Meccanica (2010) 45: 213–226 DOI 10.1007/s11012-009-9238-7

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium C.E. Nanjundappa · I.S. Shivakumara · M. Ravisha

Received: 19 November 2008 / Accepted: 1 July 2009 / Published online: 22 July 2009 © Springer Science+Business Media B.V. 2009

Abstract The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The BrinkmanLapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in C.E. Nanjundappa () Department of Mathematics, Dr. Ambedkar Institute of Technology, Bangalore 560 056, India e-mail: [email protected] I.S. Shivakumara UGC-Centre for Advanced Studies in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore 560 001, India e-mail: [email protected] M. Ravisha Department of Mathematics, East Point College of Engineering and Technology, Bangalore 560 049, India e-mail: [email protected]

the Biot number Bi, porous parameter σ , viscosity ratio , magnetic susceptibility χ, and decrease in the magnetic number M1 and non-linearity of magnetization M3 is to delay the onset of ferroconvection in a porous medium. Further, increase in M1 , M3 , and decrease in χ, , σ and Bi is to decrease the size of convection cells. Keywords Ferrofluid · Porous medium · Viscosity ratio · Paramagnetic · Magnetic susceptibility · Galerkin method

1 Introduction The ferromagnetic fluids are colloidal suspensions of fine magnetic particles with typical dimensions of about 3–10 nm dispersed in a non-conducting carrier liquids like water, kerosene, ester and hydrocarbons etc. Since 1960’s, when these fluids were initially synthesized, their technological applications have been stepped up over the years. These fluids are found to have numerous applications, for example in loud speakers, rotatory exclusion seals, bearings, dampers, shock absorbers, medicine drug targeting, and in many other thermal transport applications. A detailed introduction to ferrofluids along with their diverse applications is well documented in the books by Berkovsky et al. [1], Rosensweig [2] and Bashtovoy et al. [3].

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When a horizontal ferrofluid layer in the presence of a uniform magnetic field is heated from below, convective motions can take place called ferroconvection, analogous to classical Benard convection. Ferroconvection in a horizontal layer of ferromagnetic fluid has been studied extensively. Finlayson [4] was the first to study the linear stability of ferroconvection in a horizontal layer of ferrofluid heated from below in the presence of a uniform vertical magnetic field studied. Thermoconvective instability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi [5], whereas Shliomis [6] has analyzed the linear relation for magnetized perturbed quantities at the limit of instability. A similar analysis but with the fluid confined between ferromagnetic plates has been carried out by Gotoh and Yamada [7] using linear stability analysis. Schwab et al. [8] have conducted experiments and their results are found to be in good agreement with theoretical predictions. Stiles and Kagan [9] have extended the problem to allow for the dependence of effective shear viscosity on temperature and colloidal concentration. The effect of different forms of basic temperature gradients on the onset of ferroconvection driven by combined surface tension and buoyancy forces has been discussed by Shivakumara et al. [10] with the sole motto of understanding control of ferroconvection. Kaloni and Lou [11] have theoretically investigated the convective instability problem in a thin horizontal layer of magnetic fluid heated from below under alternating magnetic field by considering the quasi stationary model with internal rotation and vortex viscosity. Ganguly et al. [12] have characterized the heat transfer augmentation due to the thermomagnetic convection under the influence of a line dipole and correlated it with the properties of the imposed magnetic field. The influence of magnetic field on heat and mass transport in ferrofluids has been discussed by Volker et al. [13]. The effect of magnetic field–dependent (MFD) viscosity on a layer of ferromagnetic fluid heated from below subject to a transverse uniform magnetic field has been investigated theoretically by Sunil et al. [14]. Sunil and Amit Mahajan [15] have performed nonlinear stability analysis for a magnetized ferrofluid layer heated from below, in the stress-free boundaries. Recently, Nanjundappa and Shivakumara [16] have investigated the effect of velocity and temperature boundary conditions on convective instability in a ferrofluid layer.

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Thermal convection of ferrofluids saturating a porous medium has also attracted considerable attention in the literature owing to its importance in controlled emplacement of liquids or treatment of chemicals, and emplacement of geophysically imageable liquids into particular zones for subsequent imaging etc. Rosensweig et al. [17] have studied experimentally the penetration of ferrofluids in the Heleshaw cell. The stability of the magnetic fluid penetration through a porous medium in high uniform magnetic field oblique to the interface is studied by Zahn and Rosensweig [18]. The thermal convection of a ferrofluid saturating a porous medium in the presence of a vertical magnetic field is studied by Vaidyanathan et al. [19]. Their analysis is limited to free-free boundaries and to the case of effective viscosity equal to fluid viscosity. Qin and Chadam [20] have carried out the non-linear stability analysis of ferroconvection in a porous layer by including the inertial effects to accommodate high velocity. The laboratory–scale experimental results of the behavior of ferrofluids in porous media consisting of sands and sediments are presented by Borglin et al. [21]. The onset of centrifugal convection in a magnetic-fluid- saturated porous medium under zero gravity condition is investigated by Saravanan and Yamaguchi [22]. Recently, Shivakumara et al. [23] have studied thermomagnetic convection in a magnetic nanofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field. However, porous materials used in many technological applications of practical importance possess high permeability values (see Nield et al. [24] and references therein). For a high porosity porous medium Givler and Altobelli [25] have demonstrated that the effective viscosity is about ten times the fluid viscosity. Under the circumstances, a theoretical solution, which is general enough to yield accurate results for ferroconvection in porous media, is of fundamental and practical interest. All the above mentioned studies dealt with isothermal boundary conditions at the surfaces of the ferrofluid layer. However, consideration of actual situations suggests that these conditions may be too restrictive. For instance, if the heating at the lower surface is by passing an electric current through a thin metallic foil, then the appropriate boundary condition would be a fixed heat flux rather than a fixed temperature. Therefore, the aim of the present study is to investigate numerically the condition for the onset

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of ferroconvection in a high permeability ferromagnetic fluid saturated porous layer by employing a nonDarcian model for rigid-rigid paramagnetic and rigidfree paramagnetic boundaries with fixed heat flux and convective-radiative exchange conditions at lower and upper boundaries respectively. To achieve the above objectives, the paper is organized as under. Section 2 is devoted to mathematical formulation. The method of solution is discussed in Sect. 3. In Sect. 4, the numerical results presented are discussed and some important conclusions follow in Sect. 5.

2 Formulation of the problem The physical configuration is as shown in Fig. 1. The system considered is an initially quiescent magnetic fluid saturated horizontal porous layer of characteristic thickness d in the presence of an applied magnetic field H0 in the vertical direction. The horizontal extension of the porous layer is sufficiently larger so that edge effects may be neglected. A Cartesian coordinate system (x, y, z) is used with the origin at the bottom of the porous layer and z-axis is directed vertically upward. Gravity acts in the negative z-direction, ˆ where kˆ is the unit vector in the z-direction. g = −g k, We assume that the fluid is incompressible and the fluid density variation based on Boussinesq approximation can be expressed as ρ = ρ0 [1 − αt (T − T0 )]

(1)

where T is the temperature, αt is the co-efficient of thermal expansion and ρ0 is the density at the reference temperature T0 . At the lower boundary z = 0 a constant heat flux condition of the form −k1

∂T = q0 ∂z

(2)

Fig. 1 Physical configuration

is used, while at the upper boundary z = d a radiativetype of condition of the form −k1

∂T = ht (T − T∞ ) ∂z

(3)

is invoked. In the above equations, q0 is the conductive thermal flux, k1 is the overall thermal conductivity, ht is the heat transfer coefficient and T∞ is the temperature in the bulk of the environment. The governing equations are [2, 4]: ∇ · q = 0   μf 1 ∂ q 1 ρ0 + 2 ( q q · ∇) q = −∇p + ρ g − ε ∂t k ε

(4)

2 + μ˜ f ∇ q + ∇ · (B H )

(5)    DT ∂M ε ρ0 CV ,H − μ0 H · ∂T V ,H Dt   ∂T ∂M D H + μ0 T · + (1 − ε)(ρ0 C)S ∂t ∂T V ,H Dt 



= k1 ∇ 2 T ∇ · B = 0,

(6) ∇ × H = 0

or H = ∇ϕ

(7a, b)

 + H ) B = μ0 (M

(8)

 = M (H, T )H M H

(9)

M = M0 + χ(H − H0 ) − K(T − T0 ),

(10)

where q = (u, v, w) is the velocity vector, p is the pressure, H = (Hx , Hy , Hz ) is the magnetic field intensity, B = (Bx , By , Bz ) is the magnetic induction,  = (Mx , My , Mz ) is the magnetization, μf is the M dynamic viscosity, μ˜ f is the effective viscosity, k is the permeability of the porous medium. ε is the porosity of the porous medium, C the specific heat, CV ,H is the specific heat at constant volume and magnetic field, μ0 is the magnetic permeability of vacuum. Further, M0 is the reference magnetization at H = H0 and T = T0 , χ = (∂M/∂H )H0 ,T0 is the magnetic susceptibility, K = −(∂M/∂T )H0 ,T0 is the pyromagnetic co-efficient, ϕ is the magnetic scalar potential, ∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z2 is the Laplacian operator and the subscript s represents the solid.

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It is clear that there exists the following solution for the basic state: qb = 0,

1 pb (z) = p0 − ρ0 gz − ρ0 αt gβz2 2

μ0 M0 κβ μ0 κ 2 β 2 2 z z− 1+χ 2(1 + χ)2 (11)   Kβz ˆ Tb (z) = T0 − βz, Hb (z) = H0 − k, 1+χ    b (z) = M0 + Kβz kˆ M 1+χ −

where β = q0 /k1 is the temperature gradient, T0 = T∞ + q0 (1 + ht d/k1 )/ ht and the subscript b denotes the basic state. To study the stability of the system, we perturb all the variables in the form q = q  ,

p = pb (z) + p  ,

T = Tb (z) + T  , (12)  =M  b (z) + M  M

H = Hb (z) + H  , q ,

p ,

T ,

H 

 M

where and are perturbed variables and are assumed to be small. Substituting (12) into (8) and (9) and using (7a, b), we obtain (after dropping the primes) Hx + Mx = (1 + M0 /H0 )Hx , Hy + My = (1 + M0 /H0 )Hy ,

(13)

Hz + Mz = (1 + χ)Hz − KT . Again substituting (12) into (5), linearizing, eliminating the pressure term by operating curl twice and using (13) the z-component of the resulting equation can be obtained as (after dropping the primes): 

∂  2  ∇ ϕ ∂z h

2

μ0 K β 2 2 ∇h T + ρ0 αt g∇h T + 1+χ

where (ρ0 C)1 = ερ0 CV ,H + εμ0 H0 K + (1 − ε)(ρ0 C)s and (ρ0 C)2 = ερ0 CV ,H + εμ0 H0 K. Equations (7a, b), after substituting (12) and using (13), may be written as (after dropping the primes)   M0 ∂ 2ϕ ∂T 2 1+ ∇h ϕ + (1 + χ) 2 − K = 0. (16) H0 ∂z ∂z Since assuming the principle of exchange of stability is valid (Finlayson [4]), the normal mode expansion of the dependent variables is assumed in the form {w, T , ϕ} = {W (z), (z), (z)} exp[i(x + my)] (17) where  and m are wave numbers in the x- and y-directions, respectively. On substituting (17) into (14)–(16) and non-dimensionalizing the variables by setting z∗ =

z , d

∗ =

w∗ =

d w, νA

∗ =

(1 + χ)κ

Kβvd 2

(14)

where ∇h2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the horizontal Laplacian operator.

κ βvd

and (18)

where v = μf /ρ0 is the kinematic viscosity, κ = k1 /(ρ0 C)2 is the effective thermal diffusivity and A = (ρ0 C)1 /(ρ0 C)2 , we obtain (after dropping the asterisks) [(D 2 − a 2 ) − σ 2 ](D 2 − a 2 )W = −a 2 R[M1 D − (1 + M1 ) ]

 μf ∂ ρ0 + − μ˜ f ∇ 2 ∇ 2 w ∂t k = −μ0 Kβ

The energy (6) after using (12) and linearizing, takes the form (after dropping the primes):   ∂T ∂ ∂ϕ − μ 0 T0 K (ρ0 C)1 ∂t ∂t ∂z   μ 0 T0 K 2 2 wβ (15) = k1 ∇ T + (ρ0 C)2 − 1+χ

(19)

(D 2 − a 2 ) = −(1 − M2 A)W

(20)

(D 2 − a 2 M3 ) − D = 0.

(21)

Here,√ D = d/dz is the differential operator, a = 2 + m2 is the overall horizontal wavenumber, R = αt gβd 4 /νκA is the thermal Raleigh number representing the ratio of buoyancy force to viscous force, M1 = μ0 K 2 β/(1 + χ)αt ρ0 g is the magnetic number representing the ratio of magnetic force

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to gravitational force, N = RM1 = μ0 K 2 β 2 d 4 /(1 + χ)μκA is the magnetic Rayleigh number representing the ratio of magnetic force to viscous force, M2 = μ0 T0 K 2 /(1 + χ)(ρ0 C)1 is the magnetic parameter, M3 = (1 + M0 /H0 )/(1 + χ) is the measure of nonlinearity of magnetization, √  = μ˜ f /μf is the ratio of viscosities and σ = d/ k is the porous parameter. The typical value of M2 for magnetic fluids with different carrier liquids turns out to be of the order of 10−6 and hence its effect is neglected as compared to unity. Equations (19)–(21) are to be solved using the appropriate boundary conditions. The lower boundary (z = 0) is assumed to be rigid-paramagnetic with prescribed heat flux condition, while the upper paramagnetic boundary (z = 1) is assumed to be either rigid or free with radiative- type of condition. Thus the boundary conditions are: W = DW = (1 + χ)D − a = D = 0 at z = 0

(22a)

where Ai , Ci and Di are unknown constants to be determined. The base functions Wi (z), i (z) and i (z) are generally chosen such that they satisfy the corresponding boundary conditions. Substituting (23) into (19)–(21), multiplying the resulting momentum equation by Wj (z), energy equation by j (z) and magnetic potential equation by j (z); performing the integration by parts with respect to z between z = 0 and z = 1 and using the boundary conditions (22), we obtain the following system of linear homogeneous algebraic equations: Cj i Ai + Dj i Ci + Ej i Di = 0

(24)

Fj i Ai + Gj i Ci = 0

(25)

Hj i Ci + Ij i Di = 0.

(26)

The coefficients Cj i − Ij i involve the inner products of the basis functions and are given by Cj i =  D 2 Wj D 2 Wi + (2a 2  + σ 2 ) DWj DWi

+ a 2 (a 2  + σ 2 ) Wj Wi

W = DW = (1 + χ)D + a = D + Bi = 0 at z = 1 (if the boundary is rigid)

(22b)

Dj i = −a 2 R(1 + M1 ) Wj i ,

or

Ej i = a 2 RM1 Wj D i ,

W = D 2 W = (1 + χ)D + a = D + Bi = 0

Gj i = D j D i + a 2 j i ,

at z = 1 (if the boundary is free)

(22c)

where Bi = ht d/k1 is the Biot number. The case Bi = 0 and Bi → ∞ respectively correspond to constant heat flux and isothermal conditions at the upper boundary.

3 Method of solution Equations (19)–(21) together with the chosen boundary conditions (22) constitute an eigenvalue problem, which has been solved by the Galerkin Method. Accordingly, W , and are written as

Fj i = − j Wi

Hj i = − D j i ,

a Ij i =

j (1) i (1) + j (0) i (0) 1+χ + D j D i + a 2 M3 j i

1 where the inner product is defined as · · · = 0 (· · ·)dz. The above set of homogeneous algebraic equations can have a non-trivial solution if and only if Cj i Fj i 0

Dj i Gj i Hj i

Ej i 0 = 0. Ij i

(27)

The trial functions are chosen as follows: W=

N 

Ai Wi (z),

i=1

(z) =

N  i=1

(z) =

N  i=1

Di i (z)

(i) Rigid-rigid paramagnetic boundaries: Ci i (z), (23)

∗ Wi = (z4 − 2z3 + z2 )Ti−1 , ∗ , i = z2 (1 − 2z/3)Ti−1 ∗

i = (z − 1/2)Ti−1 .

(28)

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(ii) Rigid-free paramagnetic boundaries: ∗ , Wi = (z4 − 5z3 /2 + 3z2 /2)Ti−1 ∗ i = z(1 − z/2)Ti−1 ,

(29)

∗

i = (z − 1/2)Ti−1 .

Here, Ti∗ ’s are the modified Chebyshev polynomials. It is seen that the velocity trial function satisfies the boundary conditions but the temperature and magnetic potential trial functions do not satisfy their respective boundary conditions. However, the residuals from the temperature and magnetic potential conditions are included as residuals from the differential equations. Equation (27) leads to a characteristic equation from which the critical Rayleigh number as a function of wave number a is extracted numerically for various values of physical parameters , σ 2 , Bi, M1 , χ and M3 .

4 Numerical results and discussion The numerical procedure used is validated first by comparing the critical Rayleigh number Rc and the corresponding critical wave number ac as a function of Bi with those obtained by Sparrow et al. [26] in Figs. 2a and 2b, respectively, and tabulated in Table 1 for different boundary conditions for an ordi-

nary viscous fluid in the absence of magnetic field (M1 = 0 = M3 ) and in the absence of porous medium (σ = 0 and  = 1). We note that there is an excellent agreement between both the approaches and thus verifies the accuracy of the numerical procedure used in the investigation. The critical Rayleigh number Rc and the corresponding wave numbers ac obtained for different values of , σ 2 , Bi, M1 , M3 and χ are presented graphically in Figs. 3–7. The values of Rc and ac are shown in Figs. 3a and 3b, respectively as a function of σ 2 for two values of χ (= 0 and 9999) and Bi (= 0 and 2) with M1 = 10, M3 = 1 and  = 5. The results are shown for the cases of both boundaries rigid as well as lower boundary rigid and upper boundary free. We note that an increase in the value of σ 2 , Bi and χ is to increase the value of Rc and hence their effect is to delay the onset of ferroconvection (see Fig. 3a). From the figure it is evident that the variation of Bi from 0 to 2 significantly increases the critical Rayleigh number for different velocity boundary conditions considered; the least being for Bi = 0 and the highest values correspond to those for Bi = 2. Thus the system is found to be more unstable for the upper heat insulating boundary as compared to isothermal condition at the upper boundary. This behavior is not surprising as the nature of the upper boundary changes drastically from an insulated surface to a conductive boundary with an increase in the value of Bi. It is thus evident that with

Table 1 Comparison of Rc and ac for different values of Bi for M1 = 0 = M3 (i.e. in the absence of magnetic field),  = 1and σ = 0 (i.e. in the absence of porous medium) Bi

Sparrow et al. [26] Rigid-Rigid Rc

Present analysis

Rigid-Free

Rigid-Rigid

ac

Rc

ac

Rc

Rigid-Free ac

Rc

ac

0

720.000

0.00

320.000

0.00

720.000

0.000

320.000

0.000

0.01

747.765

0.71

338.905

0.58

747.765

0.7126

338.904

0.5831

0.03

768.153

0.93

353.176

0.76

768.155

0.9283

353.158

0.7624

0.1

807.676

1.23

381.665

1.015

807.676

1.2281

381.665

1.0151

0.3

869.231

1.57

428.290

1.03

869.208

1.5571

428.290

1.2992

1

974.173

1.94

513.792

1.64

974.172

1.9427

513.790

1.6438

3

1093.744

2.24

619.666

1.92

1093.74

1.2419

619.666

1.9211

10

1204.571

2.44

725.150

2.11

1204.57

2.4367

725.147

2.1055

30

1259.884

2.51

780.240

2.18

1259.91

2.5110

780.237

2.1760

100

1284.263

2.53

804.973

2.20

1284.28

2.5394

804.972

2.2029

∞

1295.781

2.55

816.748

2.21

1295.78

2.5490

816.744

2.2147

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Fig. 2 (a) Comparison of Rc as a function of Bi when M1 = 0 = M3 , σ = 0 and  = 1. (b) Comparison of ac as a function of Bi when M1 = 0 = M3 , σ = 0 and  = 1

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Fig. 3 (a) Variation of Rc as a function of σ 2 with two values of Bi for M3 = 1, M1 = 10 and  = 5. (b) Variation of ac as a function of σ 2 with two values of Bi for M3 = 1, M1 = 10 and  = 5

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Fig. 4 (a) Variation of Rc as a function of σ 2 with two values of  for M3 = 1, M1 = 10 and Bi = 2. (b) Variation of ac as a function of σ 2 with two values of  for M3 = 1, M1 = 10 and Bi = 2

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Fig. 5 (a) Variation of Rc as a function of σ 2 with two values of M1 for M3 = 1,  = 5 and Bi = 2. (b) Variation of ac as a function of σ 2 with two values of M1 for M3 = 1,  = 5 and Bi = 2

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Fig. 6 (a) Variation of Rc a as function of σ 2 with two values of M3 for M1 = 10,  = 5 and Bi = 2. (b) Variation of ac a as function of σ 2 with two values of M3 for M1 = 10,  = 5 and Bi = 2

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Fig. 7 Variation of Rc as a function of Nc with two values of M3 for σ 2 = 100,  = 5 and Bi = 2

an increase in the value of Bi the temperature perturbations will not grow so easily and therefore higher values of Rc are needed for the onset of ferroconvection. Further inspection of the figure reveals that the critical values of Rayleigh number Rc for different boundaries is predominant for Bi = 2 when compared with Bi = 0 and the nature of boundaries plays an important role on the stability of the system. The boundaries with large magnetic susceptibility (χ = 9999) are more stable compared to low magnetic susceptibility (χ = 0). In addition, increase in Bi, σ 2 and χ is to increase ac and thus their effect is to decrease the dimension of the convection cells (see Fig. 3b). As pointed out in the introduction, Brinkman’s model rests on an effective viscosity μe different from fluid viscosity μf denoted through  in dimensionless form and it has a determining influence on the onset of ferroconvection in porous media. Figures 4a and 4b respectively indicates the variation of Rc and ac as a function of σ 2 for different values of  and χ with M1 = 10, M3 = 1 and Bi = 2. From Fig. 4a one sees that an increase in the value of  is to increase Rc and hence its effect is to delay the onset of ferroconvection. This is due to an increase in viscous diffusion. Also, increase in the value of  is to increase ac and hence

its effect is to decrease the size of convection cells (see Fig. 4b). In Fig. 5a plotted the critical Rayleigh number as a function of σ 2 for two values of M1 = 0 and 5 and two values of χ = 0 and 9999 with M3 = 1,  = 5 and Bi = 2. It is observed that an increase in the value of M1 is to decrease the value of Rc and thus leads to a more unstable system due to an increase in the magnetic force. From Fig. 5b, it can be seen that increase in M1 is to decrease ac and thus its effect is to increase the size of convection cells. Figure 6a depicts Rc as a function of σ 2 for different values of nonlinearity of magnetization represented through the parameter M3 . The results presented here are for two values of χ when M1 = 10,  = 5 and Bi = 2. It can be seen that an increase in the value of M3 is to decrease Rc and thus it has a destabilizing effect on the stability of the system. This may be due to the fact that the application of magnetic field makes the ferrofluid to acquire larger magnetization which in turn interacts with the imposed magnetic field and releases more energy to drive the flow faster. Hence, the system becomes unstable with a smaller temperature gradient as the value of M3 increases. Besides, the critical Rayleigh numbers for rigid-rigid and rigid-free paramagnetic boundaries become closer as

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the value of M3 increases. As expected on physical grounds, the critical Rayleigh number for rigid-rigid boundaries is found to be greater than those of rigidfree boundaries. Whereas, an increase in M3 is to decrease ac indicating its effect is to increase the dimension of convection cells (see Fig. 6b). Further, the critical wave numbers for rigid-rigid boundaries are higher than those of rigid-free boundaries. The complementary effects of the buoyancy and the magnetic forces are made clear in Fig. 7 by displaying the loci of the critical Rayleigh number Rc and the critical magnetic Rayleigh number Nc , where N (= RM1 ) is the magnetic Rayleigh number, for different values of M3 for σ 2 = 100,  = 5 and Bi = 2 for both types of velocity boundary conditions considered. We note that Rc is inversely proportional to Nc . As M3 → ∞, irrespective of boundaries considered for two values of χ , the data fits the following linear relation exactly Rc Nc + =1 Rc0 Nc0 where Rc0 is the critical Rayleigh number in the nonmagnetic case (N = 0) and Nc0 is the critical magnetic Rayleigh number in the non-gravitational case (R = 0).

5 Conclusions From the foregoing numerical study, it is observed that an increase in the value of , σ 2 , χ and Bi is to stabilize the ferrofluid motion against convection in a porous medium. The system becomes destabilized as the values of M1 and M3 increase. Furthermore, an increase in Bi, σ 2 ,  and χ as well as decrease in M1 and M3 is to decrease the dimension of the convection cells. As M3 → ∞, irrespective of the nature of bounding surfaces of the porous layer, the critical Rayleigh (Rc ) and critical magnetic Rayleigh (Nc ) numbers fit into a straight line. Also, it is noted that (Rc and ac )rigid–rigid > (Rc and ac )rigid–free . Acknowledgements The work reported in this paper was supported by UGC under CAS Programme. The authors (CEN) and (MR) wish to thank respectively, the Management and Principal of Dr. Ambedkar Institute of Technology and East Point College of Engineering and Technology, Bangalore for the encouragement.

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