PHYSICAL REVIEW E 84, 056318 (2011)

Sharp nonlinear stability for centrifugal filtration convection in magnetizable media

S. Saravanan* and D. BrindhaDepartment of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu, India

(Received 10 May 2011; revised manuscript received 15 October 2011; published 21 November 2011)

A nonlinear stability theory is adopted to study centrifugal thermal convection in a magnetic-fluid-saturatedand differentially heated porous layer placed in a zero-gravity environment. The axis of rotation of the layeris placed within its boundaries that leads to an alternating direction of the centrifugal body force. An analysisthrough the variational principles is made to find the unconditional and sharp nonlinear limits. The compoundmatrix method is employed to solve the eigenvalue problems of the nonlinear and corresponding linear theories.The importance of nonlinear theory is established by demonstrating the failure of the linear theory in capturingthe physics of the onset of convection.

DOI: 10.1103/PhysRevE.84.056318 PACS number(s): 47.20.Qr, 47.54.Bd, 47.55.pb, 47.65.Cb

I. INTRODUCTION

The amazing combination of a normal fluid behavior andthe sensitivity for a magnetic field made magnetic fluids fas-cinating soon after their synthesis by Rosensweig and Kaiser[1]. Following a series of pioneering works by Rosensweigand his co-workers, which was later included in his book[2], a systematic scientific foundation for further researchin magnetic fluids took place. Magnetic fluids are colloidaldispersions of nano-sized single-domain magnetic particlesstabilized by surfactants in an electrically nonconducting car-rier liquid such as water, hydrocarbons, mineral oils, or siliconeliquids. Thermal agitation due to Brownian motion keeps theparticles suspended and a much thicker coat of surfactantprevents their agglomeration so that no external force canseparate the particles from the carrier liquid. Once they areexposed to an external magnetic field they become magnetized(magnetically polarized) to a very large value comparable withthe magnetization of solid ferromagnets. Since magnetizationis a function of temperature, it induces an internal field gradientin the presence of temperature inhomogeneities, which inturn interacts with the magnetization and produces magneticbody force. This enables the use of magnetic fields to controlthe flow of the fluid, giving rise to a great variety of newphenomena and numerous technical applications (see Ref. [2]).In this way the continuum description of magnetic-fluid flow,termed ferrohydrodynamics, differs from magnetohydrody-namics in terms of lack current flow and from electrohy-drodynamics in terms of the absence of the free flow ofcharges.

The problem of studying convective instability in a flatlayer of ordinary fluid that is heated from below and cooledfrom above in the presence of gravity is more than a centuryold. Unlike this, the convective instability in magnetic fluidsin the presence of a magnetic field is affected not onlyby the perturbations of the temperature field but also bythose of the magnetic field inside the fluid. Finlayson [3]investigated this in the presence of a uniform vertical magneticfield and found that the magnetic mechanism compensatesfor the effect of buoyancy and predominates it in very thin(≈1 m−3) layers which were later confirmed experimentally.

*sshravan@lycos.com

Following this, several other works were done on convectiveinstabilities occurring in magnetic fluids when both buoyancyand magnetic mechanisms are operative. The same problemin a magnetic-fluid-saturated porous medium of very largepermeability was studied by Vaidyanathan et al. [4]. Qin andChadam [5] performed a nonlinear analysis to study convectionin a magnetic-fluid-filled porous medium by including inertiaeffects to accommodate a high velocity flow. Their emphasiswas on controlling the nonlinear term caused by the magneticbody force. Recently, Sunil and Mahajan [6] made a nonlinearstability analysis in a magnetic-fluid layer heated from belowand demonstrated the coupling between the buoyancy andmagnetic mechanisms.

In rotating fluid systems, thermal buoyancy can alsobe driven by the resulting centrifugal acceleration in thedirection perpendicular to the axis of rotation and modifiesthe distribution of temperature field. This type of convec-tion, referred to as centrifugal convection, dominates thegravitational one in rapidly rotating containers or in low tomoderately rotating zero-gravity flows. It has received lessattention than its gravitational counterpart, though there arepromising applications. The initial works of Vadasz [7,8]dealt with the onset of centrifugal convection in a Darcianporous medium. The conditions imposed at the boundarieswere such that the resulting temperature gradient was collinearwith the centrifugal body force. These studies accounted forthe effect of the centrifugal body force while neglectinggravity. The same problem in a magnetic-fluid-filled porousmedium in the presence of a magnetic field was then studiedby Saravanan and Yamaguchi [9] and it was found that themagnetic field can be suitably adjusted depending on particlemagnetization to enhance convection. Recently Om et al. [10]analyzed the effect of rotation speed modulation on the onset ofcentrifugal convection and found that by applying modulationof a proper frequency to the rotation speed, it is possibleto delay or advance the onset of centrifugal convection. Allthese works were carried out in the framework of lineartheory.

To have a better understanding of the problem, one hasto perform a nonlinear analysis which provides a thresholdfor stability. In principle, stability should take into accountthe growth of all disturbances which are physically possiblerather restricting purely to infinitesimal ones. The main feature

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S. SARAVANAN AND D. BRINDHA PHYSICAL REVIEW E 84, 056318 (2011)

of the nonlinear analysis is that it provides rigorous suffi-cient conditions for stability without exploiting linearization,expansion, weakly nonlinear approximation, etc. Earlier wefound unconditional nonlinear limits for centrifugal convec-tion in an ordinary fluid-saturated porous layer [11]. We shallnow extend it to a magnetic-fluid-saturated porous mediumin the presence of a magnetic field. For this we shall employthe energy method which was initiated in its modern formby Serrin [12], developed by Joseph [13] and employedextensively by Straughan [14]. It does not impose any initialconditions on the unknown disturbances. It was believed in thepast that the classical energy method leads to uncertain resultsdue to incomparable and very low stability bounds. Later itwas established that by introducing a coupling parameter inthe energy method and by selecting it optimally, it is possibleto sharpen the stability bound in many physical problems (seeRefs. [13,14]).

II. PROBLEM FORMULATION

We consider an infinitely tall vertical porous layer −L/2 <

x < L/2 subject to a constant rotation rate ω about a verticalaxis in a zero-gravity environment (see Fig. 1). The porouslayer is saturated with a magnetic fluid. The layer is heated onits right boundary (Th) and cooled on the left boundary (Tc),and as a result of these imposed thermal boundary conditionsa uniform temperature gradient β is acting across the layer.This arrangement makes the centrifugal acceleration collinearwith the temperature gradient. A constant uniform externalmagnetic field H0 is applied parallel to the x axis.

x = 0

x = x0

L

W

y y'

H0

z z'

x = L/2

x

x'

FIG. 1. Physical configuration.

The axis of rotation is placed within the boundaries ofthe porous domain and at a dimensionless distance x0 fromthe center of the layer. The offset distance is presentedin a dimensionless form representing the ratio between thedimensional offset distance and the thickness of the porouslayer in the form x0 = x∗

0/L. Free convection occurs as a resultof the centrifugal body force. The Coriolis effect is consideredsmall, because most of the inertial forces are neglected whenusing Darcy’s law. The only inertial effect considered is thecentrifugal acceleration, as far as the changes in density areconcerned. These assumptions are compatible with Darcy’slaw, which is applicable as long as the Reynolds number isnot too large. The thickness of the layer W is assumed toobey W � L, which enables us to neglect the y componentof the centrifugal acceleration and hence to use a Cartesiancoordinate system. The fluid is assumed incompressible andthe Boussinesq approximation is employed to account for theeffects of the density variations.

For this configuration, the conservation equations in therotating frame of reference take the form (see Refs. [2,15])

∇ · v = 0, (1a)μ

Kp

v = −∇p + μ0M∇H + ρω2(x − x0)i, (1b)

ε

[ρCV,H − μ0H

(∂M

∂T

)V,H

]DT

Dt+ (1 − ε)(ρC)S

∂T

∂t

(1c)

+μ0T

(∂M

∂T

)V,H

DH

Dt= κ1∇2T ,

where μv/Kp denotes the Darcy resistance and μ0M∇H

represents the magnetic force density. Here ε is the porosity,v the filtration velocity which is related to the intrinsic fluidvelocity q by the Dupuit-Forchheimer relationship v = εq,μ the dynamic viscosity of the fluid, Kp the permeabilityof porous medium, p the pressure, ρ the fluid density, i theunit vector in the x direction, t the time, T the temperature,κ1 the thermal conductivity of the fluid, C the specific heatcapacity, CV,H the specific heat capacity at constant volumeand magnetic field, and the subscript S refers to the solid matrixof the porous medium. One should note that we dropped theunsteady inertial component in Eq. (1b) because all transientsdie down rapidly while using Darcy’s law.

Since the applied magnetic field is static and the magneticfluid is not electrically conducting, the electric field vectorvanishes. The appropriate magnetostatic limit of Maxwell’sequations are

∇ · B = 0,(2)∇ × H = 0,

where B is the magnetic induction. In the absence of a magneticfield, the constituent magnetic particles are randomly orientedwith no net magnetization. In the presence of an externalmagnetic field, the magnetic fluid becomes magnetized withintensity M given by

B = μ0(H + M), (3)

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SHARP NONLINEAR STABILITY FOR CENTRIFUGAL . . . PHYSICAL REVIEW E 84, 056318 (2011)

where μ0 is the magnetic permeability of free space. Therelaxation time of the magnetization is assumed to be sosmall that its dynamics can be disregarded in the analysis ofhydrodynamic phenomena. Thus the magnetization is alwaysaligned with the magnetic field and depends on the magnitudeof the magnetic field and temperature:

M = M(H,T )

HH. (4)

The density and magnetic equations of state are linearizedabout the external magnetic field strength H0 and an averagetemperature T0 = (Th + Tc)/2 as

ρ(T ) = ρ0[1 − α(T − T0)], (5a)

M(H,T ) = M0 + χ (H − H0) − K(T − T0), (5b)

where ρ0 is the reference density, α the coefficient of thermalexpansion of the fluid, χ = ( ∂M

∂H)T |H0,T0 the magnetic sus-

ceptibility, K = −( ∂M∂T

)H |H0,T0 the pyromagnetic coefficient,and M0 = M(H0,T0). This problem becomes well posed oncewe append the following velocity and magnetic boundaryconditions: The normal component of the filtration velocityvanishes at the boundaries. The tangential components ofthe magnetic field and normal component of the magneticinduction are continuous across the boundaries.

We seek a quiescent basic state of the form T = Tb(x),p = pb(x), H = Hb(x), and M = Mb(x). Under this setup

the governing equations (1) admit a basic state describedby

vb = (0,0,0), Tb(x) = βx + T0,

Hb =(

H0 + Kβx

1 + χ

)i, Mb =

(M0 − Kβx

1 + χ

)i, (6)

pb = p0 − ρ0ω2x

(x

2− x0

)+ ρ0ω

2αβx2(x

3− x0

2

)+ μ0Kβx

1 + χ

[M0 − Kβx

2(1 + χ )

],

where β = (Th − Tc)/L.To assess the stability of this steady solution, a perturbation

(v′,p′,T ′,H′,M′) is introduced such that

(v,p,T ,H,M) = (vb,pb,Tb,Hb,Mb) + (v′,p′,T ′,H′,M′).(7)

Substituting Eq. (6) into Eqs. (4) and (5b) and using Eq. (3),we arrive at

H ′1 + M ′

1 = (1 + χ )H ′1 − KT ′

1,

H ′i + M ′

i =(

1 + M0

H0

)H ′

i , i = 2,3,

under the assumption KβL � (1 + χ )H0. Substituting Eq. (7)into (1) and noting that H′ = ∇φ′, where φ′ is the perturbedmagnetic potential, we obtain the nonlinear perturbed equa-tions as

∇ · v = 0,

μ

Kp

v = −∇p − ρ0 αω2 (x − x0)T i + μ0Kβ

1 + χ

[(1 + χ )

∂φ

∂xi − KT i

]

+μ0χ∂φ

∂x∇ ∂φ

∂x− μ0KT ∇ ∂φ

∂x+ μ0M0

H0

[∂φ

∂y∇ ∂φ

∂y+ ∂φ

∂z∇ ∂φ

∂z

],

(ρC)1∂T

∂t+ (ρC)2

[vx

∂T

∂x+ vz

∂T

∂z

]− μ0T0K

[vx

∂2φ

∂x2+ vz

∂2φ

∂z∂x+ ∂2φ

∂t∂x

]= κ1∇2T +

[μ0T0K

2β

1 + χ− (ρC)2β

]vx, (8)

(1 + M0

H0

)∂2φ

∂z2+ (1 + χ )

∂2φ

∂x2− K

∂T

∂x= 0,

where the primes have been omitted for convenience. Here(ρC)1 = ερ0CV,H + εμ0H0K + (1 − ε)ρSCS and (ρC)2 =ερ0CV,H + εμ0H0K . The above equations are supplementedwith the conditions corresponding to impermeable, isothermal,and isomagnetic potential boundaries:

vx = T = φ = 0, on x = ± 12 .

We also assume that vx , T , and φ satisfy a periodic shape inthe (y,z) plane and the ensuing convective cell will be denotedby �.

III. LINEAR INSTABILITY ANALYSIS

The linear analysis approach assumes that the disturbanceis small and so neglects terms of quadratic and higherorder. Because it has already been discussed in detail bySaravanan and Yamaguchi [9], we will describe it only brieflyhere.

In the rest of this paper we use the symbol D to denotethe operator d/dx. Eliminating the pressure, introducing thescales for the dimensional physical quantities, and takingthe normal mode expansion vx(x,z) = U (x)eσ t+ikz, T (x,z) = (x)eσ t+ikz, φ(x,z) = �(x)eσ t+ikz the linearized perturbed

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S. SARAVANAN AND D. BRINDHA PHYSICAL REVIEW E 84, 056318 (2011)

system reduces to

(D2 − k2)U − [Rm + Rc (x − x0)]k2 + Rmk2D� = 0,

(9a)

[(D2 − k2) − Prσ ] + (NM2 − 1)U + PrM2σD� = 0,

(9b)

(D2 − M1k2)� − D = 0, (9c)

where Rc = (ρC)2αβω2KpL3

νκ1is the centrifugal Rayleigh number,

Pr = (ρC)1ν/κ1 the Prandtl number, Rm = (ρC)2μ0K2β2KpL2

κ1μ(1+χ )

the magnetic Rayleigh number, M1 = 1+M0/H0

(1+χ ) the nonlinear-

ity in magnetization, M2 = μ0T0K2

(ρC)1(1+χ) , and N = (ρC)1/(ρC)2.

Since M2 is of order 10−6 [3], it is neglected in the subsequentanalysis. Here the growth rate σ can be shown to be realas follows. We combine the first two equations in Eq. (9)by eliminating U , multiply the resulting equation by ∗, thecomplex conjugate of , and then integrate it over �. Thisresults in, after using the boundary conditions,

‖D2 ‖2 + (2k2 + Prσ )‖D ‖2

= k2(Rm − k2 − Prσ )‖ ‖2 − Rmk2∫

�

∗D�d�

+ Rck2∫

�

(x − x0)| |2d�.

Here∫�

(·)d� denotes the integration over � and ‖ · ‖ denotesthe L2(�) norm.

Multiplying Eq. (9c) by �∗, integrating over � and addingwith the above equation we have

Prσ [‖D ‖2 + k2‖ ‖2]

= −‖D2 ‖2 − 2k2‖D ‖2 + k2(Rm − k2)‖ ‖2

+ Rmk2‖D�‖2 + Rck2∫

�

(x − x0)| |2d�

+ Rmk4M1‖�‖2 − Rmk2∫

�

[ ∗D� + D�∗]d�,

whose imaginary part yields

Pr Im(σ )[‖D ‖2 + k2‖ ‖2] = 0.

This implies Im(σ ) = 0 and hence the stationary mode ofconvection alone has been analyzed at the marginal state whichis governed by Eq. (9) with M2 = σ = 0. Let us denote thelowest eigenvalue of this system with the appropriate boundaryconditions by RcL. The critical linear centrifugal Rayleighnumber of the linear instability theory is then defined as

RcL,cr = mink

RcL(k). (10)

IV. ENERGY ANALYSIS

The exponential growth of unstable disturbances presumedin Sec. III may not give a satisfactory description of thebehavior of evolution of the disturbance over a period of time.On the contrary, the energy analysis which plays an importantrole in the nonlinear stability of fluid motions is based on thestudy of time evolution of the disturbance energy itself andleads to a threshold below which the energy decays to zeroglobally and monotonically.

To obtain the nonlinear stability bound, we nondimension-alize the perturbed system of Eq. (8) using the known scales(see [9]) together with μν/Kp for pressure as

∇ · v = 0,

v = − 1

N∇P − [Rm + Rc (x − x0)]T i + Rm

∂φ

∂xi

− RmPrT ∇ ∂φ

∂x+Rm Pr

1 + χ

[χ

∂φ

∂x∇ ∂φ

∂x−M0

H0

∂φ

∂z∇ ∂φ

∂z

],

(11)∂T

∂t= 1

Pr(NM2 − 1)vx −

[vx

∂T

∂x+ vz

∂T

∂z

]+ 1

Pr∇2T

+M2

[Nvx

∂2φ

∂x2+ Nvz

∂2φ

∂z∂x+ ∂2φ

∂t∂x

], (12)

M1∂2φ

∂z2+ ∂2φ

∂x2− ∂T

∂x= 0. (13)

Adopting the standard nonlinear energy approach in thestability measure L2(�), we multiply Eqs. (11)–(13) by v,T , and φ, respectively, and integrate over � to obtain

‖v‖2 = −∫

�

[Rm + Rc (x − x0)]T vx d�

+∫

�

Rm∂φ

∂xvxd�, (14)

Pr

2

d‖T ‖2

dt= (NM2 − 1)

∫�

T vxd� − ‖∇T ‖2 + M2Pr∫

�

T

×[Nvx

∂2φ

∂x2+ Nvz

∂2φ

∂z∂x+ ∂2φ

∂t∂x

]d�, (15)

M1

∫�

∣∣∣∣∂φ

∂z

∣∣∣∣2

d� +∫

�

∣∣∣∣∂φ

∂x

∣∣∣∣2

d� +∫

�

φ∂T

∂xd� = 0. (16)

To study the nonlinear stability of the basic state, a kineticlike energy E(t) = λ1Pr

2 ‖T (t)‖2 is constructed using Eqs. (14)–(16) and its evolution is given by

dE

dt= −D + I, (17)

where

I = −λ2

∫�

φ∂T

∂xd� −

∫�

[Rm + Rc (x − x0)]T vx d�

+∫

�

Rm∂φ

∂xvx d� + (NM2 − 1)λ1

∫�

T vx d�

+M2λ1Pr∫

�

T

[Nvx

∂2φ

∂x2+ Nvz

∂2φ

∂z∂x+ ∂2φ

∂t∂x

]d�,

D = ‖v‖2 + M1λ2

∫�

∣∣∣∣∂φ

∂z

∣∣∣∣2

d� + λ2

∫�

∣∣∣∣∂φ

∂x

∣∣∣∣2

d�

+ λ1‖∇T ‖2,

with positive coupling parameters λ1 and λ2.The idea is to optimize an inequality involving the right-

hand side of Eq. (17). Hence we define RcN by

1

RcN

= maxH

ID , (18)

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SHARP NONLINEAR STABILITY FOR CENTRIFUGAL . . . PHYSICAL REVIEW E 84, 056318 (2011)

where H is the space of admissible functions over whichwe seek a maximum. This equation leads to the followingrelation:

dE

dt� RcD

(maxH

ID

)− D

= −D(

RcN − 1

RcN

).

Hence for RcN > 1 it follows, with the aid of Poincare’sinequality, that E → 0 as t → ∞ at least exponentially intime. We notice that the decay of E is for all initial disturbances

regardless of how large they may be and hence the nonlinearstability is unconditional. The nonlinear stability threshold isnow given by the variational problem (18) with RcN = 1.

The approach with nonlinear energy stability calculations isto find a variational problem like Eq. (18), determine the Euler-Lagrange equations, and maximize the coupling parameters toobtain the best value of RcN . The Euler-Lagrange equationsfor Eq. (18) are determined from

δI − δD = 0,

where

δI = −λ2

∫�

d

dε

[(φ + εkb)

∂

∂x(T + εka)

]ε=0

d� −∫

�

d

dε{[Rm + Rc (x − x0)](T + εka)(vx + εhx)}ε=0 d�

+∫

�

Rmd

dε

[∂

∂x(φ + εkb)(vx + εhx)

]ε=0

d� + (NM2 − 1)λ1

∫�

d

dε[(T + εka)(vx + εhx)]ε=0 d�

+M2λ1Pr∫

�

d

dε

{(T + εka)

[N (vx + εhx)

(∂2φ

∂x2+ ε

∂2kb

∂x2

)+ N (vz + εhz)

(∂2φ

∂z∂x+ ε

∂2kb

∂z∂x

)

+(

∂2φ

∂t∂x+ ε

∂2kb

∂t∂x

) ]}ε=0

d�

= ka

{λ2

∫�

∂φ

∂xd� −

∫�

[Rm + Rc (x − x0) − (NM2 − 1)λ1]vx d�

+M2λ1Pr∫

�

[Nvx

∂2φ

∂x2+ Nvz

∂2φ

∂z∂x+ ∂2φ

∂t∂x

]d�

}+ h ·

{−

∫�

[Rm + Rc (x − x0) − (NM2 − 1)λ1] T i d�

+ Rm∫

�

∂φ

∂xi d� + NM2λ1Pr

∫�

T

[∂2φ

∂x2i + ∂2φ

∂z∂xk]

d�

}+ kb

{− λ2

∫�

∂T

∂xd� − Rm

∫�

∂vx

∂xd�

+M2λ1Pr∫

�

∂2T

∂x∂td�

},

and

δD =∥∥∥∥ d

dε[(v + εh)2]ε=0

∥∥∥∥ + M1λ2

∫�

d

dε

[∂φ

∂z+ ε

∂kb

∂z

]2

ε=0

d�

+ λ2

∫�

d

dε

[∂φ

∂x+ ε

∂kb

∂x

]2

ε=0

d� + λ1

∥∥∥∥ d

dε

[(∇T + ε∇ka)2

]ε=0

∥∥∥∥= ka

{−2λ1

∫�

∇2T d�

}+ 2h ·

∫�

v d� + kb

{−2λ2

(M1

∫�

∂2φ

∂z2d� +

∫�

∂2φ

∂x2d�

)}.

This with M2 = 0 leads to the following Euler-Lagrange equations:

∇ · v = 0,

2λ1∇2T − [λ1 + Rm + Rc (x − x0)]vx + λ2∂φ

∂x= 0,

−[λ1 + Rm + Rc (x − x0)]T i − 2v + Rm∂φ

∂xi = 2∇π, (19)

2λ2

[M1

∂2φ

∂z2+ ∂2φ

∂x2

]− λ2

∂T

∂x− Rm

∂vx

∂x= 0,

accompanied by the appropriate boundary conditions where π is a Lagrangian multiplier.

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S. SARAVANAN AND D. BRINDHA PHYSICAL REVIEW E 84, 056318 (2011)

To actually find the sharp limits, we solve Eq. (19) for RcN . After removing π by operating curl, the Eqs. (19) may then bereduced to

D2U = k2U + 1

2[λ1 + Rm + Rc (x − x0)] k2 − 1

2Rm k2 D�,

D2 = k2 + 1

2λ1[λ1 + Rm + Rc (x − x0)]U − λ2

2λ1D�, (20)

D2� = M1k2� + 1

2D + 1

2λ2Rm DU,

where vx(x,z) = U (x)eikz, T (x,z) = (x)eikz, φ(x,z) =�(x)eikz in which k is the wave number. The relevant boundaryconditions are

U = = � = 0 at x = ±1/2. (21)

Equations (20) and (21) constitute an eigenvalue system ofequations for the nonlinear centrifugal Rayleigh number. Thecoupling parameters λ1 and λ2 must be chosen in such a way tomake RcN as large as possible. Hence we calculate the criticalnonlinear centrifugal Rayleigh number by the optimization

RcN,cr = maxλ1,λ2>0

mink

RcN (k,λ1,λ2). (22)

V. DISCUSSION AND CONCLUSION

The critical centrifugal Rayleigh numbers of both linearand nonlinear analyses are found via the compound matrixmethod (see the Appendix) which is superior to other methodsof its kind [16]. Accordingly, eigenvalue problems for ordinarydifferential equations are usually treated by first defininga solution matrix which satisfies certain prescribed initialconditions, and the required eigenvalues are then obtained asthe roots of some minor of the solution matrix. If we attempt toevaluate this minor by computing its elements separately, as ina standard shooting method, then there may be a serious loss ofnumerical accuracy especially when the differential equationis stiff. This difficulty can be avoided, however, by consideringthe differential equation satisfied by a certain compound matrixwhose elements are the minors of the solution matrix, and inthis way we can compute the required minor directly.

The results are presented in terms of critical centrifugalRayleigh number and critical wave number for differentlocations of the axis of rotation. Before presenting thenonlinear limits we would like to check the predictions madeby the compound matrix method. Hence in Table I we first

TABLE I. Comparison of present results with those of Ref. [9]

X0 Rm M1 Rc L,cr [9] k L,cr [9] Rc L,cr k L,cr

0.5 0 80

0.25 40 80

0.0 20 20

0.15 30 50

77.0799 3.2177.0828 [8] 3.20 [8]

3276.7094 8.99

338.3656 4.17

1102.3235 6.26

77.0791 3.20

3276.7488 8.98

338.3690 4.16

1102.3290 6.25

make a comparison of linear limits of Sec. III obtained bythe method with the available result [9], which is based onthe higher-order Galerkin variant of the method of weightedresidual. An excellent agreement between the two obviouslyshows that our results are accurate enough.

Figure 2 displays the linear and nonlinear marginal curveswhen the axis of rotation is placed at the center of the porouslayer (i.e., x0 = 0). These curves clearly divide the Rc-k planeinto two regions, with the region lying below representingdiffusion state and the region above indicating convectivemotion. Each of these curves has a single minimum calledthe critical centrifugal Rayleigh number Rccr accompanied bya corresponding critical wave number. It is observed that as Rmincreases, the marginal curves are displaced downward indicat-ing the destabilizing character of the magnetic mechanism onthe basic state, in contrast to that in magnetohydrodynamics.

Rccr as a function of Rm is shown in Fig. 3 for x0 = −0.5and 0 when the induced magnetization exhibits the Langevinbehavior (M1 = 80) corresponding to ferromagnetic particles.One important result is a good coincidence of both linearand nonlinear limits implying a rigorous stability bound forx0 = −0.5, i.e., for a porous layer with its axis of rotation onthe cold wall. This seems to indicate that the linear analysisitself captures the physics of the onset and we can have goodfaith in its prediction. This result also demonstrates that thecoupling parameters can be optimized to make the nonlinearlimit sharp enough. Nevertheless it starts deviating from thelinear instability one as the axis of rotation moves towardthe hot wall and delimits the region of subcritical bifurcations.This is natural because the linear theory guarantees stability for

FIG. 2. Marginal curves of Rc for different values of Rm.

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SHARP NONLINEAR STABILITY FOR CENTRIFUGAL . . . PHYSICAL REVIEW E 84, 056318 (2011)

FIG. 3. Rccr against Rm for x0 = −0.5 and x0 = 0.0 when M1 =80.

laminar flows with Rc < RcL,cr only in the conditional sense,for small disturbances. This may be verified by a suitableexperimental setup. Moreover we may infer from Fig. 3 thatthe centrifugal acceleration can also be a physical mechanismwhich can cause the proximity of RcN,cr to RcL,cr such asgravity and thermocapillarity (see Ref. [17]).

Figure 4 shows the situation when x0 = 0.25 and M1 = 1and 80. It is clear that the linear theory over predicts the onsetcriterion, which is in fact quite low. It also shows that M1

destabilizes the base state. It is quite explicit from Figs. 3 and4 that in all the cases, Rm destabilizes the conduction stateand Rccr becomes 0 at some finite Rm, denoted by RmM1 ,beyond which the magnetic mechanism alone is sufficient toinduce convection. It should be noted that RmM1 depends onlyon M1 and not on x0, as expected. The corresponding wavenumbers plotted in Fig. 5 shows that the motion sets in withlarger convective cells for an increase in Rm. It is, however,to be noted that the wave,number remains unaffected whenx0 = −0.5.

Figures 6 and 7 show the influence of x0 on the stabilitybehavior. The results are consistent with those of Saravananand Brindha [11] in the limiting case Rm = 0 and M1 → ∞.One can observe that RcN,cr increases monotonically againstx0 and becomes unbounded as x0 → 0.5 when M1 = 1.

FIG. 4. Rccr against Rm for x0 = 0.25 when M1 = 1 and 80.

FIG. 5. kcr against Rm for different values of x0.

The corresponding secondary state is established with anunbounded wave number. However, RcN,cr remains finite forall values of x0 when the paramagnetic approximation isremoved and convection sets in with finite wave number incontrast to the linear theory (see Ref. [9]).

In conclusion, we were able to derive a rigorous nonlinearstability result for the proposed model by performing energyanalysis. The nonlinear stability threshold predicted the exis-tence of subcritical instability and hence the energy methodproduced practically useful optimal results which cannot bedetermined by the linear theory.

ACKNOWLEDGMENTS

The authors thank UGC, India, for its financial supportthrough the DRS Special Assistance Programme in Fluid Dy-namics. This work was carried out as a part of a research project(No. SR/FTP/MS-02/2007) supported by SERC, Departmentof Science and Technology, India.

APPENDIX

The eigenvalue system of Eqs. (20) and (21) is solvedusing the compound matrix method. Accordingly we define thevector U = (U,U ′, , ′,�,�′)T . Let U1, U2, and U3 be three

FIG. 6. RcN,cr against x0 for different values of Rm and M1.

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S. SARAVANAN AND D. BRINDHA PHYSICAL REVIEW E 84, 056318 (2011)

FIG. 7. kN,cr against x0 for different values of Rm and M1

independent solutions obtained by replacing the boundarycondition by the initial conditions U1(0) = (0,1,0,0,0,0)T ,U2(0) = (0,0,0,1,0,0)T , and U3(0) = (0,0,0,0,0,1)T at x =− 1

2 . We then define 6C3 new variables y1 to y20 as the 3 × 3minors of the 6 × 3 solution matrix whose first, second, andthird columns are U1, U2, and U3, respectively. For example,we define

y1 =∣∣∣∣∣∣U1 U2 U3

U ′1 U ′

2 U ′3

1 2 3

∣∣∣∣∣∣ .The idea is to define y2–y20 similarly and then obtain differen-tial equations for the yi by differentiation. By differentiatingeach yi in turn and simplifying, we arrive at the followingdifferential equations for the yi’s:

y ′1 = y2 + Rmk2

2y7,

y ′2 = 1

2[λ1 + Rm + Rc (x − x0)]k2y5 + Rmk2

2y9

+k2y1 − λ2

2λ1y4,

y ′3 = y4 + 1

2[λ1 + Rm + Rc (x − x0)]k2y6 + Rmk2

2y10,

y ′4 = 1

2[λ1 + Rm + Rc (x − x0)]k2y7 + M1k

2y3 + y2/2,

y ′5 = y11 − λ2

2λ1y7,

y ′6 = y7 + y8 + y12,

y ′7 = y9 + M1k

2y6 + y13 + y5 − Rm

2λ2y1,

y ′8 = y9 + y14 + k2y6 + λ2

2λ1y10,

y ′9 = y15 + k2y7 + M1k

2y8 − Rm

2λ2y2,

y ′10 = y16 − y8 − Rm

2λ2y3,

y ′11 = k2y5 − Rmk2

2y18 + 1

2λ1[λ1 + Rm + Rc (x − x0)]y1

− λ2

2λ1y13,

y ′12 = y13 + y14 + k2y6 − Rmk2

2y19,

y ′13 = y15 + k2y7 + M1k

2y12 + y11

2,

y ′14 = y15 + k2y8 + 1

2[λ1 + Rm + Rc (x − x0)]k2y17

−Rmk2

2y20 + k2y12

− 1

2λ1[λ1 + Rm + Rc (x − x0)]y3 + λ2

2λ1y16, (A1)

y ′15 = k2y9 + 1

2[λ1 + Rm + Rc (x − x0)]k2y18 + k2y13

− 1

2λ1[λ1 + Rm + Rc (x − x0)]y4 + M1k

2y14,

y ′16 = k2y10 + 1

2[λ1 + Rm + Rc (x − x0)]k2y19 − y14

2,

y ′17 = y18 − 1

2λ1[λ1 + Rm + Rc (x − x0)]y6 + λ2

2λ1y19,

y ′18 = − 1

2λ1[λ1+Rm+Rc (x−x0)]y7+M1k

2y17+Rm

2λ2y11,

y ′19 = y20 − y17

2+ Rm

2λ2y12,

y ′20 = k2y19 + 1

2λ1[λ1 + Rm + Rc (x − x0)]y10 + Rm

2λ2y14.

This system was numerically integrated from 0 to 1 subject tothe initial condition

y15

(−1

2

)= 1.

The appropriate final condition which satisfies (23) is found tobe

y6

(1

2

)= 0.

The value of RcN is varied such that the final condition issatisfied to a predefined accuracy.

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