Indian Journal of Pure & Applied Physics Vol. 40, March 2001, pp. 166-171

Effect of magnetic field dependent viscosity on ferroconvection in sparsely distributed porous medium*

G Vaidyanathan

Department of Physics, Pondicherry Engineering College, Pondicherry 605 0 I 4

A Ramanathan & S Maruthamanikandan

Department of Mathematics, Kanchi Mamunivar Centre for Post Graduate Studies, Pondicherry 605 008

Received I 5 May 200 I; revised 3 October 200 I; accepted 5 October 200 I

The effect of magnetic field on viscosity of ferrofluid-inducing convection in a sparsely distributed porous medium heated from below has been studied. A linear stability analysis is carried out for both stationary and oscillatory modes. It is found that stationary mode alone is favourable when compared with oscillatory mode. The variation of heat dependent viscosity with respect to magnetic field tends to stabilize the system. Numerical results are obtained and are illustrated graphically for various permeability values.

1 Introduction

The method of formation of ferrofluids takes place during the mid-sixties of the last century. This is because, it had very large important applications in various fields like nuclear fusion, high speed noiseless printing, pressure seals for compressors and blowers 1, liquid cooled loud speakers2

, the novel zero leakage rotating shaft seals are used in computer disk drives3

• Ferrohydrodynamics is being studied due to the formation of magnetic fluids.The study of physical properties of liquid having strong magnetization has induced great interest. In view of the physical properties of this type of fluid made the study of thermoconvective instability of ferrofluids. The magnetization depends on magnetic field, temperature and density of the fluid. The change of any one of these causes a variation in body force. This leads to convection in ferromagnetic fluids in the presence of magnetic field gradient. This is known as ferroconvection (Chandrasekhart

Shliomis5 gave linearized relation for normalized perturbation quantities valid at the limit of instability. Lalas & Carrni6 analyzed thermo­convective instability of ferrofluids without considering buoyancy effect. Finlayson7 dealt with convective instability of ferromagnetic fluids heated

*Presented at the National Conference on Magnetic Fluid, March 28-30, 2000, Pondicherry Engineering College, Pondicherry

from below in the presence of a vertical uniform magnetic field. Rudraiah and Sekhar8 have studied the effect of non-uniform temperature gradients on convection in magnetic fluids and the effect of internal heat generation on ferroconvection. The study of ferroconvective instability of fluids saturating a porous medium has been analysed by Vaidyanathan et al. 9. Sekar et al. 10 analyzed the ferroconvection in fluids saturating a rotating densely packed porous medium. All the above investigators did not consider the effect of magnetic field on viscosity of ferroconvection. In the present analysis the authors are interested in studying the effect of magnetic field dependent viscosity on ferroconvection in a sparsely distributed porous medium. Here, the porous medium is assumed to be bounded by stress-free boundaries on either side and is heated from below. The conditions for instability to set in by stationary and oscillatory modes have been investigated. It is found that the increase in magnetization stabilizes the system and the increase in permeability also stabilizes the system. Numerical results are obtained for various values of permeability and for different coefficients of viscosity. The results are also presented in graphical form.

2 Mathematical Formulation

An infinitely spread horizontal layer of ferromagnetic fluid saturating a sparsely distributed porous medium heated from below is considered. A

V AIDY ANA THAN et al.:FERROCONVECTION IN POROUS MEDIUM 167

uniform magnetic field H0 acts along the vertical direction which is taken as z-axis.

The fluid is assumed to be incompressible Boussinesq fuid having a variable viscosity, given by 11 = 11~ct + o.B).

The governing mathematical equations used for the above model are as follows:

The continuity equation is:

v =. q =0 ... ( 1)

The momentum equation for a ferrofluid with variable viscosity is:

p0 Dq =- Vp + pg + =V'. (HB) + J1V'2q _l:_q ... (2) Dt ko

where q, p, g = (0,0,-g), H, B, Jl, and k0 denote the velocity, pressure, acceleration due to gravity, magnetic field, magnetic induction, variable viscosity and permeability of the porous medium respectively. The above equations =V'.(HB) is a dyadic (having nine components).

The temperature equation as given by Finlayson7 is:

[p,Cv" -l',u[~~L]: + l'oT[~~]v" · dd7 = KIV 2T+ <P ... (3)

where Cv.H is the heat capacity at constant volume and magnetic field, T is the temperature, M is the magnetization, K1 is the thermal conductivity and <P is the viscous dissipation containing second order

terms in velocity. H[~~] is a dyadic (as followed

by Finlayson\

The density equation of state for Boussinesq magnetic fluid is:

. . . (4)

The magnetization depends on the magnitude the magnetic field and temperature, which can be written as:

M =_!!_ M (H,1) H

To evaluate partial derivatives of magnetization M, the linearized magnetic equation of state is:

... (5)

where H0 is the reference magnetic field, T. is the

Ta -- (To + ~) . average temperature given by 2

where T0 and T1 are the constant average temperatures of the lower and upper surfaces of the

I [aM] · h · ·b·1· ayer, X= dH IS t e magnetic susceptl 1 1ty,

Ho .To

K2 =[~~] is the pyromagnetic coefficient and Ho .To

H is magnitude of H.

Basic state is assumed to be quiescent state and the basic state quantities are obtained by substituting velocity of quiescent state in the constituent equations. Next a small perturbation is imparted on the basic state and linear theory is used to obtain perturbation equation. Further analysis is being carried out following Finlayson 7 and Vaidyanathan et a/.9.

On linearization the following components of modified Navier-Stoke's equations can be obtained:

, du dp dH1 2 f.11 Po-=--+Jlo(Mo+Ho) --+JlJV' u--u ... (6) ili ~ ~ ~

, dv dp dH 2 2 f.11 Po-=--+Jlo(Mo+Ho) --+ Jl1V' u--v ... (7) dt dy dz k0

+Jl10Jlo(M0+Ho)V'2w- f.1 1w _1!:.!_0Jlo(M0+Ho)w ... (8) ko ko

Hence the vertical component of Eq. (2) is:

168 INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002

... (9)

Following the analysis of Finlayson 7 and Sekar et a/.10

, the normal mode solutions of all dynamical variables can be written as:

f(x, y, z, t) = f(z, t) exp{ i (kxx + kyy) }

where the wave number k is given by:

e=k 2 +k 2 X y

... (10)

. . . (11)

Upon making use of Eqs (I 0) and (II), the vertical component of momentum equation can be written as:

and the modified Fourier heat conduction equation takes the form:

+ [ P f3 JloToKi /3 )w ... (1 3) < (l +X)

where

.. . (14)

Also one can have :

a2<1> [ M 0 } 2 ae o+x)---2

- 1+- e-K2 --=o az H 0 az ... (15)

Proceeding along the same line of 7 one gets the following dimensionless equations:

--+-- (D -a) w =(D -a) w + [M,D<)> [a 1) 2 2. 2 2 2. •

dt * k0 *

aT* a . ? 2·,..... v. • P, at* -P,M2 at* (D<)> )=(D--a )1 +(1 -M2)aR w

.. . (17)

... (18)

where the following non-dimensional parameters and terms are introduced:

r* = _v_~ , w • = _w_d ,<)>. = _(1_+_X_)_K_1 a_R_~1_2¢;_ ' R= ...:..g_a...:...{Xi_4

.:....p..::..c d V K2 pcf3vd- vK

1

K R11 2 :) r* = ___f!_~(J a=kd z • = ~ D= ~- k • = ~Q.. 8*

f3 d 2 ' ' d ' :). * ' II d2 ' Pc V . oz = OJ..4Ho(l +X)

p = Jlc M =( 11oKif3 ) M ==[ JloToKi ) M r K, ' I (1 + X)apog ' 2 (I+ X)PoC ' 3

[ I+~ I (1 +X) = J ... (19)

3 Exact Solution for Free Boundaries

The exact solution can be obtained to indicate the qualitative aspects of instability of• the layer heated from below. The boundary conditions for stress free non-conducting boundaries are:

* D2 * rrJ~< * D * w = w = 1 = Dw = <I> = 0

at z = - 1 /2 and z = + 1 /2 . . . (20)

Following the analysis of Finlayson 7, the exact solutions satisfying the boundary cond ition given by Eq. (20) are then given by:

* at * * ...,.;t. crr• * w =Ae cos nz , 1 =Be cos 1t z

V AIDY ANA THAN et al.:FERROCONVECTION IN POROUS MEDIUM 169

• crt• • • (c) crt• . • D<j> = C e cos nz ,<j> = ; e sm nz ... (21)

where A, B, and Care constants. Substitution of Eq. (21) into the set of Eqs ( 16)-( 18) and dropping asterisks for convenience leads to:

[(~<' +a'~~ ++~' +a'} + a'OM 3{ :, +(~<' +a' l} ] A-aR'~'(1+M 1)B+aRv,M 1C=O ... (22)

[aR'h (1-M2)]A-[(7t2+a2)+P,cr] B + M2P,crC = 0 ... (23)

... (24)

Table I - Marginal stability of magnetic field dependent viscosity of ferrotluid saturating a sparsely distributed porous medium destabilize by stationary mode having M1 = I 000, M2 =

Table 2 - Marginal stability of magnetic field dependent viscosity of ferrotluid saturating a sparsely distributed porous medium destabilize by stationary mode having, M1 = 1000, M2 = 0, (Brinkman Number k0= 0.0 I)

Coeff.of viscosity 8

0.01

0.03

0.05

Magnetiza-tion M3

I 3 5 7

I 3 5 7

I 3 5 7

Critical Thermal wave Rayleigh No. number (a c) Nc=(RMl)c

4.03 8572.19 3.50 6296.27 3.32 5798.78 3.20 5594.06

4.03 8678.12 3,46 6500.76 3.24 6093.43 3.12 5975.23

4.03 8784.12 3.43 6703.71 3.20 6383 .93 3.08 6349.17

0, (Brinkman Number k0= 0.1 0) 0.07 I 4.00 8889.53

Coeff. of viscosity 8

0.01

0.03

0.05

0.07

0.09

Magnetization Critical M3 wave

number (ac)

I 3.43 3 3.00 5 2.83 7 2.74

I 3.43 3 2.96 5 2.78 7 2.69

I 3.39 3 2.96 5 2.78 7 2.65

I 3.39 3 2.92 5 2.74 7 2.59

I 3.39 3 2.92 5 2.69 7 2.60

Thermal Rayleigh No. Nc=(RMl)c

2330.05 1584.07 1419.83 1349.30

2355.22 1628.47 1481.74 1427.85

2380.26 1672.51 1542.85 1504.98

2405.19 1716.24 1603.06 1580.94

2430.14 1759.65 1662.79 1665.89

For the existence of non-trivial solutions of the above equations, the determinant of the coefficients

3 3.39 6905.32 5 3.16 6671.04 7 3.00 6717 .07

0.09 I 4.00 8994.90 3 3.39 7105.57 5 3.12 6955.09 7 2.96 7079.51

of A, B and C in Eqs (22) - (24) must vanish . This determinant on simplification yields :

Vcr2 + Wcr +X= 0 ... (25)

where

V = P, (7t2 + a 2) [Mz7t2 - (7t2 + a2M3)] ... (26)

W = (,2 +a')' P ,M,n' { ~ + (~r ' +a' l}

- (7t2 +a2)(7t 2 + a2MJ)l

[ ~'+a')+P,{:, +~'+a')}]

.. . (27)

and

170 INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002

... (28)

26 ,......._---------------,

14

I -o- M3 = I 121

+ M3 = J N 0 20 --tr-M 1 = 5 ....

X -t-M 3 : 7 I)

z 18 __.

16

14

5

Fig. I - The effect of magnetic field on the variation of thermal Rayleigh number (Nc) versus coefficient viscosity (8) for a medium of permeability k0 = 0.10

Eq. (25) helps to obtain R for which solutions exist. If oscillatory instability exists , the angular frequency factors cr = icr1• Since V and Ware real, • Eq.(25) could be satisfied for cr = icr~> if and only if 0"1 = 0. Following Chandrasekhar

4, since 0"1 is

positive, oscillatory instability cannot occur.

Following7 the eigenvalue equation for Rayleigh number for stationary mode is obtained by taking 0" 1

= 0 and is given by:

R=

( 2 2 Y 2 2 ~( 2 2\2 I ( 2 2) 2 [ I ( 2 2)~} n +a A" +a M3~ n +a J +ko n +a +a &13 ko + n +a J

a2(I-M2)[n

2 +(I +M1)a

2M3]

... (29)

10,---------------------------------~

9 -

~0~·0~1 ----~0~-J------O~O-S----;~------O·LOS~

5

Fig. 2 - The effect of magnetic field on the variation of thermal Raylei gh number (Nc) versus coefficient viscosity (8) for a medium of permeability k0 = 0.0 I

10

o-------<>-------o-----0-....0

KQ • 0·01

j-<>- Ml ,1;·•--Ml ·2J

----------------K0 •0·10

J" . -l>-Ml , ,,...._MJ. 1 I {).--

______,,.. _..

0.01 0·03 0·05 0·09

Fig. 3 - The effect of magnetic field on the variation of thermal Rayleigh number (Nc) versus coeffici ent viscosity (8) for a medium of permeability /41= 0.10 and /41 = 0.0 I

V AIDY ANA THAN et al.:FERROCONVECTION IN POROUS MEDIUM 171

In the above expression, if one takes 8 = 0 (that is viscosity 11 is constant), one gets the Rayleigh number identical to that of Vaidyanathan et a/.9

.

Also, in the absence of porous medium and magnetization (and 8 = 0) it reduces to classical Buoyancy-induced convection.

When M 1 is very large, the critical magnetic thermal Rayle igh number N, = (RM1), for stationary mode can be obtained using:

destabilizes the system as in the earlier works.

It is also observed that from Tables I and 2, and Figs 1-3, as the permeability of the porous medium increases from small values, the thermal Rayleigh number decreases . Thi s means that the system is destabilized . The increase in viscosity with respect to magnetic field stabilizes the system. Thi s can be seen from Tables I and 2, and Figs 1-3 , as the coefficient of viscosity is increased , the critical thermal Rayleigh number increases. The above fact

N, = is also true for various permeability values. When

(n2 +} Xn2 +a 2M3 (n2 +} J +~ (n2 +a2 )+}&13 ~ + (n2 +a2) convection is further delayed and thi s can be seen ~ [ j} the medium of smaller permeability is introduced,

ko ko from Tables I and 2, and Fig. 3. 4

a (1 - M2)M3

... (30)

4 Discussion

The effect of magnetic field dependent viscosity on ferroconvection is investigated using the Brinkman model and linear stability analysis is being carried out. It has been proved that oscillatory instability cannot occur. The permeability value of the porous medium has been taken using the values given by Walker and Homsy 11

• The magnetization parameter M 1 is taken to be 1000 (Vaidyanathan et al. 12

). For such fluids M 2 will have negligible value and hence taken to be zero . M3 is increased from I to 7, because M3 cannot take values less than one. 8, the coefficient of viscosity, is varied from 0.01 to 0.09.

From Tables I and 2, and from Figs 1-3, it is seen that as the magnetization parameter M3 is increased, the thermal Rayleigh number Nc= (RM1)c is found to decrease. This means, the system is getting destabilized. So, here also the magnetization

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7 Finlayson B A, J Fluid Mech, 40 ( 1971 ) 486.

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