3where b = Bo,and U is as defined in ( 2 . 9 ) .The induction problem is most easily solved in terms of the scalar fieldP(x)defined

byB = v A v A ( X P ( X ) ) , (B 2)

which, for r > a , is related to Y byaar

Y = P + ( x V ) P = - ( r P ) .Hence,Unlike Y ,P is defined also for r < a , and there satisfies

P = (b x ) ( i + a ( : y ) ( r > a ) .

V 2 P = -iw,uocrP ( r c ) .


21/21

A suspension of particles in a periodic magnetic jield 529Moreover, continuity of B across r = a is satisfied provided

[PI = [ W / a r ] = 0 across r = a. (B 6 )The solution of (B5 ) regular at r = 0 is

P = P - (b x) j l - ( r < a ) ,(3 (6)where 7 = ( l + i ) hand sinz coszz2 zj,(z) = ---,the spherical Bessel function of order one. The continuity conditions (B 6 ) now give

Pj,(r) = g+a> Prjl(r)= B-% (B 10)so that, after some simplification,

which together with (B 8),determines @ ( A )= & ) ( A ) + di)@).The real and imaginaryparts of (B 1 1 ) give the results (2.29) and (2.30).The solution as found here was first described by Lamb (1883) as part of a generalstudy of electromagnetic induction in a spherical conductor.R E F E R E N C E S

AREF,H. 1984 Stirring by chaotic advection. J . Fluid Mech. 143, 1-21.AREF,H. & POMPHREY,. 1982 Integrable and chaotic motions of four vortices I. The case ofidentical vortices. Proc. R . Soc. Lond. A 380, 359-387.BATCHELOR,.K. 1970 The stress system in a suspension of force-free particles. J . Fluid Mech.41, 545-570.BRANCHER,.-P. 1988 Comportement dun ferrofluide dans un champ magnetique tournant etapplication. J . Mec. Theor. A p p l . 7 , 329-350.BRAUNBECK,. 1932 Eine neue Methode elektrodenloser Leitfahigkeitsmessung. 2. hys. 73,JANSONS,. M. 1983 Determination of the constitutive equations fo r a magnetic fluid. J . FluidMech. 137, 187-216.LAMB,H. 1883 On electrical motions in a spherical conductor. Phil. Trans. R. Soc. Lond. 174,5 19-549.LANDAU, . D. & LIFTSHITZ, . M. 1959 Fluid Mechanics, Pergamon.MOFFATT,H. K. 1965 On fluid flow induced by a rotating magnetic field. J . Fluid Mech. 22 .521-528 (and Corrigendum 58 (1973) 823).MOFFATT, . K. 1980 Rotation of a liquid metal under the action of a rotating magnetic field. InMHD Flows and Turbulence, 11 (ed. H. Branover & A. Yakhot), pp. 45-62. Jerusalem: Israel

University Press.

3 12-334.

ROSENSWEIGI,. E. 1985 Ferrohydrodynamics, Cambridge University Press.

H.K. Moffatt- On the behaviour of a suspension of conducting particles subjected to a time-periodic...

Documents

Transcript of H.K. Moffatt- On the behaviour of a suspension of conducting particles subjected to a time-periodic...