Applied Mathematics and Computation 218 (2011) 2785–2799

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

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

A nonlinear stability analysis of a rotating double-diffusive magnetizedferrofluid

Sunil a,⇑, Poonam Sharma a, Amit Mahajan b

a Department of Mathematics, National Institute of Technology, Hamirpur, Himachal Pradesh 177 005, Indiab Department of Mathematics and Statistics, Lambton Tower, University of Windsor, Ontario, Canada

a r t i c l e i n f o a b s t r a c t

Keywords:Magnetized ferrofluidNonlinear stabilityRotationDouble-diffusive convectionMagnetization

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.08.021

⇑ Corresponding author. Tel.: +91 1972 304134; fE-mail addresses: sunilnitham@gmail.com (Sunil

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotatingabout a vertical axis for stress-free boundaries is derived via generalized energy method.The mathematical emphasis is on how to control the nonlinear terms caused by magneticbody and inertia forces. The result is compared with the result obtained by linear instabil-ity theory. The critical magnetic thermal Rayleigh number given by energy theory isslightly less than those given by linear theory and thus indicates the existence of subcriticalinstability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear criticalstability thermal Rayleigh number coincides with that of linear critical stability thermalRayleigh number. For lower values of magnetic parameters, this coincidence is immedi-ately lost. The effect of magnetic parameter, M3, solute gradient, S1, and Taylor number,TA1 , on subcritical instability region have been analyzed. We also demonstrate couplingbetween the buoyancy and magnetic forces in the nonlinear stability analysis.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The main aim of hydrodynamic stability theory is to determine critical values of Rayleigh numbers by employing two use-ful theories namely, the linearized stability theory and the energy stability theory. In some respect the two theories comple-ment each other. The linearized stability theory provides sufficient conditions for the disturbances of a basic flow to beunstable, while energy stability theory provides sufficient conditions for the disturbances to be stable. In the former case,above the critical Rayleigh number, the system becomes linearly unstable and the convective motion begins. It is howeverpossible that convection could commence below the critical value of Rayleigh numbers and in that case energy results be-come very important. For they delimit a band of Rayleigh numbers, where possible subcritical instabilities might arise. Theenergy method is one of the oldest method for nonlinear stability and it can be traced to the work of [1,2] and its recentrevival has been inspired by the work of [3–6]. By introducing the coupling parameters in the energy method and by select-ing them optimally, it has been possible to sharpen the stability bound in many physical problems [7]. A nonlinear stabilityanalysis of non-magnetic fluids has been studied by many authors [8–15]. Recently, unconditional nonlinear energy stabilityanalysis for thermal convection with temperature-dependent viscosity in fluid/porous system has been considered by Hilland Carr [16] and established an excellent agreement between nonlinear and linear stability thresholds, as also found inthe studies of Hill and Straughan [17]. Ferrofluids are the suspensions of ferromagnetic particles of size 10 nm surroundedby surfactant dissolved in a carrier fluid such as organic solvent or water. Ferrofluids span a very wide range of applications

. All rights reserved.

ax: +91 1972 223834.), poonamnit82@gmail.com (P. Sharma), amitmahajan2006@gmail.com (A. Mahajan).

2786 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

[18,19]. Many of these are concerned with the remote positioning and control of ferrofluid using magnetic force fields. Thetheory of convective instability of ferrofluids begins with Finlayson [20] and interestingly continued by Refs. [21–30].

Double-diffusive convection in a rotating ferrofluid layer is a subject of practical interest for its applications in engineer-ing. Among the applications in engineering disciplines one can find the chemical process industry, solidification and centrif-ugal casting of metals and rotating machinery. A layer of ferrofluid heated and soluted from below has relevance andimportance in chemical technology and bio-mechanics. From the physical point of view the rotating double-diffusive caseof a mixture heated and soluted from below is interesting because two stabilizing effects are present: that of rotation andthat of concentration of solute (the heavier ‘‘salt’’ at the lower part of the layer acts to stop motion through convective over-turning). The density gradients that provide the driving buoyancy force are induced by the combined effects of temperatureand solute concentration present in the fluid. When there are two components such as this, the phenomenon of convectionwhich arises is called double-diffusive convection. The double diffusive convection problems have been studied by manyauthors [31–45]. Ryskin et al. [42] have studied the effect of solute in convection of ferrofluids and have shown that the lin-ear as well as the nonlinear convective behavior is significantly altered by the concentration field as compared to single-component systems. Ryskin et al. [43] have studied the thermal convection in binary fluid mixtures with a week concentra-tion diffusivity, but strong solutal buoyancy forces. Ryskin et al. [44] have studied the influence of a magnetic field on theSoret-effect-dominated thermal convection in fluids. Also, linear and nonlinear double-diffusive convection in a fluid satu-rated anisotropic porous layer with Soret effect has been examined analytically by Gaikwad et al. [45]. The effect of rotationon the onset of thermal convection has been studied by [26,37]. It is seen in these papers that the critical Rayleigh number isthe function of the Taylor number (which is the ratio of Coriolis forces to viscous frictional forces) and that a sufficiently largerotation stabilizes the fluid layer. A nonlinear stability problem of a rotating double-diffusive porous layer has been studiedby Guo and Kaloni [8]. More recently, Sunil and Mahajan [46] studied the nonlinear stability analysis of magnetized ferro-fluid heated from below. Also, a nonlinear stability analysis of thermo convective rotating magnetized ferrofluid is done bySunil and Mahajan [47].

In the present paper we applied the generalized energy method to study the nonlinear stability problem of a rotating dou-ble diffusive magnetized ferrofluid layer. The boundaries are taken to be stress-free [41]. Furthermore, the average velocitycondition is considered to exclude the rigid motions [48]. In case of non-ferrofluids, it is observed that the nonlinear andlinear stability results coincide. This in turn implies the exclusion of occurrence of subcritical instability. For ferrofluids,we have found that there is a difference between the values of nonlinear critical magnetic thermal Rayleigh number and lin-ear critical magnetic thermal Rayleigh number, which shows the possibility of the existence of subcritical instability. Com-parison of the results, obtained by the energy method and the linear stability analysis has been discussed finally. In thispaper we also examine the coupling between the buoyancy and magnetic forces in the nonlinear stability analysis. This prob-lem to the best of our knowledge has not been investigated yet.

2. Formulation of the problem

2.1. Governing equations and basic state

Consider an infinite, horizontal layer of thickness ‘d’ of an electrically non-conducting incompressible thin magnetizedferrofluid with constant viscosity heated and soluted from below, and confined between two horizontal plates z = �d/2and z = d/2, in the presence of uniform magnetic field H ¼ Hext

0 k. The layer is subjected to a uniform rotation with an angularvelocity X = (0,0,X) about a vertical axis.

For convective motion of an incompressible rotating double-diffusive magnetized ferrofluid, the governing equations (uti-lizing the Boussinesq approximation), are given by the following [8,47]

r � q ¼ 0; ð1Þ

q0@

@tþ q � r

� �q ¼ �rpþ lr2qþ q0½1� aðT � TaÞ þ a0ðC � CaÞ�gþ l0ðM � rÞHþ 2q0ðq�XÞ; ð2Þ

@

@tþ q � r

� �T ¼ jr2T; ð3Þ

@

@tþ q � r

� �C ¼ j0r2C; ð4Þ

r � B ¼ 0;r�H ¼ 0;B ¼ l0ðHþMÞ; ð5Þ

M ¼ HH½M0 þ vðH � H0Þ � K1ðT � TaÞ þ K2ðC � CaÞ�: ð6Þ

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2787

Here, q, q0, q, t, p, l, l0, M, B, j, j0, a and a0 are the fluid density, reference density, velocity, time, pressure, viscosity, mag-netic permeability of vacuum, magnetization, magnetic induction, thermal diffusivity, solute diffusivity, thermal expansioncoefficient and concentration expansion coefficient analogous to the thermal expansion coefficient, respectively. Ta and Ca

are the average temperature and solute concentration given by Ta ¼ ðTLþTU Þ2 ;Ca ¼ ðCLþCU Þ

2 , respectively, where TL, TU and CL,

CU are the constant average temperatures and solute concentrations of the lower and upper surfaces of the layer,b(=jdT/dzj) and b0(=jdC/dzj) are uniform temperature and solute gradients, respectively, H = jHj, M = jMj and M0 = M(H0, Ta,Ca).The magnetic susceptibility, pyromagnetic coefficient and salinity magnetic coefficient are defined by v � @M

@H

� �H0 ;Ta ;Ca

;

K1 � � @M@T

� �H0 ;Ta ;Ca

and K2 � @M@C

� �H0 ;Ta ;Ca

; respectively.

The effect of rotation contributes two terms: (a) Centrifugal force -12 gradjX� rj2 and (b) Coriolis acceleration 2(X � q). In

(2), p ¼ pf � 12 q0jX� rj2 is the reduced pressure, where pf stands for fluid pressure.

The basic state is assumed to be quiescent state and is given by

q ¼ qb ¼ ð0;0;0Þ; q ¼ qbðzÞ ¼ q0ð1þ abz� a0b0zÞ; p ¼ pbðzÞ; T ¼ TbðzÞ ¼ �bzþ Ta;

C ¼ CbðzÞ ¼ �b0zþ Ca; b ¼ TL � TU

d; b0 ¼ CL � CU

d; Hb ¼ H0 �

K1bz1þ vþ

K2b0z

1þ v

� �k

and

Mb ¼ M0 þK1bz1þ v�

K2b0z

1þ v

� �k; ð7Þ

where the subscript ‘b’ denotes the basic state.

2.2. Perturbation equations

Let q0 ¼ ðu;v ;wÞ;q0; p0; h; c;H0 ¼ H01;H02;H

03

� �and M0 ¼ M0

1;M02;M

03

� �, represent the perturbations of velocity, density, pres-

sure, temperature, concentration, magnetic field intensity and magnetization to the basic state respectively. The nonlinearperturbation equations by retaining only quadratic terms and neglecting terms of order higher than quadratic, can be writtenas

r � q0 ¼ 0; ð8Þ

q0@q0

@t¼ �rp0 þ lr2q0 � l0

K1b1þ v�

K2b0

1þ v

� �ð1þ vÞ/0zk� K1hkþ K2ckh i

þ q0gðah� a0cÞk� q0q0 � rq0

þ l0M0

H0/0xr/0x þ l0

M0

H0/0yr/0y þ l0v/0zr/0z � l0K1hr/0z þ l0K2cr/0z þ 2q0q0 �XÞ; ð9Þ

@h@tþ q0 � rh ¼ jr2hþ bw; ð10Þ

@c@tþ q0 � rc ¼ j0r2cþ b0w; ð11Þ

1þM0

H0

� �r2/0 � M0

H0� v

� �/0zz ¼ K1hz � K2cz: ð12Þ

Here, H0 =r/0 [by Eq. (5)2], where /0 is the perturbed magnetic potential depending on temperature and solute concentra-tion. In the present problem, temperature and solute are considered to have an independent impact, so we can take the po-tential /0 as the difference of two potentials /01 and /02, one analogous to temperature and other analogous to solute. Further,analysis has been carried out using these two potentials.

2.3. Non-dimensional equations

The perturbed Eqs. (8)–(12) are non-dimensionalized using the transformations as in Sunil and Mahajan [46] in additionto the following

c� ¼ S1=2

b0dc; /�2 ¼

ð1þ vÞS1=2

K2b0d2 /02;

to obtain non-dimensional set of equations (dropping ⁄),

r � q ¼ 0; ð13Þ

2788 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

1Pr

@q@t¼ �rpþr2qþ R1=2ð1þM1 �M4Þhk� 1

LeS1=2 1�M0

1 þM04

� �ck� R1=2ðM1 �M4Þ/1zk

� 1Le

S1=2 M01 �M0

4

� �/2zk�M1hr/1z þ

1Le1=2 M1=2

4 M01=24 hr/2z þ

1Le1=2 M1=2

4 M01=24 cr/1z

þ M3 �1

1þ v

� �M1/1xr/1x �

1Le1=2 M1=2

4 M01=24 ð/1xr/2x þ /2xr/1xÞ þ

1Le

M01/2xr/2x

� �

þ M3 �1

1þ v

� �M1/1yr/1y �

1Le1=2 M1=2

4 M01=24 ð/1yr/2y þ /2yr/1yÞ þ

1Le

M01/2yr/2y

� �

þ v1þ v

� �M1/1zr/1z �

1Le1=2 M1=2

4 M01=24 ð/1zr/2z þ /2zr/1zÞ þ

1Le

M01/2zr/2z

� �

� 1Pr

q � rqþ T1=2A

Prðq� kÞ � 1

LeM0

1cr/2z; ð14Þ

@h@tþ q � rh ¼ r2hþ R1=2w; ð15Þ

@c@tþ q � rc ¼ 1

Ler2cþ S1=2w; ð16Þ

M3r2/1 � ðM3 � 1Þ/1zz ¼ hz; ð17Þ

M3r2/2 � ðM3 � 1Þ/2zz ¼ cz: ð18Þ

The non-dimensional parameters used in the above equations are same as in the work of Sunil and Mahajan [46]. However,some additional parameters are introduced due to effect of solute and rotation, which are given as

S ¼ ga0b0q0d4

lj0; M0

1 ¼l0K2

2b0

a0q0gð1þ vÞ ; TA ¼2Xd2

j

!2

; Le ¼ jj0; Pr ¼ m

j;

M4 ¼l0K1K2b

0

aq0gð1þ vÞ ; M04 ¼

l0K1K2ba0q0gð1þ vÞ ; M5 ¼

M4

M1¼ M0

1

M04

¼ K2b0

K1b:

Here, S is salt Rayleigh number, M01 is the effect of magnetization due to salinity, TA is the Taylor number, Le is the Lewis

number, Pr is the Prandtl number and M5 represents the ratio of the salinity effect on magnetic field to pyromagneticcoefficient.

Taking operators k � curl and k � curlcurl on (14), we get

1Pr

@f@t¼ r2fþ 1

Le1=2 M1=24 M01=2

4 ðhx/2zy � hy/2zxÞ �M1ðhx/1zy � hy/1zxÞ �1Pr

k � curlðq � rqÞ

þ 1Le1=2 M1=2

4 M01=24 ðcx/1zy � cy/1zxÞ �

1Le

M01ðcx/2zy � cy/2zxÞ þ

T1=2A

Prwz; ð19Þ

1Pr

@r2w@t

¼ r4wþ R1=2ð1þM1 �M4Þr21h�

1Le

S1=2 1þM04 �M0

1

� �r2

1cþ1Pr

k � curlcurlðq � rqÞ

� 1Le

S1=2 M01 �M0

4

� �r2

1/2z þM1 hzr21/1z � /1zzr2

1hþ hzx/1zx þ hzy/1zy

� � R1=2ðM1 �M4Þr2

1/1z

� 1Le1=2 M1=2

4 M01=24 hzr2

1/2z � /2zzr21hþ hzx/2zx þ hzy/2zy

�

þ 1Le1=2 M1=2

4 M01=24 hx/2zzx þ hy/2zzy þ cx/1zzx þ cy/1zzy

� �M1ðhx/1zzx þ hy/1zzyÞ

�M1=24 M01=2

4

Le1=2 czr21/1z � /1zzr2

1cþ czx/1zx þ czy/1zy

� � 1

LeM0

1ðcx/2zzx þ cy/2zzyÞ

þ 1Le

M01 czr2

1/2z � /2zzr21cþ czx/2zx þ czy/2zy

� � 1

PrT1=2

A fz; ð20Þ

where f ¼ k � curlq is the z-component of vorticity.From (15) and (16), we have

@hz

@tþ qz � rhþ q � rhz ¼ r2hz þ R1=2wz; ð21Þ

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2789

@cz

@tþ qz � rcþ q � rcz ¼

1Ler2cz þ S1=2wz: ð22Þ

The functions q, h, c, /1, /2 must be subject to boundary conditions and we suppose that q, h, c, /1, /2 are periodic in x, y withperiods 2p

a1and 2p

a2, respectively, and the surfaces are stress free so that

w ¼ 0;uz ¼ 0; vz ¼ 0; h ¼ 0; c ¼ 0;/1z ¼ 0;/2z ¼ 0 at z ¼ �12: ð23Þ

To exclude the rigid motions we assume that the mean values of u, v are zero, see Ref. [48]; i.e. we require

ZV

udV ¼Z

VvdV ¼ 0; ð24Þ

where V ¼ 0; 2pa1

h � 0; 2p

a2

h � � 1

2 ;12

� �, is the typical periodicity cell.

Also Eqs. (20)–(22) are of higher order than (14)–(18). So we need extra boundary conditions [49]. As before hz, cz andr2w are periodic in x and y and, in addition to (23), we have

hzz ¼ 0; czz ¼ 0; fz ¼ 0;wzz ¼ 0 on z ¼ �12: ð25Þ

3. Energy stability analysis

To study the nonlinear stability of basic state (7), an L2 energy, E (t), is constructed using Eqs. (A.1)–(A.6) (see Appendix A),and the evolution of E (t) is given by

dEdt¼ I0 � D0 þ N0; ð26Þ

where

E ¼ 12khzk2 þ k1

2Prkrwk2 � k3

2Prkfk2 � k4

2kczk

2; ð27Þ

and the terms I0, D0 and N0 are written in Appendix A.Here, �ve sign with k3

2Pr kfk2 and k4

2 kczk2 term in the energy Eq. (27) shows that energy of the system is consumed due to

rotation and solute concentration as the system is rotated along vertical axis and soluted from below. Now, we take theassumption that the energy consumed due to solute concentration and rotation is less than the energy produced due tovelocity and temperature. This assumption will ensure the validity of principle of exchange of stability (see Appendix C).

We now define

m ¼maxH

I0

D0; ð28Þ

where H is the space of admissible solutions.Then we require m < 1 so that

dEdt6 �a0D0 þ N0; ð29Þ

with a0 = 1 �m(>0).We now introduce the generalized energy functional as

VgðtÞ ¼ EðtÞ þ b0E1ðtÞ; ð30Þ

where b0 is a positive coupling parameter to be chosen and the complementary energy E1(t) is given by

E1ðtÞ ¼12krhk2 þ 1

2Prkrqk2 þ 1

2krck2

: ð31Þ

The evolution of Vg(t) is given by

dVg

dt6 �a0D0 þ N0 þ b0I1 � b0D1 þ b0N1; ð32Þ

where the terms I1, D1 and N1 are written in Appendix B.Now, we write some easily obtainable results from Eqs. (17), (18) and recall embedding theorems

2790 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

kr/1k 6 khk; kr/1zk 6 kr2/1k 6 krhk; kr2/1zk 6 kr2hk; kr2/1xk 6 kr2hk; kr2/1yk 6 kr2hk;kr/2k 6 kck; kr/2zk 6 kr2/2k 6 krck; kr2/2zk 6 kr2ck; kr2/2xk 6 kr2ck; kr2/2yk 6 kr2ck;krwk 6 krqk; sup jFj 6 C�kr2Fk; F 2 fq; h; c;/1x;/1y;/1z;/2x;/2y;/2zg;

ð33Þ

where C⁄ is a computable positive constant depending on V, its value is given in Galdi and Straughan [49] and the statementis proved in Adams [50].

The production term I1 is estimated using Eqs. (A.8), (B.2) (see Appendix A and B) and (33), the Cauchy–Schwartz and theYoung inequalities [51], we have

b0I1 6 2b0e20D1 þ

b0

2e20p4

Rfð2þM1 �M4Þ2 þ ðM1 �M4Þ2g þ SLek4

Le� 1þM01 �M0

4

� �2 þ M01 �M0

4

� �2n o

24

35D0:

Choosing e20 ¼ 1

4 and

b0 ¼a0p4

4 Rfð2þM1 �M4Þ2 þ ðM1 �M4Þ2g þ 1Le

Sk4ðLe� 1þM0

1 �M04Þ

2 þ M01 �M0

4

� �2n oh i

and defining

D2 ¼a0

2D0 þ

b0

2D1; ð34Þ

it then follows easily that

b0I1 6 D2: ð35Þ

We next estimate nonlinear terms N1 and N0. With the help of (30), (33) and (34), we find,

N1 6 C�2b0

� �3=2

1þ 1Pr1=2 þ Le1=2 þ M1 þ

M01

Le1=2 þM1=24 M01=2

4 1þ 1Le1=2

� �� �2M3 þ

2v� 11þ v

� �� �D2V1=2

g ; ð36Þ

N0 6 C�2b0

� �3=2

1þ 2k1

Pr1=2 þ k4Le1=2�

þ

2b1=20

a1=20

k1=23

21=21

Pr1=2 þM1 þ M01Le1=2

� þ 3k1=2

1 M1 þ 1Le1=2 M0

1

� þM1=2

4 M01=24 1þ 1

Le1=2

� 3k1=2

1 þ k1=23

21=2

� �8>><>>:

9>>=>>;

2666664

3777775D2V1=2

g : ð37Þ

Using Eqs. (34)–(37) in (32), we get

_VgðtÞ 6 �D2ð1� �AV1=2g Þ; ð38Þ

where

�A ¼ C�2b0

� �3=2

1þ 2k1

Pr1=2 þ k4Le1=2�

þ b0 1þ 1Pr1=2 þ Le1=2

� þ 2k3b0

a0Pr

� 1=2

þ 2b1=20

a1=20

3k1=21 þ k1=2

3

21=2

� �þ b0 2M3 þ 2v�1

1þv

� �

M1 þ M01Le1=2 þM1=2

4 M01=24 1þ 1

Le1=2

� n o

26666664

37777775: ð39Þ

Lemma. Let

0 < m < 1; ð40ÞVgð0Þ < �A�2 ð41Þ

with À given by Eq. (39). Then, there exists a positive constant K⁄, such that

VgðtÞ 6 Vgð0Þe�K�ð1��AV1=2g ð0ÞÞt; t P 0: ð42Þ

Following the procedure as in Sunil and Mahajan [47], we obtain the result (42) with K⁄ given by

K� ¼ 1B2 1þ Leþ 2

Pr

� �ð1þ k0Þ

: ð43Þ

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2791

Since Vg(t) in Eq. (30) does not contain the terms kr/1k2 and kr/2k2, the kinetic energy terms for magnetic potential, it is worth-while checking as to what happens to kr/1k2 and kr/2k2 as t ?1. Using inequality (33)1 and (33)6, we have

kr/1k26 khk2 and kr/2k

26 kck2

: ð44Þ

Thus, Eq. (30) and inequalities (44) ensure the decay of kr /1k2 and kr/2k2, i.e. kH0k2.

3.1. The eigenvalue problem of nonlinear analysis

The Euler–Lagrange equations are obtained after taking transformations w ¼ffiffiffiffiffik1p

w; /1 ¼ffiffiffiffiffik2p

/1; f ¼ffiffiffiffiffik3p

f; c ¼ffiffiffiffiffik4p

c and/2 ¼

ffiffiffiffiffik5p

/2 (dropping caps) and assuming the plane tiling form

ðw; h; c;/1;/2; fÞ ¼ ½WðzÞ;HðzÞ;CðzÞ;U1ðzÞ;U2ðzÞ;wðzÞ�gðx; yÞ

where r21g þ a2g ¼ 0, ‘a’ being the wave number [41], pp. 106–114; [14]. The wave number is found a posteriori to be non-

zero, we see that W, H, C, U1, U2, w satisfy

2ðD2 � a2Þ2W � a2k1=21 R1=2ð1þM1 �M4ÞHþ

a2k1=21 S1=2

k1=24 Le

1�M01 þM0

4

� �Cþ R1=2

k1=21

D2Hþ a2k1=21

k1=22

R1=2ðM1 �M4ÞDU1

þ a2k1=21

k1=25 Le

S1=2 M01 �M0

4

� �DU2 �

k1=24

k1=21

S1=2D2C� k1 þ k3

k1=21 k1=2

3

!T1=2

A

PrDw ¼ 0; ð45Þ

2D2ðD2 � a2ÞH� a2k1=21 R1=2ð1þM1 �M4ÞW � k1=2

2 DU1 þR1=2

k1=21

D2W ¼ 0; ð46Þ

2Le

D2ðD2 � a2ÞC� a2k1=21 S1=2

Lek1=24

1�M01 þM0

4

� �W þ k1=2

4 S1=2

k1=21

D2W � k1=25

k1=24

DU2 ¼ 0; ð47Þ

2ðD2 � a2Þwþ ðk1 þ k3Þk1=2

1 k1=23

T1=2A

PrDW ¼ 0; ð48Þ

2ðD2 � a2M3ÞU1 þa2R1=2k1=2

1

k1=22

ðM1 �M4ÞDW � k1=22 DH ¼ 0; ð49Þ

2ðD2 � a2M3ÞU2 �a2S1=2k1=2

1

k1=25 Le

M01 �M0

4

� �DW � k1=2

5

k1=24

DC ¼ 0: ð50Þ

Thus, the exact solution to the Eqs. (45)–(50) subject to boundary conditions

W ¼ 0; D2W ¼ 0; H ¼ 0; C ¼ 0; DU1 ¼ 0; DU2 ¼ 0; Dw ¼ 0 at z ¼ �12; ð51Þ

is written in the form

W ¼ A1 cospz; H ¼ A2 cos pz; C ¼ A3 cos pz; DU1 ¼ A4cospz; DU2 ¼ A5 cospz; Dw ¼ A6 cospz; ð52Þ

where A1, A2, A3, A4, A5 and A6 are constants. Substituting solution (52) in Eqs. (45)–(50), we get the equations involving coef-ficients of A1, A2, A3, A4, A5, A6. For the existence of non-trivial solutions, the determinant of the coefficients of A1, A2, A3, A4, A5

and A6 must vanish. This determinant on simplification yields

Re ¼ maxk1 ;k

02 ;k3 ;k4 ;k

05

minx1

R0e k1; k02; k3; k4; k

05; x1;M1;M3;M

01;M5; S1; TA1 ; Le;Pr

� �; ð53Þ

where

R0e ¼Rp4 ; S1 ¼

Sp4 ; TA1 ¼

TA

p4 ; x1 ¼a2

p2 ; k02 ¼k2

p4 ; k05 ¼k5

p4 :

To achieve this requires careful selection of k1; k02; k3; k4 and k05 are found to be

k1 ¼ k3 ¼1

x1f1þM1ð1�M5Þg; k02 ¼

ð1þ x1ÞM1ð1�M5Þ1þM1ð1�M5Þ

; k4 ¼1þM0

1ð 1M5� 1Þ

Lef1þM1ð1�M5Þg; k05 ¼

ð1þ x1ÞM01

1M5� 1

� Le2f1þM1ð1�M5Þg

:

ð54Þ

2792 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

Using (54) in Eq. (53), we have

Re ¼4ð1þ x1M3Þ � M1ð1�M5Þ

1þM1ð1�M5Þ

h iTA1

Pr2 þ ð1þ x1Þ3 þ x1S1 1þM01

1M5� 1

� n oh ix1½4ð1þ x1M3Þf1þM1ð1�M5Þg � 2M1ð1�M5Þ�

: ð55Þ

For M1 sufficiently large, we obtain the magnetic thermal Rayleigh number

Ne ¼ M1Re ¼ð3þ 4x1M3Þ

TA1

Pr2 þ ð1þ x1Þ3 þ x1S1f1þM01

1M5� 1

� g

h ix1ð2þ 4x1M3Þð1�M5Þ

: ð56Þ

The magnetic thermal Rayleigh number Ne is minimized with respect to x1 and we use Newton–Raphson iteration scheme toobtain the values of critical wave number and the corresponding critical magnetic thermal Rayleigh number.

In the absence of solute and rotation, (56) reduces to Ne ¼ ð1þx1Þ3ð3þ4x1M3Þx1ð2þ4x1M3Þ

,

which is in good agreement with previous published work [46].For fixed values of S1 ¼ 100; TA1 ¼ 100;Pr ¼ 1;M0

1 ¼ 0:1;M5 ¼ 0:1, the critical wave number, xce, and critical magneticthermal Rayleigh number, Nce, depend on the parameter M3, taking the values

Nce ¼ 288:52; xce ¼ 3:734 for M3 ¼ 1;Nce ¼ 271:54; xce ¼ 3:258 for M3 !1;

and intermediate values for intermediate M3. Here we can rearrange Eq. (55) to demonstrate the interaction of buoyancy andmagnetic modes of instability.

Re

Rceþ Ne

Nce

Nceð2þ 4x1M3Þð1�M5Þ154:39ð3þ 4x1M3Þ

�¼

TA1Pr2 þ ð1þ x1Þ3 þ x1S1 1þM0

11

M5� 1

� n o154:39x1

: ð57Þ

When M3 is very large, (57) reduces to ReRceþ Ne

Nceð1:58Þ ¼ 1:58.

Here we remark that in the absence of rotation and solute gradient there is tight coupling between the buoyancy andmagnetic forces for nonlinear energy stability analysis [46], whereas in the presence of rotation and solute gradient, eachindividual convective mechanism yields the different wave number, so tight coupling is not possible in the present case.

3.2. Linear analysis

For analyzing the linear instability results, we return to the non-dimensionalized Eqs. (13)–(18), neglecting the nonlinearterms. We again perform the standard, stationary, mode analysis and look for the solution of these equations in the form(52). The boundary conditions in the present case are same, i.e. (51). Following the procedure as stated earlier in the energystability case, we have

R‘ ¼ð1þ x1M3Þ

TA1Pr2 þ ð1þ x1Þ3h i

þ x1S1 1þ x1M3 þ x1M01M3

1M5� 1

� h ix1½1þ x1M3 þ x1M1M3ð1�M5Þ�

: ð58Þ

We again consider magnetic thermal Rayleigh number N‘ depends on the parameter M3. For M1 sufficiently large, the criticalmagnetic thermal Rayleigh number, in linear case, is

N‘ ¼ð1þ x1M3Þ

TA1

Pr2 þ ð1þ x1Þ3h i

þ x1S1 1þ x1M3 þ x1M01M3

1M5� 1

� h ix2

1M3ð1�M5Þ: ð59Þ

In the absence of solute and rotation, (59) reduce to

N‘ ¼ð1þ x1Þ3ð1þ x1M3Þ

x21M3

;

which is in good agreement with the previous published work [20].There are instances in which the two theories coincide. This is true for the classical Bénard problem. In the absence of

magnetic parameters M1 ¼ 0;M01 ¼ 0 and M3 ¼ 0

� �, we obtain

R‘ ¼ð1þ x1Þ3 þ

TA1Pr2

x1þ S1 ¼ Re:

In the absence of solute and rotation ðS1 ¼ 0; TA1 ¼ 0Þ, this further simplifies to

R‘ ¼ð1þ x1Þ3

x1¼ Re;

i.e., in both the cases, the linear instability boundary � the nonlinear stability boundary.

Table 1The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with magnetic parameter (M3) for S1 ¼ 100;M0

1 ¼ 0:1; Pr ¼ 1;M5 ¼ 0:1; TA1 ¼ 100.

M3 xc‘ Nc‘ xce Nce Nc‘ � Nce

1 4.41 316.16 3.73 288.52 27.645 3.58 281.57 3.39 275.51 6.06

10 3.43 276.67 3.33 273.57 3.1015 3.38 274.99 3.31 272.91 2.0820 3.35 274.14 3.29 272.57 1.5725 3.33 273.63 3.28 272.37 1.26

Table 2The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with solute gradient (S1) for M3 ¼ 5; TA1 ¼ 100; Pr ¼ 1;M0

1 ¼ 0:1;M5 ¼ 0:1.

S1 xc‘ Nc‘ xce Nce Nc‘ � Nce

100 3.58 281.57 3.39 275.51 6.06200 3.77 498.73 3.49 489.60 9.13300 3.94 715.61 3.58 703.62 11.99400 4.09 932.25 3.67 917.56 14.66500 4.25 1148.69 3.75 1131.45 17.24

Table 3The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with Taylor number ðTA1 Þ for M3 ¼ 5; S1 ¼ 100;Pr ¼ 1;M0

1 ¼ 0:1;M5 ¼ 0:1.

TA1xc‘ Nc‘ xce Nce Nc‘ � Nce

1 1.83 239.19 1.35 230.29 8.910 2.13 244.75 1.75 236.89 7.8620 2.38 250.11 2.07 242.85 7.2640 2.77 259.38 2.52 252.71 6.6760 3.09 267.47 2.86 261.10 6.3780 3.35 274.79 3.15 268.61 6.18

100 3.58 281.57 3.39 275.51 6.06200 4.45 310.52 4.29 304.68 5.84400 5.59 356.36 5.46 350.49 5.87600 6.41 394.55 6.29 388.55 6.00800 7.07 428.47 6.96 422.31 6.16

1000 7.63 459.51 7.52 453.20 6.312000 9.66 590.44 9.57 583.43 7.01

10000 16.73 1259.11 16.65 1249.09 10.02

260

270

280

290

300

310

320

1 6 11 16 21

Linear

3M

Nonlinear

Instability region

Stability region

Subcritical instability region

Fig. 1. The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with magnetic parameter (M3) for S1 ¼ 100; Pr ¼ 1;M01 ¼ 0:1;

M5 ¼ 0:1; TA1 ¼ 100.

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2793

Here, the energy method leads to the strong result that arbitrary sub-critical instabilities are not possible, which is ingood agreement with the previous published work [4,5]. Thus, for lower values of magnetic parameters, this coincidenceis immediately lost.

200

300

400

500

600

700

800

900

1000

1100

1200

100 150 200 250 300 350 400 450 500

1S

Instability region

Stability region

Subcritical instability region

Linear

Nonlinear

Fig. 2. The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with solute gradient (S1) for M3 ¼ 5;M01 ¼ 0:1;M5 ¼ 0:1;

TA1 ¼ 100; Pr ¼ 1.

200

250

300

350

400

450

500

100010010

Stability region

Instability region

Subcritical instability region

1AT

Linear

Nonlinear

Fig. 3. The variation of the critical magnetic thermal Rayleigh numbers (Nc‘ and Nce) with Taylor number ðTA1 Þ for M3 ¼ 5;M01 ¼ 0:1;M5 ¼ 0:1;

S1 ¼ 100;Pr ¼ 1.

2794 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

4. Discussion of results

We begin with the important observation that the values of critical magnetic thermal Rayleigh number calculated by en-ergy method i.e. Nce are always smaller than those obtained by linear stability analysis i.e. Nc‘. Thus the difference betweenthe values of Nc‘ and Nce reveals that there is a band of Rayleigh numbers where subcritical instabilities may arise. The var-iation of xc‘, xce and Nc‘, Nce with various parameters are given in Tables 1–3 and the results are further illustrated in Figs. 1–3. Fig. 1 represents the plot of critical magnetic thermal Rayleigh numbers Nc‘ and Nce versus magnetic parameter M3. Thisfigure indicates that the magnetic parameter M3 has a destabilizing effect because as M3 increases, the values of Nc‘ and Nce

decreases. We also note that the values of Nc‘ are always higher than those of Nce and this is quite understandable becausethe linear stability theory gives sufficient conditions for instability, while the energy stability theory gives the sufficient con-dition for stability. Thus, the difference between the values of Nc‘ and Nce reveals that there is a band of Rayleigh numberswhere subcritical instabilities may arise. We note that this band decreases as M3 increases (Table 1).

Fig. 2 represents the plot of critical magnetic thermal Rayleigh numbers Nc‘ and Nce versus solute gradient S1. It is observedfrom figure that the solute gradient S1 has a stabilizing effect because the values of Nc‘ and Nce increase with the increase in thevalues of solute parameter. Here, in double-diffusive convection two diffusing components heat and salt are present thatproduce the density differences required to derive the motion. The components make opposing contributions to the verticaldensity gradient as motion is encouraged due to heating and solute acts to prevent motion through convective overturning.Thus, these two physical effects are competing against each other. Due to this competition, it means that the linear theory ofinstability does not always capture the physics of instability completely and (subcritical) instabilities might arise before thethreshold is reached. It is also noted that subcritical instability region increases with the increase of solute gradient.

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2795

From Fig. 3 it is clear that as Taylor number TA1 increases the values of Nc‘ and Nce increase indicating the stabilizing effectof rotation. In order to investigate our results, we must review the results and physical explanations. It is well known thatrotation introduces vorticity into the fluid in case of Newtonian fluid [41]. Then the fluid moves in the horizontal planes withhigher velocities. On account of this motion, the velocity of the fluid perpendicular to the plane reduces, and hence delays theonset of convection. We also note that the subcritical instability region expands a little for small values of Taylor number andexpands significantly for large values of Taylor number.

5. Conclusions

The principal conclusions from the above analysis are as under:

1. The result we establish is that the boundaries of nonlinear stability and linear instability analyses do not coincide.2. It is found that the solute gradient and Taylor number always delay the onset of convection, whereas magnetization has-

tens the onset of convection.3. Comparison between the results obtained by nonlinear and linear theory shows that linear critical magnetic thermal Ray-

leigh number is higher in values than the nonlinear critical magnetic thermal Rayleigh number and thus indicates theexistence of subcritical instability region.

4. It is found that subcritical region of instability can be induced by magnetic mechanism alone.5. It is noted that subcritical instability region decreases as magnetization increase whereas as solute gradient increases, the

gap between the linear and energy stability results widens.6. It is noted that subcritical instability region decreases for small values of Taylor number and expands for large values of

Taylor number.7. In non-ferrofluids, a best possible is verified in that we show that the global nonlinear stability Rayleigh number is exactly

the same as that for linear instability.

Acknowledgments

The financial assistance to Dr. Sunil in the form of Research and Development Project [No. 25(0165)/08/EMR-II] from theCouncil of Scientific and Industrial Research (CSIR) New Delhi, to Miss Poonam Sharma in the form of Research Fellowship(RF) from NIT Hamirpur and to Dr. Amit Mahajan in the form of Post-Doctoral Fellowship [Grant No. 8258 of NSERC] fromUniversity of Windsor, Canada are gratefully acknowledged.

Appendix A

12Pr

dkrwk2

dt¼ �kr2wk2 þ R1=2ð1þM1 �M4Þhr1w � r1hi �

1Le

S1=2 1�M01 þM0

4

� �hr1w � r1ci

� R1=2ðM1 �M4Þhr1w � r1/1zi �1Le

S1=2 M01 �M0

4

� �hr1w � r1/2zi þ

T1=2A

Prhwfzi �M1½hwhzx/1zxi

þ hwhzy/1zyi þ hwhzr21/1zi � hwr2

1h/1zzi� þ1

Le1=2 M1=24 M01=2

4 ½hwhzx/2zxi þ hwhzy/2zyi

þ hwhzr21/2zi � hw/2zzr2

1hi� þ1

Le1=2 M1=24 M01=2

4 ½hwczx/1zxi þ hwczy/1zyi þ hwczr21/1zi

� hw/1zzr21ci� �

1Le

M01½hwczx/2zxi þ hwczy/2zyi þ hwczr2

1/2zi � hwr21c/2zzi�

� 1Le1=2 M1=2

4 M01=24 ½hwhx/2zzxi þ hwhy/2zzyi þ hwcx/1zzxi þ hwcy/1zzyi� þM1½hwhx/1zzxi

þ hwhy/1zzyi� þ1Le

M01½hwcx/2zzxi þ hwcy/2zzyi� �

1Prhwk � curlcurlðq � rqÞi; ðA:1Þ

12Pr

dkfk2

dt¼ �krfk2 �M1½hfhx/1zyi � hfhy/1zxi� �

1Le

M01½hfcx/2zyi � hfcy/2zxi� þ

1Le1=2 M1=2

4 M01=24 ½hfhx/2zyi

� hfhy/2zxi þ hfcx/1zyi � hfcy/1zxi� þT1=2

A

Prhfwzi �

1Prhfk � curlðq � rqÞi; ðA:2Þ

2796 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

12

dkhzk2

dt¼ �krhzk2 þ R1=2hwzhzi þ hhzzq � rhi; ðA:3Þ

12

dkczk2

dt¼ � 1

Lekrczk

2 þ S1=2hwzczi þ hczzq � rci; ðA:4Þ

M3kr/1k2 � ðM3 � 1Þk/1zk

2 þ hhz/1i ¼ 0; ðA:5ÞM3kr/2k

2 � ðM3 � 1Þk/2zk2 þ hcz/2i ¼ 0; ðA:6Þ

I0 ¼R1=2k1ð1þM1 �M4Þhr1w � r1hi �k1

LeS1=2 1�M0

1 þM04

� �hr1w � r1ci þR1=2hwzhzi �

k1

LeS1=2 M0

1 �M04

� �� hr1w � r1/2zi � k2hhz/1i � k4S1=2hwzczi þ k5hcz/2i �R1=2k1ðM1 �M4Þhr1w � r1/1zi þ ðk1 þ k3Þ

T1=2A

Prhwfzi; ðA:7Þ

D0 ¼ krhzk2 þ k1kr2wk2 þ k2M3kr/1k2 � k2ðM3 �1Þk/1zk

2 � k3krfk2 � k4

Lekrczk

2 � k5M3kr/2k2 þ k5ðM3 �1Þk/2zk

2;

ðA:8Þ

N0 ¼ hhzzq � rhi þ k3M1½hfhx/1zyi � hfhy/1zxi� þk3

LeM0

1½hfcx/2zyi � hfcy/2zxi� �k3

Le1=2 M1=24 M01=2

4 ½hfhx/2zyi

� hfhy/2zxi þ hfcx/1zyi � hfcy/1zxi� þk1

Le1=2 M1=24 M01=2

4 ½hwhzx/2zxi þ hwhzy/2zyi þ hwhzr21/2zi � hw/2zzr2

1hi�

þ k1

Le1=2 M1=24 M01=2

4 ½hwczx/1zxi þ hwczy/1zyi þ hwczr21/1zi � hw/1zzr2

1ci� �k1

Le1=2 M1=24 M01=2

4 ½hwhx/2zzxi

þ hwhy/2zzyi þ hwcx/1zzxi þ hwcy/1zzyi� � k1M1½hwhzx/1zxi þ hwhzy/1zyi þ hwhzr21/1zi � hw/1zzr2

1hi�

� k1

LeM0

1½hwczx/2zxi þ hwczy/2zyi þ hwczr21/2zi � hw/2zzr2

1ci� þ k1M1½hwhx/1zzxi þ hwhy/1zzyi�

þ k1

LeM0

1½hwcx/2zzxi þ hwcy/2zzyi� � k4hczzq � rci þ k3

Prhfk � curlðq � rqÞi � k1

Prhwk � curlcurlðq � rqÞi; ðA:9Þ

with k1, k2, k3, k4, k5 being five positive coupling parameters.We also assume that the energy dissipated by the solute concentration and rotation is less than the energy dissipated by

the velocity, temperature and magnetization. This assumption will ensure that all the terms on the right hand side of equa-tions (A.8) are always less than the left hand side of that equation.

Appendix B

I1 ¼ R1=2ð2þM1 �M4Þhrw � rhi � S1=2

Le1�M0

1 þM04

� �hrw � rci � R1=2ðM1 �M4Þhrw � r/1zi

� S1=2

LeM0

1 �M04

� �hrw � r/2zi þ S1=2hrw � rci; ðB:1Þ

D1 ¼ kr2qk2 þ kr2hk2 þ 1Lekr2ck2

; ðB:2Þ

N1 ¼ hq � rhr2hi þ 1Prhq � rq � r2qi þ hq � rcr2ci þM1hhr2q � r/1zi þ

1Le

M01hcr2q � r/2zi

� 1Le1=2 M1=2

4 M01=24 ½hhr2q � r/2zi þ hcr2q � r/1zi� �

M01

LeM3 �

11þ v

� �h/2xr2q � r/2xi

þM1=24 M01=2

4

Le1=2 M3 �1

1þ v

� �½h/1xr2q � r/2xi þ h/2xr2q � r/1xi� �

v1þ v

� �M1h/1zr2q � r/1zi

þM1=24 M01=2

4

Le1=2 M3 �1

1þ v

� �½h/1yr2q � r/2yi þ h/2yr2q � r/1yi� �

v1þ v

� �M0

1

Leh/2zr2q � r/2zi

�M1 M3 �1

1þ v

� �½h/1xr2q � r/1xi þ h/1yr2q � r/1yi� �

M01

LeM3 �

11þ v

� �h/2yr2q � r/2yi

þM1=24 M01=2

4

Le1=2

v1þ v

� �½h/1zr2q � r/2zi þ h/2zr2q � r/1zi�: ðB:3Þ

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2797

Appendix C

We shall prove Principle of Exchange of Stabilities to be valid, to this end we take w = wert, h = hert, /1 = /1 ert, /2 = /2ert,f = fert, c = cert, in linearized form of Eqs. (15)–(20), to obtain

rPrr2w ¼ r4wþ R1=2ð1þM1 �M4Þr2

1h�1Le

S1=2 1þM04 �M0

1

� �r2

1c�1Le

S1=2 M01 �M0

4

� �r2

1/2z

� R1=2ðM1 �M4Þr21/1z �

1Pr

T1=2A fz; ðC:1Þ

rPr

f ¼ r2fþ T1=2A

Prwz; ðC:2Þ

rr21h ¼ r2

1r2hþ R1=2r21w; ðC:3Þ

rr21c ¼

1Ler2

1r2cþ S1=2r21w; ðC:4Þ

M3r2/1 � ðM3 � 1Þ/1zz ¼ hz; ðC:5ÞM3r2/2 � ðM3 � 1Þ/2zz ¼ cz: ðC:6Þ

Multiplying Eqs. (C.1)–(C.6) by w�; f�; h�; c�;/�1;/�2 (⁄ denoting the complex conjugate) respectively, we get

rPrkrwk2 ¼ �kr2wk2 þ R1=2ð1þM1 �M4Þhr1h � r1w�i � S1=2

Le1þM0

4 �M01

� �hr1c � r1w�i

þ S1=2

LeM0

1 �M04

� �hr2

1/2zw�i þ R1=2ðM1 �M4Þhr21/1zw�i þ T1=2

A

Prhfzw�i; ðC:7Þ

� rPrkfk2 ¼ krfk2 þ T1=2

A

Prhwf�zi; ðC:8Þ

rð1þM1 �M4Þkr1hk2 ¼ �ð1þM1 �M4Þkr1rhk2 þ R1=2ð1þM1 �M4Þhr1w � r1h�i; ðC:9Þ

�r1þM0

4 �M01

� �Le

kr1ck2 ¼1þM0

4 �M01

� �Le2 kr1rck2 � S1=2 1þM0

4 �M01

� �Le

hr1w � r1c�i; ðC:10Þ

M3kr/1k2 � ðM3 � 1Þk/1zk

2 ¼ hh/�1zi; ðC:11Þ

M3kr/2k2 � ðM3 � 1Þk/2zk

2 ¼ hc/�2zi: ðC:12Þ

Now,

S1=2hr21/2zw�i ¼ S1=2h/2zr2

1w�i ¼ h/2zðr�r21c� � 1

Ler2

1r2c�Þi½using equationðC:4Þ� ¼ �r�hr21/2c�zi þ

1Leh/2r2

1r2c�zi

¼ �r�M3krr1/2k2 þ r�ðM3 � 1Þkr1/2zk

2 �M3

Lekr2r1/2k

2 þ ðM3 � 1ÞLe

krr1/2zk2 ðC:13Þ

[by using Eq. (5) and integrating by parts]Similarly, one can obtain

R1=2hr21/1zw�i ¼ �r�M3krr1/1k

2 þ r�ðM3 � 1Þkr1/1zk2 �M3kr2r1/1k

2 þ ðM3 � 1Þkrr1/1zk2 ðC:14Þ

Eq. (C.7), using Eqs. (C.13), (C.14) becomes

rPrkrwk2 � r�

M3 M04 �M0

1

� �Le

krr1/2k2 þ r�ðM3 � 1Þ ðM

04 �M0

1ÞLe

kr1/2zk2 þ r�M3ðM1 �M4Þkrr1/1k

2

� r�ðM3 � 1ÞðM1 �M4Þkr1/1zk2

¼ �kr2wk2 þ R1=2ð1þM1 �M4Þhr1h � r1w�i � S1=2

Le1þM0

4 �M01

� �hr1c � r1w�i

�ðM3 � 1Þ M0

4 �M01

� �Le2 krr1/2zk

2 þ ðM1 �M4Þ½ðM3 � 1Þkrr1/1zk2 �M3kr2r1/1k

2�

þM3 M0

4 �M01

� �Le2 kr2r1/2k

2 þ T1=2A

Prhfzw�i: ðC:15Þ

2798 Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799

Adding equations (C.15), (C.8), (C.9) and, (C.10), we obtain

rPrkrwk2 � r� M3ðM0

4 �M01Þ

Lekrr1/2k

2 þ r�ðM3 � 1Þ ðM04 �M0

1ÞLe

kr1/2zk2 þ r�M3ðM1 �M4Þkrr1/1k

2

� r�ðM3 � 1ÞðM1 �M4Þkr1/1zk2 þ rð1þM1 �M4Þkr1hk2 � r ð1þM0

4 �M01Þ

Lekr1ck2 � r

Prkfk2

¼ �kr2wk2 þ ð1þM04 �M0

1ÞLe2 kr1rck2 þ krfk2 � ðM1 �M4Þ½M3kr2r1/1k

2 � ðM3 � 1Þkrr1/1zk2�

� ð1þM1 �M4Þkr1rhk2 þ ðM04 �M0

1ÞLe2 ½M3kr2r1/2k

2 � ðM3 � 1Þkrr1/2zk2�

þ R1=2ð1þM1 �M4Þ½hr1h � r1w�i þ hr1w � r1h�i� þ T1=2

A

Pr½hfzw�i þ hwf�zi�

� S1=2

Le1þM0

4 �M01

� �½hr1w � r1c�i þ hr1c � r1w�i�: ðC:16Þ

The right hand side of (C.16) is real and so if we let r = rr + iri, then taking the imaginary part of (C.16), we find

ri

1Pr krwk2 þ M3 M04�M01ð Þ

Le krr1/2k2 � ðM3 � 1Þ M04�M01ð Þ

Le kr1/2zk2 � 1

Pr kfk2

�M3ðM1 �M4Þkrr1/1k2 þ ðM3 � 1ÞðM1 �M4Þkr1/1zk

2 þ ð1þM1 �M4Þkr1hk2

� 1þM04�M01ð ÞLe kr1ck2

26664

37775 ¼ 0:

Since we have assumed that the energy consumed due to solute concentration and rotation is less than the energy produceddue to velocity and temperature. So under the assumption, quantity within the bracket is positive definite.

Thus ri = 0 and r 2 R. So Principle of Exchange of Stabilities holds.

References

[1] O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criteria, Phil. Trans. R. Soc. A 186 (1895) 123–164.[2] W.McF. Orr, Stability or instability of the steady motions of perfect liquid, Proc. Roy. Irish Acad. A. 37 (1907) 9–68.[3] J. Serrin, On the stability of viscous fluid motions, Arch. Rat. Mech. Anal. 3 (1959) 1–13.[4] D.D. Joseph, On the stability of the Boussinesq equations, Arch. Rat. Mech. Anal. 20 (1965) 59–71.[5] D.D. Joseph, Nonlinear stability of the Boussinesq equations by the method of energy, Arch. Rat. Mech. Anal. 22 (1966) 163–184.[6] D.D. Joseph, Sability of Fluid Motions, vol. I–II, Springer, Berlin, 1976.[7] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer Verlag, New York, 2004.[8] J. Guo, Y. Qin, P.N. Kaloni, Nonlinear stability problem of a rotating doubly diffusive fluid layer, Int. J. Eng. Sci. 32 (1994) 1207–1219.[9] P.N. Kaloni, Z. Qiao, Nonlinear stability of convection in a porous medium with inclined temperature gradient, Int. J. Heat Mass Trans. 40 (1997) 1611–

1615.[10] B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin, 1998.[11] L.E. Payne, B. Straughan, Unconditional nonlinear stability in temperature-dependent diffusive layer, Stud. Appl. Math. 105 (2000) 59–81.[12] B. Straughan, A sharp nonlinear stability threshold in rotating porous convection, Proc. Roy. Soc. Lond. A 457 (2001) 87–93.[13] P.N. Kaloni, Z. Qiao, Non-linear stability of convection in a porous medium with inclined temperature gradient and variable gravity effects, Int. J. Heat

Mass Trans. 44 (2001) 1585–1591.[14] P.N. Kaloni, J.X. Lou, Weakly nonlinear instability of a ferromagnetic fluid rotating about a vertical axis, J. Magn. Magn. Mater. 284 (2004) 54–68.[15] B. Straughan, A note on convection with nonlinear heat flux, Ric. Mat. 56 (2007) 229–239.[16] A.A. Hill, M. Carr, Sharp global nonlinear stability for a fluid overlying a highly porous material, Proc. Roy. Soc. A 466 (2010) 127–140.[17] A.A. Hill, B. Straughan, Global stability of thermal convection in a fluid overlying a highly porous material, Proc. Roy. Soc. A 465 (2009) 207–217.[18] R.E. Rosenweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985.[19] S. Odenbach, Magnetoviscous Effects in Ferrofluids, Springer Verlag, Berlin, 2002.[20] B.A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40 (1970) 753–767.[21] D.P. Lalas, S. Carmi, Thermoconvective stability of ferrofluids, Phys. Fluids 14 (2) (1971) 436–437.[22] M.D. Gupta, A.S. Gupta, Convective instability of a layer of ferromagnetic fluid rotating about a vertical axis, Int. J. Eng. Sci. 17 (1979) 271–277.[23] L. Schwab, U. Hildbrandt, K. Stierstadt, Magnetic Bénard convection, J. Magn. Magn. Mater. 39 (1–2) (1983) 113–114.[24] P.J. Stiles, M.J. Kagan, Thermoconvective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field, J. Magn. Magn. Mater. 85 (1990)

196–198.[25] P. Blennerhasset, F. Lin, P. Stiles, Heat transfer through strongly magnetized ferrofluids, Proc. Roy. Soc. Lond. A 433 (1991) 165–177.[26] S. Venkatasubramanian, P.N. Kaloni, Effects of rotation on thermoconvective instability of horizontal layer of ferrofluids, Int. J. Eng. Sci. 32 (1994) 237–

256.[27] M.I. Shilomis, Convective instability of magnetized ferrofluids, J. Magn. Magn. Mater. 252 (2002) 197–202.[28] Sunil, D. Sharma, V. Sharma, Effect of dust particles on rotating ferromagnetic fluid heated from below saturating a porous medium, J. Coll. Int. Sci. 291

(2005) 152–161.[29] D. Laroze, J. Martinez-Mardones, J. Bragard, P. Vargas, Convection in rotating binary ferrofluid, Phys. A: Statis. Mech. Appl. 371 (1) (2006) 46–49.[30] Sunil, A. Sharma, D. Sharma, P. Kumar, Effect of magnetic field dependent viscosity on thermal convection in ferromagnetic fluid, Chem. Eng. Commun.

195 (2008) 571–583.[31] G. Veronis, Effect of a stabilizing gradient of solute on thermal convection, J. Fluid Mech. 34 (1968) 315–336.[32] D.D. Joseph, Global stability of the conduction-diffusion solution, Arch. Rat. Mech. Anal. 36 (1970) 285–292.[33] E.H. Huppert, J.S. Turner, Double diffusive convection, J. Fluid Mech. 106 (1981) 299–329.[34] R.W. Griffiths, Layered double-diffusive convection in porous media, J. Fluid Mech. 102 (1981) 221–248.[35] N. Rudraiah, I.S. Shiva kumara, R. Friedrich, The effect of rotation on linear and nonlinear double-diffusive convection in a sparsely packed porous

medium, Int. J. Heat Mass Trans. 29 (9) (1986) 1301–1307.[36] J. Guo, P.N. Kaloni, Double-diffusive convection in a porous medium, nonlinear stability and Brinkman effect, Stud. Appl. Math. 94 (1995) 341–358.

Sunil et al. / Applied Mathematics and Computation 218 (2011) 2785–2799 2799

[37] Sunil, D. Sharma, R.C. Sharma, Effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium, J. Geophy. Eng. 1(2) (2004) 116–127.

[38] A.A. Hill, Double-diffusive convection in a porous medium with a concentration based internal heat source, Proc. Roy. Soc. A. 461 (2005) 561–574.[39] Sunil, A. Sharma, R.C. Sharma, Effect of dust particles on ferrofluid heated and soluted from below, Int. J. Therm. Sci. 45 (4) (2006) 347–358.[40] F. Zhao, D. Liu, G. Tang, Natural convection in an enclosure with localized heating and salting from below, Int. J. Heat Mass Trans. 51 (2008) 2889–2904.[41] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York, 1981.[42] A. Ryskin, H.-W. Müller, H. Pleiner, Thermodiffusion effects in convection of ferrofluids, Magnetohydrodynamics 39 (2003) 51–55.[43] A. Ryskin, H.-W. Müller, H. Pleiner, Thermal convection in binary fluid mixtures with week concentration diffusivity, but strong solutal buoyancy

forces, Phys. Rev. E 67 (2003) 046302.[44] A. Ryskin, H.-W. Müller, H. Pleiner, Influence of a magnetic field on the Soret-effect- dominated thermal convection in fluids, Phys. Rev. E 69 (2004)

046301.[45] S.N. Gaikwad, M.S. Malashetty, K. Rama Prasad, An analytic study of linear and nonlinear double-diffusive convection in a fluid saturated anisotropic

porous layer with Soret effect, Appl. Math. Model. 33 (2009) 3617–3635.[46] Sunil, A. Mahajan, A nonlinear stability analysis for magnetized ferrofluid heated from below, Proc. Roy. Soc. Lond. A 464 (2008) 83–92.[47] Sunil, A. Mahajan, A nonlinear stability analysis for rotating magnetized ferrofluid heated from below, Appl. Math. Comput. 204 (2008) 299–310.[48] P. Kloden, R. Wells, An explicit example of hopf bifurcation in fluid mechanics, Proc. Roy. Soc. Lond. A 390 (1983) 293–320.[49] G.P. Galdi, B. Straughan, A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem, Proc. Roy. Soc. Lond. A 402 (1985) 257–283.[50] R. Adams, Sobolev Spaces, Academic Press, New York, 1975.[51] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1994.