Effect of magnetic field dependent viscosity and rotation on ferroconvection saturating a porous medium in the presence of dust particles

International Communications in Heat and Mass Transfer 32 (2005) 1387 – 1399 www.elsevier.com/locate/ichmt

Effect of magnetic field dependent viscosity and rotation on ferroconvection saturating a porous medium in the presence of dust particlesB Sunil a,T, Anu Sharma a, R.G. Shandil b, Urvashi Gupta c a

Department of Applied Sciences, National Institute of Technology, Hamirpur (H.P.)-177 005, India Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla (H.P.)-171 005, India c Department of Chemical Engineering and Technology, Panjab University, Chandigarh-160 014, India

b

Available online 8 August 2005

Abstract This paper deals with the theoretical investigation of the combined effect of magnetic field dependent (MFD) viscosity and rotation on ferroconvection saturating a porous medium in the presence of dust particles subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis has been carried out to study the onset of ferroconvection. The cases of stationary convection and oscillatory modes have been discussed. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of overstability. D 2005 Elsevier Ltd. All rights reserved. Keywords: Ferroconvection; Magnetic field dependent viscosity; Rotation; Dust particles; Porous medium

1. Introduction An authoritative introduction to ferrohydrodynamics and the research on magnetic liquid has been discussed in detail in the celebrated monograph by Rosensweig [1]. The Be´nard convection in ferromagnetic fluids has been considered by many authors [2–12]. The ferromagnetic fluid has been considered to be clean in all the above studies. In many geophysical situations, the fluid is often not pure Communicated by J.P. Hartnett and W.J. Minkowycz. T Corresponding author. E-mail addresses: [email protected], [email protected] (Sunil).

B

0735-1933/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.07.001

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but contains suspended/dust particles. Scanlon and Segel [13] have considered the effects of suspended particles on the onset of Be´nard convection. Convection in ferromagnetic fluids is gaining much importance due to their astounding physical properties. One such property is viscosity of ferromagnetic fluid. Sunil et al. [14,15] have studied the effect of dust particles on thermal convection in ferromagnetic fluid saturating a porous medium by assuming isotropic viscosity and MFD viscosity separately. There is great interest in the rotation of ferromagnetic fluids in a rotating magnetic field, where large rotational velocities can be achieved. The thermoconvective instability in a ferromagnetic fluid saturating a porous medium of very large permeability subjected to a vertical magnetic field has been discussed using Brinkman model by Vaidyanathan et al. [16]. The porous medium of very low permeability allows us to use the Darcy model. The effect of temperature, rotation and porous medium on ferromagnetic fluids as a single-component fluid has been studied by Sekar et al. [17], Sekar and Vaidyanathan [18] and Sharma et al. [19]. In view of the above investigations, it is attempted to discuss the influence of rotation on medium permeability and how MFD viscosity affects the magnetization in ferromagnetic fluid heated from below in the presence of dust particles saturating a porous medium of very low permeability using Darcy’s model. The present study can serve as a theoretical support for experimental investigations, e.g., evaluating the influence of impurities in a ferromagnetic fluid on thermal convection phenomena in porous medium.

2. The perturbation equations and dispersion relation Consider an infinite, horizontal layer of thickness d of an electrically non-conducting incompressible Boussinesq ferromagnetic fluid, embedded in dust particles (neutral), having a variable viscosity, given by l = l 1(1 + d.B), heated from below saturating a porous medium. Here l 1 is taken as viscosity of the fluid when the applied magnetic field intensity is absent. The whole system is assumed to be rotating with angular velocity 6 = (0, 0, X) along the z-axis. The gravity field g = 0, 0,
g) and uniform vertical magnetic field intensity H = (0, 0, H 0) pervade the system. This fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium of porosity e and medium permeability k 1. Let qV = (u, v, w), q1V = (S , r, s), pV, qVh, HV and MV denote, respectively, the small perturbations in ferromagnetic fluid velocity q, particle velocity qd, pressure p, density q, temperature T, magnetic field intensity H and magnetization M. Then the linearized perturbation Eqs. become
 q0 Bu BpV BH1V 2q0 l1 mN 0 Bu ¼ L0
þ l0 ðM0 þ H0 Þ ; ð1Þ þ Xv
u
L0 Bx Bz e Bt e k1 e Bt
 q0 Bv BpV BH2V 2q0 l1 mN 0 Bv L0 ¼ L0
þ l0 ðM0 þ H0 Þ ; ð2Þ
Xu
v
By Bz e Bt e k1 e Bt
q Bw BpV BH3V l1 l K2 b L0 0 ¼ L0
þ l0 ðM0 þ H0 Þ
w
0 fH3Vð1 þ vÞ
K2 hg þ gq0 ðahÞ Bz 1þv Bz e Bt k1  l1 mN 0 Bw ; ð3Þ
dl0 ðM0 þ H0 Þw
k1 e Bt

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Bu Bv Bw þ þ ¼ 0; Bx By Bz

ð4Þ




   Bh B BU1V
l0 T0 K2 e L0 qC1 þ emN 0 Cpt Bt Bt Bz
   l T0 K22 b ¼ L0 K1 j2 h þ qC2 b
0 w þ mN 0 bCpt w; ð1 þ vÞ

ð5Þ

where qC1 ¼ eq0 CV ;H þ ð1
eÞqs Cs þ el0 K2 H0 ;qC2 ¼ q0 CV ;H þ l0 K2 H0 ;L0 ¼   B2 U1V M0 Bh j21 U1V
K2 þ 1þ ð1 þ vÞ ¼ 0: 2 Bz H0 BZ

m

B K Bt

þ1 ; ð6Þ

Eliminating u, v, pV between Eqs. (1)–(3) and using Eq. (4), we obtain     q0 B l1 mN 0 B þ þ j2 w L0 e Bt k1 e Bt
   l0 K2 b 2 B 2q0 Bf l1 2 2 ¼ L0
j ð1þvÞ ðU1 VÞ
K2 h þq0 gj1 ðahÞ
X
dl0 ðM0 þH0 Þj1 w : 1þv 1 Bz Bz k1 e ð7Þ The vertical component of the vorticity equation is

 q0 mN 0 Bf 2q0 X Bw l1 ¼ L0
f : þ L0 Bt e Bz e e k1

ð8Þ

Further the analysis has been carried out using the techniques of Refs. [10,11,14,15]. We analyze the normal mode technique. This can be written as

f ð x;y;z;t Þ ¼ f ð z;t Þexpi kx x þ ky y ;

ð9Þ

where f(z, t) represents W(z, t), H(z, t), Z(z, t) and U 1(z, t). Here we consider the case where both boundaries are free as well as perfect conductors of heat. The exact solution subject to the boundary conditions 1 W 4 ¼ D2 W 4 ¼ T 4 ¼ DU41 ¼ 0 at z ¼ F ; 2 is written in the form rt4

rt4

W 4 ¼ A1 e cospz4;T4 ¼ B1 e cospz4;DU41 ¼ C1 ert4 cospz4;U41 ¼

ð10Þ 

 C1 rt4 e sinpz4; p

ð11Þ

where A 1, B 1, C 1 are constants and r is the growth rate which is, in general, a complex constant. Substituting Eq. (11) in linearized perturbation dimensionless equations and dropping asterisks for convenience, we get the equations involving coefficients of A 1, B 1, C 1. For the existence of non-trivial

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solutions, the determinant of the coefficients of A 1, B 1, C 1 must vanish. This determinant on simplification yields iT 5 r51 þ T4 r41
iT 3 r31
T2 r21 þ iT 1 r1 þ T0 ¼ 0:

ð12Þ

Here bs21 L2 ; e2 i s1 h  s1 s1 x1 T4 ¼ b 2L2 L4 þ bL1 þ dM3 L2 ; e e P
  2s1 s1 1 x1 s1 L2 þ 2L1 L4 b2 þ TA1 s21 L2
x1 R1 s21 L3 ð1
M2 Þ þ dM3 T3 ¼ b L24 þ e eP e P    1 s1 bL1 ; þ L4 L2 þ  e e T5 ¼

ð13Þ ð14Þ

ð15Þ


   2s1 2bL2 L4 s1 2 2 þ þ TA1 s1 ðs1 bL1 þ 2L2 Þ
x1 R1 L3 s1 ð1
M2 ÞL4 þ L5 T2 ¼ b L1 L4 þ eP P e      x1 dM3 1 s1 s1 bL1 L4 þ þ L2 L4 þ ; ð16Þ þ e P P 
 s b L2 1 ð1
M2 Þ þ L5 L4 þ 2bL1 L4 þ TA1 ðL2 þ 2L1 bs1 Þ
x1 R1 L3 T1 ¼ P P P   dM3 x1 s1 bL1 þ L2 bL1 L4 þ ; ð17Þ þ P P
   b x1 dM3 x1
R1 L3 L5 ; T0 ¼ bL1 2 þ TA1 þ P P2 P

ð18Þ

where R a2 r TA ; x ¼ ; ir1 ¼ 2 ; P ¼ p2 k1 ;s1 ¼ sp2 ; TA1 ¼ 2 ; b ¼ 1 þ x1 ; L1 ¼ ð1 þ x1 M3 Þ 1 4 2 p p p p L2 ¼ ½fðPrV
e Pr M2 Þ þ x1 PrVM3 g þ L1 ehPr ; L3 ¼ f1 þ x1 M3 ð1 þ M1 Þg;   s1 1þf L4 ¼ and L5 ¼ h þ 1
M2 : þ e P R1 ¼

Non-dimensional parameters used in the above equations are same as in the work of Sunil et al. [15]; however, two additional parameters are introduced due to the effect of rotation, which are given as d2 Z4 ¼ Z and TA ¼ m



2Xd 2 me

2 :

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3. The case of stationary convection When the instability sets in as stationary convection (and M 2 i 0), the marginal state will be characterized by r 1 = 0, then the Rayleigh number is given by   ð1 þ x1 Þð1 þ x1 M3 Þ ð1 þ x1 Þ þ TA1 P2 þ x1 dM3 R1 ¼ : ð19Þ x1 h1 Pfð1 þ x1 M3 Þ þ x1 M3 M1 g To investigate the effects of medium permeability, non-buoyancy magnetization, dust particles, rotation and MFD viscosity, we examine the behaviour of (dR 1)/(dP), (dR 1)/(dM 3), (dR 1)/(dh 1), (dR 1)/ (dTA 1) and (dR 1)/(dd) analytically.

(a)

2 Curve 1

1.8

Curve 2

xc

1.6

Curve 3

1.4

Curve 4

1.2 1 0.8 0.6 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

δ

(b) 2600 2400

Curve 1 Curve 2

Nc

2200

Curve 3

2000

Curve 4

1800 1600 1400 1200 1000 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

δ Fig. 1. (a) Variation of x c versus d. (b) Marginal instability curve for variation of N c versus d for M 5 = 0.1, h 1 = 3, TA 1 = 100, P = 0.001; M 3 = 4 for curve 1, M 3 = 8 for curve 2, M 3 = 12 for curve 3 and M 3 = 16 for curve 4.

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Eq. (19) gives   dR1 1 ð1 þ x1 Þð1 þ x1 M3 Þ x1 þ x1 dM3 þ ð1
TA1 P2 Þ ¼
2 ; x1 fð1 þ x1 M3 Þ þ x1 M3 M1 g P dP

ð20Þ

which is always negative when TA1 b

1 : P2

ð21Þ

This shows that the medium permeability always has a destabilizing effect when (21) holds. In the absence of rotation, (20) yields the destabilizing effect of medium permeability. Medium permeability

(a) 1.5 Curve 1

1.4

Curve 2

1.3

xc

Curve 3

1.2

Curve 4

1.1

Curve 5

1 0.9 0.8 0.7 0.6 4

6

8

10

12

14

16

18

20

M3

(b) 2800 Curve 1

2600

Curve 2 Curve 3

2400

Curve 4

2200

Curve 5

Nc 2000 1800 1600 1400 4

6

8

10

12

14

16

18

20

M3 Fig. 2. (a) Variation of x c versus M 3. (b) Marginal instability curve for variation of N c versus M 3 for M 5 = 0.1, h 1 = 3, TA 1 = 100, P = 0.001; d = 0.01 for curve 1, d = 0.03 for curve 2, d = 0.05 for curve 3 and d = 0.07 for curve 4 and d = 0.09 for curve 5.

Sunil et al. / Int. Commun. Heat and Mass Transf. 32 (2005) 1387–1399

(a)

100

1393

Curve 1 Curve 2 Curve 3

xc

10

1 100

1000

10000

100000

1000000

10000000

TA1

(b) 100000 Curve 1 Curve 2 Curve 3

10000

Nc 1000

100 100

1000

10000

100000

1000000

10000000

TA1 Fig. 3. (a) Variation of x c versus TA 1. (b) Marginal instability curve for variation of N c versus TA 1 for M 3 = 1, M 5 = 0.1, h 1 = 3, d = 0.01; P = 0.003 for curve 1, P = 0.005 for curve 2 and P = 0.007 for curve 3.

may have a duel role in the presence of rotation. The medium permeability has a stabilizing effect if 3Þ . Thus, for higher values of Taylor number TA 1, the stabilizing effect of medium TA1 N 1þx1 ðP1þdM 2 permeability has been predicted. Eq. (19) yields n o3 2 2 2 2 ð 1 þ x þ T P ÞM
d ð 1 þ x M Þ þ x M M 1 A 1 1 3 1 1 1 3 ð1 þ x1 Þ 4 dR1 5: ¼
dM3 P h1 f1 þ x1 M3 þ x1 M3 M1 g2 2

ð22Þ

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(a) 35 30

Curve 1 Curve 2

25

Curve 3

20

xc

15 10 5 0 0.001

0.002

0.003

0.004

0.005

0.006

0.007

P

(b) 100000 Curve 1 Curve 2 Curve 3

10000

Nc 1000

100 0.001

0.002

0.003

0.004

0.005

0.006

0.007

P Fig. 4. (a) Variation of x c versus P. (b) Marginal instability curve for variation of N c versus P for M 3 = 1, M 5 = 0.1, h 1 = 3, d = 0.01; TA 1 = 105 for curve 1, TA 1 = 106 for curve 2 and TA 1 = 107 for curve 3.

Eq. (22) yields that (dR 1)/(dM 3) is always negative in the absence of MFD viscosity, thus indicating the destabilizing effect of the non-buoyancy magnetization, whereas in the presence of MFD viscosity, non-buoyancy magnetization may have a destabilizing or a stabilizing effect. Eq. (19) also yields   ð1 þ x1 Þð1 þ x1 M3 Þ ð1 þ x1 Þ þ P2 TA1 þ x1 dM3 dR1 ; ¼
dh1 x1 Ph21 fð1 þ x1 M3 Þ þ x1 M3 M1 g

ð23Þ

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dR1 ð1 þ x1 Þð1 þ x1 M3 ÞP ¼ ; x1 h1 f1 þ x1 M3 þ x1 M3 M1 g dTA1

ð24Þ

dR1 ð1 þ x1 Þð1 þ x1 M3 ÞM3 ¼ : dd Ph1 fð1 þ x1 M3 Þ þ x1 M3 M1 g

ð25Þ

This shows that dust particles are found to have a destabilizing effect, whereas rotation and MFD viscosity have a stabilizing effect. The destabilizing effect of dust particles on non-

(a)

3 Curve 1

2.8

Curve 2

2.6

xc

Curve 3

2.4

Curve 4

2.2

Curve 5

2 1.8 1.6 1.4 1.2 1 1

2

3

4

5

6

7

h1

(b) Curve 1

7500

Curve 2

6500

Curve 3 Curve 4

5500

Curve 5

4500

Nc

3500 2500 1500 500 1

2

3

4

5

6

7

h1 Fig. 5. (a) Variation of x c versus h 1. (b) Marginal instability curve for variation of N c versus h 1 for P = 0.001, M 5 = 0.1, d = 0.01, TA 1 = 100; M 3 = 1 for curve 1, M 3 = 2 for curve 2, M 3 = 3 for curve 3, M 3 = 4 for curve 4 and M 3 = 5 for curve 5.

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magnetic fluid is accounted by many authors [19–22] and is found to be valid for a ferromagnetic fluid also. For M 1 sufficiently large, we obtain the results for the magnetic mechanism operating in porous medium   ð1 þ x1 Þð1 þ x1 M3 Þ ð1 þ x1 þ x1 dM3 Þ þ TA1 P2 ; Nm ¼ R1 M1 ¼ x21 h1 PM3

ð26Þ

where N m is the magnetic thermal Rayleigh number. The values of critical wave number for the onset of instability are determined numerically using Newton–Raphson method by the condition (dN m )/(dx 1) = 0. The critical wave number (x c) and magnetic thermal Rayleigh number (N c), depend on M 3, P, TA 1, d and h 1. The variation of x c and N c with various parameters are illustrated in Figs. 1–5. Fig. 1(a) and (b) indicate that the system stabilizes as the MFD viscosity parameter d increases, however, the cell size is found to shrink. Fig. 1(b) also leads to the conclusion that the non-buoyancy magnetization has a destabilizing effect for lower values of d, whereas for higher values of d, the stabilizing effect of non-buoyancy magnetization has been predicted. Fig. 2(a) indicates a decrease in cell shape x c as M 3 increases. Fig. 2(b) illustrates that as M 3 increases, N c decreases for small values of d, whereas for higher values of d, N c decreases for lower values of M 3 and then increases for higher values of M 3. Fig. 3(a) and (b) indicate the stabilizing nature of cell size x c and corresponding N c as TA 1 increases. Fig. 3(b) also shows that the medium permeability has a destabilizing effect for lower values of TA 1, whereas for higher values of TA 1, the stabilizing effect of medium permeability has been observed. Fig. 4(a) illustrates the stabilizing nature of cell size x c as P increases. Fig. 4(b) illustrates that as P increases, N c always decreases for small values of TA 1, whereas for higher values of TA 1, N c decreases for lower values of P and then increases for higher values of P. Fig. 5(a) and (b) show the stationary nature of cell size x c and destabilizing nature of N c as h 1 increases. The magnetic Rayleigh number N c shows a drastic decrease in the presence of dust particles because the heat capacity of clean fluid is supplemented by that of the dust particles.

4. The case of oscillatory modes Equating the imaginary parts of Eq. (12), we obtain  
 2    s1 2s1 x1 s1 4 bs1 2 2 2 r1 r1 2 L2
r1 2L4 L1 b þ b L4 þ L2 þ TA1 s21 L2 þ dM3 e e eP P  
    2 1 s1 s b L2 þ L4 L2 þ 2bL1 L4 þ 
x1 R1 1 L3 ð1
M2 Þ þ e P e e P    s1 dM3 x1 s1 bL1 þ L2 bL4 L1 þ ¼ 0: þ TA1 ð2s1 L1 b þ L2 Þ
x1 R1 L3 L4 L5 þ ð1
M2 Þ þ P P P ð27Þ

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It is evident from Eq. (27) that r 1 may be either zero or non-zero, meaning that the modes may be either non-oscillatory or oscillatory. In the absence of rotation and dust particles, we obtain the result (after simplification) as
 1 1 r1 fðPrV
ePr M2 Þ þ x1 PrVM3 gð1 þ x1 þ dM3 x1 Þ þ ð1 þ x1 Þð1 þ x1 M3 Þf2ð1 þ x1 Þ þ dM3 x1 g P e ¼ 0:

ð28Þ

r1 ¼ 0;

ð29Þ

Hence

which implies that oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied for a porous medium. Thus, from Eq. (27), we conclude that the oscillatory modes are introduced due to the presence of the rotation and dust particles, which were non-existent in their absence.

5. The case of overstability Equating real and imaginary parts of (12) and eliminating R 1 between them, we obtain A3 c31 þ A2 c21 þ A1 c1 þ A0 ¼ 0;

ð30Þ

where c1 ¼

r21 ; A3





 s41 ð1
M2 ÞL1 2 L2 s41 ð1
M2 Þ s31 f f ð1
M2
hÞg þ ¼ b þ 2 b P e e3 e

dM3 s41 x1 ð1
M2 ÞL2 ; e2 P

  L1 s1 h ð1 þ f ÞL5 L2 L5 s1 TA1 L1 ðh þ L5 Þ dM3 s1 x1 L1 h 2 þ b þ b þ þ A0 ¼ 2 P P3 P e P3 P
 L2 bL1 ð1 þ f Þ dM3 x1 L2 L5 þ TA1 L5 :
þ P P3 e þ

ð31Þ

ð32Þ

Since r 1 is real for overstability, the three values of c 1 (=r 12) are positive. The product of roots of (30) is
(A 0)/(A 3), and if this is to be negative, then A 3 N 0 and A 0 N 0. Eqs. (31) and (32) show that the product is negative if fN

h Pb L1 ð1 þ f Þ; ; and L2 N e ð1
M2 Þ

ð33Þ

which implies that

q0 CV ; H þ l0 K2 H0 ð1
M2 ÞNq0 Cpt and PrVN




 ePr M2 Pð1 þ x1 Þð1 þ f Þ þ : e ð1 þ x1 M3 Þ

ð34Þ

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h i

Pð1þx1 Þð1þf Þ ePr M2 Thus, for q0 CV ;H þ l0 K2 H0 ð1
M2 ÞNq0 Cpt and Pr VN ð1þx þ , overstability canM Þ e 1 3 not occur and the principle of the exchange of stabilities is valid. Hence, the above conditions are the sufficient conditions for the non-existence of overstability. In the absence of magnetic field, the above conditions, as expected, reduces to C v N C pt and m N (K 1)/(q 0C v ), which are in good agreement with the results obtained earlier [19,20,22]. Nomenclature Latin symbols C V, H Specific heat at constant volume and magnetic field, kJ/m3K Specific heat of solid (porous matrix) material, kJ/m3K Cs Specific heat of dust particles, kJ/m3K C pt d Thickness of the ferromagnetic fluid layer, m g Acceleration due to gravity, m/s2; g = (0, 0,
g) H Magnetic field intensity, amp/m; (0, 0, H 0) External magnetic field intensity, amp/m Hext HV The perturbation in magnetic field intensity, amp/m Uniform magnetic field intensity, amp/m H0 The wave number along the x-direction, m
1 kx ky The wave number along the y-direction, m
1 k1 Medium permeability, m2 K1 Thermal conductivity, W/mK The pyromagnetic coefficient, amp/mK;
ð BM K2 BT ÞH0 ;Ta m mass of the dust particle, kg M Magnetization, amp/m MV The perturbation in the magnetization, amp/m The magnetization when magnetic field is H 0 and temperature Ta , amp/m M0 Uniform particle distribution (1/m3); constant N0 p The reduced pressure, N/m2 pV The perturbation in reduced pressure, N/m2 q Darcian (filter) velocity of the ferromagnetic fluid, m/s qV The perturbation in velocity, m/s (u, v, w) = (0, 0, 0) Velocity of the dust particles, m/s qd The perturbation in velocity, m/s (S , r, s) = (0, 0, 0) q1V t Time, s T Temperature, K Greek a b m l l1 l0 d q

letters Coefficient of thermal expansion, K
1 A uniform temperature gradient, K/m; |dT/dz| Kinematic viscosity, m2/s Dynamic viscosity (variable), kg/ms2 Viscosity (in the absence of applied magnetic field intensity), kg/ms2 Magnetic permeability of free space, H/m Linear measure of the viscosity variations with the applied magnetic field, T
1 Fluid density, kg/m3

Sunil et al. / Int. Commun. Heat and Mass Transf. 32 (2005) 1387–1399

q0 qs f v h 6 qV j e r U 1V

1399

Reference density, kg/m3 Density of solid (porous matrix) material, kg/m3
1 The z-component of the vorticity, BM s ; (Bv)/(Bx)
(Bu)(By) The magnetic susceptibility; BH H0 ;Ta The perturbation in temperature T, K Angular velocity, m/s The perturbation in density q, kg/m3 Del operator, m
1 Medium porosity, m3/m3 The growth rate, s
1 The perturbed magnetic potential, amp

Acknowledgement Financial assistance to Dr. Sunil in the form of a Research and Development Project [No. 25(0129)/ 02/EMR-II] of the Council of Scientific and Industrial Research (CSIR), New Delhi, is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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