Journal of Magnetism and Magnetic Materials 324 (2012) 1523–1527

Contents lists available at SciVerse ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/jmmm

On stationary convection and oscillatory motions in ferromagneticconvection in a ferrofluid layer

Jyoti Prakash

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla 171005, India

a r t i c l e i n f o

Article history:

Received 17 August 2011

Received in revised form

8 November 2011Available online 13 December 2011

Keywords:

Ferrofluid

Complex growth rate

Principle of exchange of stabilities

Oscillatory motion

53/$ - see front matter & 2011 Elsevier B.V. A

016/j.jmmm.2011.11.058

ail address: jpsmaths67@gmail.com

a b s t r a c t

It is shown analytically that the ‘principle of the exchange of stabilities’ (PES), in general, is not valid in

ferromagnetic convection in a ferrofluid layer, for the case of free boundaries and hence a sufficient

condition is derived for the validity of the PES. Upper bounds for the complex growth rate are then

obtained. It is proved that the complex growth rate s¼ srþ isi (where sr and si are, respectively, the

real and imaginary parts of s) of an arbitrary oscillatory motion of growing amplitude, in ferromagnetic

convection in a ferrofluid layer, for the case of free boundaries lies inside a semicircle in the right half of

the srsi-plane whose center is at the origin and ðradiusÞ2 ¼ RM1=Pr , where R is the Rayleigh number,M1

is the magnetic number and Pr is the Prandtl number. Further, bounds for the case of rigid

ferromagnetic boundaries are also derived separately.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

A Ferrofluid is a colloid suspension containing magneticnanoparticles covered by a surfactant for preventing their aggre-gation and suspended in a nonconducting fluid. In recent yearsferrofluids have attracted many researchers due to their practicalapplications in various fields like viscous damping system, med-ical sciences (drug targeting, endoscopic analysis, magneticseparation of cells and magnetic resonance imaging (MRI), noise-less printing system, etc. [1]. Several researchers studied ferro-magnetic convection both theoretically and experimentally invarious geometries and magnetic conditions [2–6]. In the presentcommunication the main emphasis is given to the fundamentalpaper of Finlayson [2].The results derived by Finlayson (Eq. 19) forthe nonexistence of oscillatory motion, for the case of freeboundaries, are very limiting, limited only to the case of neutralinstability(when sr¼0) and does not conclude anything whensr40 (unstable mode) i.e.when s is not purely imaginary. He,further, pointed out that for the case of rigid boundaries, theoscillatory instability must be considered, but it may be thelimitation of the Galerkin method that he had to restrict himselfto stationary convection in this case also. Venkatasubramanianand Kaloni [6] also tried to prove, by an indirect technique, thatthe ‘principle of the exchange of stabilities’ is valid for aferromagnetic fluid layer heated from below (by taking the limitas z-0 in a rotatory ferromagnetic fluid layer heated from below,p. 242) but their mathematics seems to be incorrect while

ll rights reserved.

equating the imaginary part of Eq. (35), since we cannot sayabout the integrated part ½D2wD3w�DwD4w�1=2

�1=2 (where w iscomplex conjugate of w) of Eq. (35) whether it is real or imaginary(since w is an unknown function) and hence cannot be ignoredfrom the imaginary part without justification.

By using famous Pellew and Southwell technique [7], we canprove that, for the case of free boundaries, the ‘principle of theexchange of stabilities’ is not, in general, valid for a ferromagneticfluid layer heated from below but a sufficient condition can bederived for the validity of the PES and this is an open problem forthe case of rigid boundaries. However, bounds for the complexgrowth rates for both free and rigid boundaries can be derivedusing the above mentioned technique. The present paper isprecisely based on this contention and provides a sufficientcondition for the validity of the PES, and also provides upperbounds to the complex growth rates of arbitrary oscillatorymotions of growing amplitude in ferromagnetic convection in aferrofluid layer heated from below for the cases of free andferromagnetic rigid boundaries.

2. Governing equations and boundary conditions

Consider a layer of a ferromagnetic fluid confined between twohorizontal surfaces at z¼�d/2 and z¼d/2 in the presence of avertical magnetic field. A constant temperature gradient is main-tained across the surfaces with lower and upper surfaces main-

tained, respectively, at constant temperatures T0þ(DT/2) and

T0�(DT/2). The relevant non-dimensional governing equations

J. Prakash / Journal of Magnetism and Magnetic Materials 324 (2012) 1523–15271524

of a ferromagnetic convection are given by Finlayson [2] (withM2¼0 and all the perturbed quantities having dependence of the

form exp. [i(kxþky)þst], where k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

xþk2y

qis the wave number

and s represents the complex growth rate):

ðD2�a2ÞðD2

�a2�sÞw¼�R1=2a½M1D|�ð1þM1Þy� ð1Þ

ðD2�a2�PrsÞy¼�R1=2aw, ð2Þ

ðD2�a2M3Þ|¼Dy, ð3Þ

together with the boundary conditions

w¼ 0¼ y¼DؼD2w at z¼�1=2 and z¼ 1=2 ð4Þ

(Both the boundaries dynamically free [2])

or w¼ 0¼ y¼Ø¼Dw at¼�1=2 and z¼ 1=2 ð5Þ

(both the boundaries rigid ferromagnetic [6])where z is the real independent variable such that

�1/2rzr1/2, D¼d/dz is differentiation w.r.t. z, a2 is a constant,Pr40 is a constant, R40 is a constant, M140 is a constant,M340 is a constant, s¼ srþ isi is a complex constant such thatsr and si are real constants and as a consequence the dependentvariables w(z)¼wr(z)þ iwi(z), yðzÞ ¼ yrðzÞþ iyiðzÞ and fðzÞ ¼frðzÞþ

ifiðzÞ are complex valued functions of the real variable z such thatwr(z), wi(z), yrðzÞ,yiðzÞ,frðzÞ and fiðzÞ are real valued functions ofthe real variable z.

The meaning of the symbols from the physical point of view isas follows : D¼d/dz is differentiation along the vertical direction,a2 is the square of the wave number, Pr is the Prandl number,R¼gabd4/nk is the Rayleigh number, M1 ¼ m0K2b=ð1þwÞarog ismagnetic number, K ¼�ð@M=@TÞH0, T0

is the pyromagnetic coeffi-cient, w¼ ð@M=@HÞH0, T0

is the magnetic susceptibility, M3 ¼ 1þðM0=H0Þ=1þw is nonlinearity of magnetization parameter, s is thecomplex growth rate, w is the vertical velocity, y is the tempera-ture and | is amplitude of magnetic potential,. It may further benoted that Eqs. (1)–(5) describe an eigen value problem for s andgovern ferromagnetic convection in a ferrofluid layer.

3. Mathematical analysis

First of all it is shown that the PES is not, in general, valid forthe case of free boundaries. It is shown as follows: MultiplyingEq. (1) by wn (n denotes the complex conjugation) throughoutand integrating the resulting equation over the vertical range of z,we getZ 1=2

�1=2wnðD2

�a2ÞðD2�a2�sÞwdz¼�R1=2aM1

Z 1=2

�1=2wnD|dz

þR1=2að1þM1Þ

Z 1=2

�1=2wny dz ð6Þ

Using Eqs. (2) and (3) and the boundary conditions (4), we canwrite

�R1=2aM1

R 1=2�1=2 wnD|dz¼M1

R 1=2�1=2 D|ðD2

�a2�PrsnÞyndz

¼M1

Z 1=2

�1=2D|D2yndz�M1ða

2þprsnÞ

Z 1=2

�1=2ynD|dz

¼M1

Z 1=2

�1=2D|D2yndzþM1ða

2þprsnÞ

Z 1=2

�1=2|Dyndz

¼M1

Z 1=2

�1=2D|D2yndzþM1ða

2þprsnÞ

Z 1=2

�1=2|ðD2�a2M3Þ|

ndz:

ð7Þ

(using Eq. (3))

R1=2að1þM1Þ

Z 1=2

�1=2wnydz¼�ð1þM1Þ

Z 1=2

�1=2yðD2�a2�PrsnÞyndz:

ð8Þ

Combining Eqs. (6)–(8), we obtain

Z 1=2

�1=2wnðD2

�a2ÞðD2�a2�sÞwdz

¼M1

Z 1=2

�1=2D|D2yndzþM1ða

2þprsnÞ

Z 1=2

�1=2|ðD2�a2M3Þ|

ndz

�ð1þM1Þ

Z 1=2

�1=2yðD2�a2�PrsnÞyndz: ð9Þ

Integrating the various terms of Eq. (9), by parts, for anappropriate number of times and making use of the boundaryconditions (4) and the equality

Z 1=2

�1=2cnD2ncdz¼ ð�1Þn

Z 1=2

�1=29Dnc92

dz,

where

c¼wðn¼ 1,2Þor c¼ y,|,zðn¼ 1Þ, ð10Þ

we may write Eq. (9) in the form:

Z 1=2

�1=2ð9D2w92

þ2a29Dw92þa49w92

ÞdzþsZ 1=2

�1=2ð9Dw92

þa29w92Þdz

¼�M1

Z 1=2

�1=2D2|Dyndz�M1ða

2þPrsnÞ

Z 1=2

�1=2ð9D|92

þa2M39|92Þdz

þð1þm1Þ

Z 1=2

�1=2ð9Dy92

þa29y92þPrsn9y92

Þdz ð11Þ

Now multiplying the complex conjugate of Eq. (3) by | andintegrating over the vertical range of z, we obtain

Z 1=2

�1=2ð9D|92

þa2M39|92Þdz¼�

Z 1=2

�1=2|Dyndz,

which clearly implies thatR 1=2�1=2 |Dyndz is real.

Now multiplying Eq. (3) by Dyn and integrating over thevertical range of z, we obtain

Z 1=2

�1=2D2|Dyndz�a2M3

Z 1=2

�1=2|Dyndz¼

Z 1=2

�1=29Dy92

dz:

SinceR 1=2�1=2 |Dyndz is real, therefore

R 1=2�1=2 D2|Dyndz in the above

equation is also real as right hand side is real.Equating imaginary parts on both sides of Eq. (11), using the

fact thatR 1=2�1=2 D2|Dyndz is real, we get

si

"Z 1=2

�1=2ð9Dw92

þa29w92Þdz�M1Pr

Z 1=2

�1=2ð9D|92

þa2M39|92Þdz

þð1þM1ÞPr

Z 1=2

�1=29y92

dz

#¼ 0, ð12Þ

which clearly implies that si, in general, is not equal to zero.Hence the required conclusion directly follows from this.

Now we derive a sufficient condition for the validity of the PESin the present case.

Theorem 1. If R40, srZ0, RM1Pr/p4r1, then a condition for theexistence of solution (w, y, |, s) of Eqs. (1)–(3) together withboundary conditions (4) is that si¼0.

J. Prakash / Journal of Magnetism and Magnetic Materials 324 (2012) 1523–1527 1525

Proof. Equating imaginary parts on both sides of Eq. (11), using the

fact thatR 1=2�1=2 D2|Dyndz is real and canceling si(a0) throughout, we

can writeZ 1=2

�1=2ð9Dw92

þa29w92Þdz¼M1Pr

Z 1=2

�1=2ð9D|92

þa2M39|92Þdz

�ð1þM1ÞPr

Z 1=2

�1=29y92

dz ð13Þ

Multiplying Eq. (3) by fn (complex conjugate of j) throughoutand integrating first term on the left hand side and the right handside, by parts, by making use of appropriate boundary conditionson f and y, we have from the final equationR 1=2�1=2ð9Df9

2þa2M39f9

2Þdz¼

R 1=2�1=2ðDf

nÞy dz

rZ 1=2

�1=2ðDfnÞydz

����������

rZ 1=2

�1=29ðDfn

Þy9dz

rZ 1=2

�1=29Dfn99y9dz

rZ 1=2

�1=29Df99y9dz

rZ 1=2

�1=29Df92

dz

!1=2 Z 1

09y92

dz

!1=2

ð14Þ

(utilizing the Schwartz inequality), which implies that

Z 1=2

�1=29Df92

dzrZ 1=2

�1=29Df92

dz

!1=2 Z 1=2

�1=29y92

dz

!1=2

:

Thus

Z 1=2

�1=29Df92

dz

!1=2

rZ 1=2

�1=29y92

dz

!1=2

Using this inequality in inequality (14), we obtainZ 1=2

�1=2ð9Df92

þa2M39f92Þdzr

Z 1=2

�1=29y92

dz: ð15Þ

We note that since w and y satisfy w(�1/2)¼0¼w(1/2) andy(�1/2)¼0¼y(1/2), we have by Rayleigh–Ritz inequality [8]Z 1=2

�1=29Dw92

dzZp2

Z 1=2

�1=29w92

dz ð16Þ

andZ 1=2

�1=29Dy92

dzZp2

Z 1=2

�1=29y92

dz: ð17Þ

Now multiplying Eq. (2) by yn (complex conjugate of y )throughout and integrating the resulting equation, by parts, foran appropriate number of times,by making use of the boundaryconditions on y, namely y(�1/2)¼0¼y(1/2), we have from thereal part of the final equationZ 1=2

�1=2ð9Dy92

þa29y92þPrsr9y9

2Þdz¼ Real part of R1=2a

Z 1=2

�1=2ynwdz

rR1=2a

Z 1=2

�1=2ynwdz

����������

rR1=2a

Z 1=2

�1=29ynw9dz

rR1=2a

Z 1=2

�1=29yn99w9dz

rR1=2a

Z 1=2

�1=29y99w9dz

rR1=2a

Z 1=2

�1=29y92

dz

!1=2 Z 1=2

�1=29w92

dz

!1=2:

(using Schwarz inequality).Combining this inequality with the inequality (17) and the fact

that sr Z0, we obtain

p2

Z 1=2

�1=29y92

dzrR1=2a

Z 1=2

�1=29y92

dz

!1=2 Z 1=2

�1=29w92

dz

!1=2

,

which implies that

Z 1=2

�1=29y92

dz

!1=2

rR1=2a

p2

Z 1=2

�1=29w92

dz

!1=2

and thus

Z 1=2

�1=2ð9Dy92

þa29y92þPrsr9y9

2Þdzr

Ra2

p2

Z 1=2

�1=29w92

dz

which, upon using inequality (16), gives

Z 1=2

�1=29y92

dzrR

p4

Z 1=2

�1=29Dw92

dz ð18Þ

Using inequalities (15) and (18) in Eq. (13), we obtain

Z 1=2

�1=2ð9Dw92

þa29w92Þdzr

RM1Pr

p4

Z 1=2

�1=29Dw92

dz

�Rð1þM1Þa2Pr

Z 1=2

�1=29y92

dz,

which gives

1�RM1Pr

p4

� �Z 1=2

�1=29Dw92

dzþa2

Z 1=2

�1=29w92

dz

þRð1þM1ÞPra2

Z 1=2

�1=29y92

dzr0

and thus we necessarily have

RM1Pr

p441 ð19Þ

Hence if RM1Pr/p4r1 , we must have si¼0.

Theorem 1 may be stated in an equivalent form as: for ferromag-netic convection in a ferrofluid layer, a necessary condition for theoccurrence of oscillatory motions is that RM1Pr/p441 and hence asufficient condition for the validity of the ‘principle of theexchange of stabilities’ is that RM1Pr/p4r1.

Now we obtain upper bounds for the complex growth rates ofarbitrary oscillatory motions of growing amplitude in ferromag-netic convection for both free and rigid boundaries separately.

We prove the following theorems:

Theorem 2. If R40, srZ0,sia0 then a necessary condition forthe existence of a nontrivial solution (w,y,|,s) of the Eqs. (1)–(3)together with the boundary conditions (4) is that

9s92oRM1

Pr:

J. Prakash / Journal of Magnetism and Magnetic Materials 324 (2012) 1523–15271526

Proof. Multiplying Eq. (2) by their respective complex conjugatesand integrating over the vertical range of z for an appropriatenumber of times and using boundary conditions (4), we getZ 1=2

�1=2½9D2y92

þ2a29Dy92þa49y92

�dzþ2Prsr

Z 1=2

�1=2ð9Dy92

þa29y92Þdz

þP2r 9s9

2Z 1=2

�1=29y92

dz¼ Ra2

Z 1=2

�1=29w92

dz, ð20Þ

Since sr Z0, we obtain from Eq. (20), thatZ 1=2

�1=29y92

dzrRa2

P2r 9s9

2

Z 1=2

�1=29w92

dz: ð21Þ

Now using inequalities (15)and (21) in Eq. (13), we obtainZ 1=2

�1=2ð9Dw92

þa29w92ÞdzþRa2ð1þM1ÞPr

Z 1=2

�1=29y92

dz

rRa2M1

Pr9s92

Z 1=2

�1=29w92

dz,

which can be rearranged as:Z 1=2

�1=29Dw92

dzþ 1�RM1

Pr9s92

!a2

Z 1=2

�1=29w92

dz

þRa2ð1þM1ÞPr

Z 1=2

�1=29y92

dzr0 ð22Þ

It follows from inequality (22) that

9s92oRM1

Pr: ð23Þ

This completes the proof of the theorem.The above theorem may be stated in an equivalent form as :

the complex growth rate of an arbitrary oscillatory motion ofgrowing amplitude in ferromagnetic convection in a ferrofluidlayer, for the case of free boundaries, lies inside a semicircle in theright half of the srsi�plane whose center is at the origin andðradiusÞ2 ¼ RM1=Pr :

Theorem 3. If R40, srZ0,sia0 then a necessary condition forthe existence of a nontrivial solution (w, y, |, s) of the Eqs. (1)–(3)together with the boundary conditions (5) is that

9s92s2i o

RM1

Pr

� �2

Proof. Multiplying Eq. (1) by wn (the complex conjugate of w)throughout and integrating the resulting equation over thevertical range of z, we getZ 1=2

�1=2wnðD2

�a2ÞðD2�a2�sÞwdz¼�R1=2aM1

Z 1=2

�1=2wnD|dz

þR1=2að1þM1Þ

Z 1=2

�1=2wnydz:

ð24Þ

Using Eq. (2) and the boundary conditions (5), we can write

R1=2að1þM1Þ

Z 1=2

�1=2wnydz¼�ð1þM1Þ

Z 1=2

�1=2yðD2�a2�PrsnÞyndz

ð25Þ

Combining Eqs. (24) and (25), we obtainZ 1=2

�1=2wnðD2

�a2ÞðD2�a2�sÞwdz¼�R1=2aM1

Z 1=2

�1=2wnD|dz

�ð1þM1Þ

Z 1=2

�1=2yðD2�a2�PrsnÞyndz ð26Þ

Integrating the various terms of Eq. (26) by parts for anappropriate number of times and making use of the boundaryconditions (5) and the equality (10), we obtainZ 1=2

�1=2ð9D2w92

þ2a29Dw92þa49w92

ÞþsZ 1=2

�1=2ð9Dw92

þa29w92Þdz

¼�R1=2aM1

Z 1=2

�1=2wnD|dz

þð1þM1Þ

Z 1=2

�1=2ð9Dy92

þa29y92þPrsn9y92

Þdz ð27Þ

Equating imaginary parts of both sides of Eq. (27) and dividingthe resulting equation throughout by si (a0), we getZ 1=2

�1=2ð9Dw92

þa29w92Þdz¼

�R1=2aM1

siimaginary part of

Z 1=2

�1=2wnD|dz

�ð1þM1ÞPr

Z 1=2

�1=29y92

dz ð28Þ

Now

�R1=2aM1

siimaginary part of

Z 1=2

�1=2wnD|dz

r �R1=2aM1

siimaginary part of

Z 1=2

�1=2wnD|dz

����������

rR1=2aM11

si

Z 1=2

�1=2wnD|dz

����������

rR1=2aM1

9si9

Z 1=2

�1=2wnD|dz

����������

rR1=2aM1

9si9

Z 1=2

�1=29wnD|9dz

rR1=2aM1

9si9

Z 1=2

�1=29w99D|9dz

rR1=2aM1

9si9

Z 1=2

�1=29w92

dz

!1=2 Z 1=2

�1=29D|92

dz

!1=2

ð29Þ

(utilizing Schwartz inequality).From inequalities (15) and (21), we have

Z 1=2

�1=29D|92

dz

!1=2

rR1=2a

pr9s9

Z 1=2

�1=29w92

dz

!1=2

: ð30Þ

Using inequality (30) in (29), we obtain

�R1=2aM1

siimaginary part of

Z 1=2

�1=2wnD|dz

rRa2M1

pr9s99si9

Z 1=2

�1=29w92

dz: ð31Þ

Utilizing inequalities (31) in Eq. (28), we finally haveZ 1=2

�1=29Dw92

dzþa2 1�RM1

pr9s99si9

!Z 1=2

�1=29w92

dz

þPrð1þM1Þ

Z 1=2

�1=29y92

dzr0,

which clearly implies that

9s92s2i o

RM1

Pr

� �2

ð32Þ

or ðs2r þs

2i Þs

2i o

RM1

Pr

� �2

: ð33Þ

J. Prakash / Journal of Magnetism and Magnetic Materials 324 (2012) 1523–1527 1527

The above theorem may be stated in an equivalent form as: thecomplex growth rate ðsr ,siÞ of an arbitrary oscillatory motion ofgrowing amplitude in ferromagnetic convection in a ferrofluidlayer, for the case of ferromagnetic rigid boundaries, lies insidethe right half of the region given by Eq. (33).

4. Conclusions

The linear stability theory is used to derive sufficient conditionfor the validity of the ‘principle of the exchange of stabilities’ forthe case of free boundaries and to obtain the bounds for thecomplex growth rates in ferromagnetic convection, in a ferrofluidlayer heated from below in the presence of a uniform verticalmagnetic field. These bounds are important especially when boththe boundaries are not dynamically free so that exact solutions inthe closed form are not obtainable. Further, since the resultsderived herein involve only the non-dimensional quantities and

are independent of the wave number, are thus of uniformapplicability.

References

[1] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge,MA, USA, 1985.

[2] B.A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40(1970) 753–767.

[3] A. Zebib, Thermal convection in a magnetic fluid, J. Fluid Mech 321 (1996)121–136.

[4] A. Lange, Thermal convection of magnetic fluids in a cylindrical geometry, J.Magn. Magn. Mater. 252 (2002) 194–196.

[5] M.D. Gupta, A.S. Gupta, Convective instability of a layer of a ferromagneticfluid rotating about a vertical axis, Int. J. Eng. Sci. 17 (3) (1979) 271–277.

[6] S. Venkatasubramanian, P.N. Kaloni, Effects of rotation on the thermoconvec-tive instability of a horizontal layer of ferrofluids, Int. J. Eng. Sci. 32 (2) (1994)237–256.

[7] A Pellew, R.V. Southwell, On the maintained convective motion in a fluidheated from below, Proc. R. Soc. London, Ser. A A176 (1940) 312–343.

[8] M.H. Schultz, Spline Analysis, Prentice Hall Inc., Englewood Cliffs, NJ, USA,1973.