Effect of temperature/gravity modulation on the onset of magneto-convection in electrically...

*Corresponding author. Tel.: #91-080-2220483; fax: #91-080-2259843.

E-mail address: [email protected] (P.G. Siddhesh-war).

Journal of Magnetism and Magnetic Materials 219 (2000) L153}L162

Letter to the Editor

E!ect of temperature/gravity modulation on the onset ofmagneto-convection in electrically conducting #uids with

internal angular momentum

P.G. Siddheshwar!,*, S. Pranesh"

!Department of Mathematics, UGC-DSA Center in Fluid Mechanics, Central College Campus, Bangalore University,Bangalore 560 001, India

"Department of Mathematics, Christ College, Hosur Road, Bangalore 560 029, India

Received 28 May 1999; received in revised form 23 May 2000

Abstract

The e!ect of time-periodic temperature/gravity modulation on the onset of magneto-convection in electricallyconducting #uids with internal angular momentum is investigated by making a linear stability analysis. The results of thepresent study are presented against the background of the results of weak electrically conducting #uids. The qualitative"ndings of Siddheshwar and Pranesh (J. Magn. Magn. Mater. 192 (1999) 159) are found to be true in the present case alsoexcept that the eigenvalue is found to be magnitudewise less than that obtained in the case of a weak electricallyconducting #uid. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Rayleigh}Benard; Natural convection; Modulation; Magnetic "eld; Linear stability

1. Introduction

The thermal/gravity modulation of convection isone of the important methods of controlling con-vection (see Ref. [1]). Siddheshwar and Pranesh [1]examined the e!ect of a transverse magnetic "eldon thermal/gravity convection in a weak electri-cally conducting #uid with internal angular mo-mentum. In this paper, we study the thermal andgravity modulation problems by extending ourearlier work [1] to electrically conducting #uids of

"nite conductivity. Fluids with internal angu-lar momentum such as the one considered herehave practical applications [1}3], more so in thepresence of external mechanisms like magnetic"eld/rotation, in regulating convection. It appearsthat experimental work relating to the present pa-per has not been performed.

2. Mathematical formulation

Consider a layer of a Boussinesquian, "nitelyelectrically conducting #uid with internal angularmomentum con"ned between two "nite horizontalwalls distant &¸' apart. It should be made clear herethat the #uid suspension comprises of two phases:

LETTER TO THE EDITOR

0304-8853/00/$-see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 4 3 8 - 8

Fig. 1. Physical con"guration.

electrically conducting carrier liquid and electri-cally inert solid particles of very small size. This isin contrast to ferro#uids in which the carrier liquidis electrically non-conducting and the solid par-ticles are "ne magnetic particles. The solid particlesin the present case are coated suitably to preventagglomeration. The suspension is assumed to sub-scribe to the Eringen microcontinuum descriptionas assumed in Ref. [1]. In the literature [4,5] the#uids under consideration are also called &mag-netomicropolar #uids'. The uniform magnetic "eldis directed along the Z-axis. A cartesian co-ordinatesystem is taken with origin in the lower boundaryand Z-axis vertically upwards (see Fig. 1). Thegoverning equations are [1,2]:Continuity equation

+ ) q"0, (1)

Conservation of linear momentum

oRCRqRt#(q )+)qD"!+P!ogkK #(2f#g)+2q

#f+]x#k.(H )+)H. (2)

Conservation of angular momentum

oRICRxRt #(q )+)xD"(j@#g@)+(+ )x)#g@+2x

#f(+]q!2x). (3)

Conservation of energy

R¹Rt #Aq!

boRC

V

+]xB )+¹"s+2¹. (4)

Equation of state

o"oR[1!a(¹!¹

R)]. (5)

Magnetic induction equation

RHRt #(q )+)H"(H )+)q#c

.+2H. (6)

Here q is the velocity, x is the spin, ¹ is thetemperature, H is the magnetic "eld, P is the hydro-magnetic pressure, o is the density, o

Ris the density

of the #uid at reference temperature ¹"¹R, g is

the acceleration due to gravity, f is the couplingviscosity coe$cient or vortex viscosity, g is theshear kinematic viscosity co-e$cient, I is the mo-ment of inertia, j@ and g@ are the bulk and shear spinviscosity coe$cients, b is the micropolar heat con-duction co-e$cient, C

Vis the speci"c heat, s is the

thermal conductivity, a is the coe$cient of thermalexpansion and c

."1/k

.p (k

.: magnetic permeab-

ility, p: electrical conductivity). All the viscositycoe$cients, heat conduction coe$cient and ther-mal conductivity are thermodynamically restrictedon the assumption of Clausius}Duhem inequalityand are all positive quantities.

The no-spin condition is assumed for micro-rotation at the stress-free boundary. The assump-tion of a spin-vanishing condition at a free surfaceis no doubt debatable but nevertheless is of greatutility in a qualitative understanding of the prob-lem with managable mathematics. This intentionleads us into assuming a spin-vanishing condition.The boundary conditions on magnetic "eld do nothave any in#uence on the convection as in the caseof a classical Newtonian #uid which is electricallyconducting (see Ref. [6]). The lower wall z"0 andthe upper wall z"¸ are subjected to the temper-atures

¹"¹R#1

2*¹[1#e cos ct] (7)

and

¹"¹R!1

2*¹[1!e cos(ct#u)], (8)

respectively, where e is the small amplitude, c is thefrequency and u is the phase angle.

L154 P.G. Siddheshwar, S. Pranesh / Journal of Magnetism and Magnetic Materials 219 (2000) L153}L162


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2.1. Basic state

The basic state of the #uid is quiescent and isdescribed by

qH"0, x

H"0, ¹"¹

H(z, t), P"P

H(z, t),

o"oH(z, t), H

H"H

0kK . (9)

The temperature ¹H, pressure P

Hand density

oH

satisfy

R¹HRt "s

R2¹H

Rz2 , (10)

!

RPHRz "o

Hg (11)

and

oH"o

0[1!a(¹

H!¹

R)]. (12)

The solution of Eq. (10) that satis"es the thermalboundary conditions (7) and (8) is

¹H"¹

R#

*¹

2¸(¸!2z)#e ReM[a(j)ejz@L

#a(!j)e~jz@L]e~*ctN, (13)

where

j"(1!i)Ac¸2

2s B1@2

, (14)

a(j)"*¹

2 Ce~*(!e~jej!e~j D

and Re stands for the real part.

2.2. Linear stability analysis

Let the basic state be disturbed by an in"nitesi-mal thermal perturbation. We now have

q"qH#q@, x"x

H#x@, o"o

H(z)#o@,

P"PH(z)#P@,

¹"¹H#h@, H"H

0kK #H@. (15)

The prime indicates that the quantities are in"ni-tesimal perturbations.

Substituting Eq. (15) into Eqs. (1)}(6) and usingthe basic state solution, we get the linearized equa-

tions governing the in"nitesimal perturbations inthe form

+ ) q@"0, (16)

oRARq@Rt B"!+P@#o@gkK #(2f#g)+2q@

#f+]x@#k.H

0

RH @zRz kK , (17)

oRIARx@Rt B"(j@#g@)+(+ )x@)#g@+2x@

#f(+]q@!2x@), (18)

Rh@Rt "A

R¹HRz BC

bX @z

oRC

V

!=@D#s+2h@, (19)

o@"!aoRh@, (20)

RH@Rt "H

0

R=@Rz kK #c

.+2H@. (21)

The perturbation Eqs. (16)}(21) are non-dimen-sionalized using the following de"nitions:

(xH, yH, zH)"Ax

¸

,y

¸

,z

¸B, tH"t

¸2/s,

=H"=@s/¸

, XH"[+]x@]

zs/¸3

,

hH"h

*¹

, HH"H@H

0

. (22)

Using Eq. (20) in Eq. (17) and operating curltwice on the resulting equation, operating curl onceon Eq. (18) and non-dimensionalizing the tworesulting equations and Eqs. (19) and (21), usingEq. (22), we get

1

Pr

RRt (+

2=)"R+21h#(1#N

1)+4=

#N1+2X

z#Q

Pr

Pm+2RH

zRz , (23)

N2

Pr

RXzRt "N

3+2X

z!N

1+2=!2N

1X

z, (24)

RhRt"A

R¹0Rz B [N

5X

z!=]#+2h, (25)

P.G. Siddheshwar, S. Pranesh / Journal of Magnetism and Magnetic Materials 219 (2000) L153}L162 L155


RHzRt "

R=Rz #

Pr

Pm+2H

z, (26)

where the asterisks have been dropped for simpli-city . The dimensionless groups are

M"Sk2.pH2

0¸2

f#g(Hartmann number),

N1"

ff#g

(coupling parameter) (0)N1)1),

N2"

I

¸2(inertia parameter) (0)N

2)r),

N3"

g@(f#g)¸2

(couple stress parameter)

(0)N3)m),

N5"

bo0C

V¸2

(micropolar heat conduction

parameter) (0)N5)n),

Pm"

f#gc.

(magnetic Prandtl number),

Pr"f#goRs

(Prandtl number),

Q"M2 (Chandrasekhar number)

and

R"

ag *¹oR¸3

(f#g)s(Rayleigh number).

In the above, the quantities r, m, and n are "nite,positive real numbers. The range of values of N

1,

N2, N

3and N

5speci"ed above is guided by the

Clausius}Duhem inequality discussed earlier.More discussion on these is presented in Section 6.The range of values for the other parameters is as inthe classical magnetoconvection problem involvinga Newtonian #uid (see Ref. [5]).

In Eq. (25), (R¹0/Rz) is the non-dimensional form

of (R¹H/Rz),

where

R¹0Rz "!1#e Re M[A(j)ejz

#A(!j)e~jz] e~*ctN (27)

and

A(j)"j2C

e~*r!e~jej!e~j D.

Eqs. (23)}(26) are solved subject to the conditions

="

R2=Rz2 "X

z"h"

RHzRz "0 at z"0, 1. (28)

Eliminating Xz, h and H

zfrom Eqs. (23)}(26), we

get a di!erential equation of order 10 for= in theform

G>1 CARRt!

Pr

Pm+2B A

1

Pr

RRt!(1#N

1)+2B>2

#ARRt!

Pr

Pm+2BN2

1+2!>

3D!R+2

1AR¹

0Rz B A

RRt!

Pr

Pm+2B A!N

5N

1+2

!

N2

Pr

RRt#N

3+2!2N

1BH="0 (29)

where

>1"A

RRt!+2B+2,

>2"A

N2

Pr

RRt!N

3+2#2N

1Band

>3"Q

Pr

Pm>

2

R2Rz2 .

In the dimensionless form, the velocity boundaryconditions for solving Eq. (29) are obtainable fromEqs. (23)}(26) and (28), by following Chandra-sekhar [7], in the form

="

R2=Rz2 "

R4=Rz4 "

R6=Rz6 "

R8=Rz8 "0

at z"0, 1. (30)

3. Stability analysis

We now seek the eigenvalues R of Eq. (29) for thebasic temperature distribution (27) that departs



from the linear pro"le R¹0/Rz"!1 by quantities

of order e. Thus, the eigenvalues of the presentproblem di!er from those of the ordinary Benardconvection by quantities of order e. We seek thesolution of Eq. (29) in the form

(R,=)"(R0,=

0)#e(R

1,=

1)#e2(R

2,=

2)

#2. (31)

The expansion (31) is substituted into Eq. (29)and the powers of e are separated. FollowingVenezian [8], we can obtain R

0, R

1and R

2in the

form

R0"

(1#N1)N

3K8

1#N

1(2#N

1)K6

1#Q[N

3K2

1#2N

1]K2

1p2

a2[(N3!N

5N

1)K2

1#2N

1]

,

(32)

R1"0 (33)

and

R2"!

R20a20

2Re

]+DA

2D2DB

n(j)D2 [¸

1(c, n)#¸H

1(c, n)]

2D¸1(c, n)D2

, (34)

where

A2"C

Pr

PmK2

n(N

1N

5K2

n!N

3K2

n!2N

1)

#

c2N2

Pr D#ic CN

2K2

nPm

!N1N

5K2

n

#N3K2

n#2N

1D ,

Bn(j)

"A(j)gn1

(j)#A(!j)gn1

(!j)

"

!2np2j2 [ej!e~j#(!1)n(e~j~*r!ej~*r)](ej!e~j) [j2#(n#1)2p2] [j2#(n!1)2p2]

,

K2n"n2p2#a2,

¸1(c, n)"(¹

1X

5#c2¹

2X

6)

#ic(¹2X

5!¹

1X

6),

¹1"!K2

n(K2

nX

1!c2X

2)

!QPr

PmK2

nn2p2X

3!R

0a2X

5,

¹2"K2

n(X

1#K2

nX

2)#Q

Pr

PmK2

nn2p2X

4

!R0a2X

6,

X1"

Pr

PmK2

n C(N3K2

n#2N

1)(1#N

1)K2

n

!

c2N2

Pr2!N2

1K2

nD!

c2Pr2

[N3K2

n#2N

1#N

2(1#N

1)K2

n],

X2"(N

3K2

n#2N

1)(1#N

1)K2

n!

c2N2

Pr2!N2

1K2

n

#

K2n

Pm(N

3K2

n#2N

1#N

2(1#N

1)K2

n),

X3"K2

n(N

3K2

n#2N

1)!

c2N2

Pr,

X4"N

3K2

n#2N

1#

K2nN

2Pr

,

X5"

Pr

PmK2

n(N

1N

5K2

n!N

3K2

n!2N

1)#

c2N2

Pr,

and

X6"

N2K2

nPm

!N1N

5K2

n#N

3K2

n#2N

1.

¸H1(c, n) in Eq. (34) is the conjugate of ¸

1(c, n).

4. Minimum Rayleigh number for convection

The value of R obtained by this procedure is theeigenvalue corresponding to the eigenfunction= that, though oscillating, remains bounded intime. Since R is a function of the horizontal wavenumber &a'. and the amplitude of perturbations e,we have

R(a, e)"R0(a)#e2R

2(a)#2. (35)



It was shown by Venezian [8] that the critical valueis determined to O(e2) by evaluating R

0and R

2at

a"a0. It is only when one wishes to evaluate

R4

that a2

must be taken into account wherea"a

2minimizes R

2. To evaluate the critical value

of R2

(denoted by R2#

) one has to substitute a"a0

in R2, where a

0is the value of a at which R

0given

by Eq. (32) is minimum.We now evaluate R

2#for three cases:

(a) When the oscillating "eld is symmetric so thatwall temperatures are modulated in phase with/"0. In this caseBn(j)"b

nor 0, accordingly as n is even or

odd.(b) When the wall temperature "eld is antisym-

metric corresponding to out-of-phase modula-tion with /"p. In this caseBn(j)"0 or b

n, accordingly as n is even or

odd.(c) When only the temperature of the bottom wall

is modulated, the upper plate being held atconstant temperature with /"!iR. In thiscase,

Bn(j)"

bn2

, for integer values of n

and

bn"

!4np2j2[j2#(n#1)2p2][j2#(n!1)2p2]

(36)

The variable j de"ned in Eq. (14), in terms of thedimensionless frequency, reduces to

j"(1!i) Ac2B

1@2

and thus

DbnD2"

16n2p4c2[c2#(n#1)4p4] [c2#(n!1)4p4]

. (37)

Hence, from Eq. (35) and using the above expres-sion of B

n(j), we can obtain the following expres-

sion for R2#

in the form

R2#"!

R20a20

2

]+DA

2D2DB

n(j)D2[¸

1(c, n)#¸H

1(c, n)]

2D¸1(c, n)D2

. (38)

In Eq. (38) the summation extends over evenvalues of n for case (a), odd values of n for case (b)and for all values of n for case (c). The in"nite series(38) converges rapidly in all cases.

In the next section we address ourselves to an-other important problem of gravity modulationusing the analysis of the previous section.

5. Gravity modulation

Under the in#uence of a periodically varyingvertical gravity "eld

g"g0(1#d cos c

1t), (39)

where g0

is the mean gravity, d is the small ampli-tude of gravity modulation, c

1is the frequency and

t is the time. The #uctuating gravity is referred to asg-jitter.

The governing equations for the Boussinesquian,electrically conducting #uid with internal angularmomentum are essentially the same as Eqs. (1)}(6)but with &g' given by Eq. (39).

Following the analysis in the previous section,the non-dimensional form of the linearized per-turbation equations, on using Eq. (39), is

1

Pr

RRt (+

2=)"R(1#d cos c1t)+2

1h

#(1#N1)+4=#N

1+2X

z

#QPr

Pm+2RH

zRz , (40)

N2

Pr

RXzRt "N

3+2X

z!N

1+2=!2N

1X

z, (41)

RhRt"=!N

5X

z#+2h, (42)

RHzRt "

R=Rz #

Pr

Pm+2H

z, (43)

where the parameters N1, N

2, N

3, N

5, R, Pr, Pm

and Q are as de"ned earlier.



Following the analysis of the previous section,we obtain R

2#in the form

R2#"

R20a20

2+

DA2D2[¸

1(c

1, n)#¸H

1(c

1, n)]

2D¸1(c

1, n)D2

, (44)

where ¸1(c

1, n)"(!¹

1X

5!c2

1¹

2X

6)#

ic1(¹

1X

6!¹

2X

5). R

0is essentially the same as

that of the previous section.The expressions for X

1, X

2, X

3, X

4, X

5, K2

nare

essentially the same as in the previous section, butwherever c is there it is to be replaced by c

1.

6. Results and discussion

In this paper we made an analytical study of thee!ects of temperature/gravity modulation andtransverse magnetic "eld at the onset of convectionin an electrically conducting #uid, of "nite conduct-ivity, with internal angular momentum. This is anextension of our earlier work on weak electricallyconducting #uids [1].

The solution obtained in this paper is based onthe assumption that the amplitude of the temper-ature/gravity modulation is small. Furthermore,the validity of the results depends on the valueof the modulation frequency c. When c(1 (i.e.,the period of modulation is large) the temper-ature/gravity modulation a!ects the entire volumeof #uid and hence the disturbance grows large. Onthe other hand, the e!ect of modulation disappearsfor large frequencies. This is due to the fact that thebuoyancy force takes a mean value leading to theequilibrium state of the unmodulated case. It mustbe noted here that the presence of suspended par-ticles is to increase the viscosity of the #uid accord-ing to the well-known Einstein [9] relation forsuspensions, namely k"k

0(1#2.5a/), where

k and k0

are the viscosities, respectively, of the #uidwith suspended particles (suspension) and #uidwithout suspended particles, a is the shape factorand / is the volume fraction of the suspendedparticles. The shape factor a is 1 for spherical par-ticles and greater than 1 for other shapes such asellipsoids, rods, etc. In order to understand theincrease in the viscosity of a suspension with theconcentration, it is helpful to realize that the viscos-

ity of any #uid is associated with dissipation ofmechanical energy into heat within the #uid. Theviscosity of a suspension also depends on variousother factors such as viscosity of the carrier liquid(suspending medium), the interaction between theparticles, on whether the particles are rigid or de-formable and on the temperature and electricalconductivity. In this paper we keep the analysis tothe bare minimum and do not venture into delinea-ting the above individual contributions to viscosity.We just assume that the viscosity of a suspension ishigher than that of the carrier liquid. The assumedviscosity}concentration relation is valid for lowconcentrations. In the case of higher concentrationsone will have to add a quadratic term in / to theEinstein relation. In the light of the above dis-cussion, we consider values of Prandtl number of#uids with suspended particles to be higher thanthose of the #uids without suspended particles.

Before embarking on a discussion of the resultsdepicted by the "gures let us make some commentson the parameters N

1, N

2, N

3and N

5arising due

to the suspended particles. Assumption of theClausius}Duhem inequality presents certain ther-modynamic restrictions leading to non-negative-ness of N

1, N

2, N

3and N

5. For f"0 (N

1"0) it is

clear that Eq. (23) for= becomes independent of G,i.e. it is uncoupled. As fPR, we see that N

1P1

and N3P0. This is the Stokesian description [10]

of suspension. Thus, it is obvious that couple stresscomes into play only at small values of N

3. This

supports the contention that N13 [0, 1] and that

N3

is a small positive real number. Coming toN

2and N

5, they have to be "nite because the

increasing of concentration has to practically stopsomewhere and hence N

2and N

5have to be posit-

ive, "nite real numbers. We now discuss the resultsarrived in the paper.

Fig. 2 is the plot of critical Rayleigh numberR

#versus frequency c for di!erent values of Chan-

drasekhar number Q and Prandtl number Pr, inrespect of in-phase temperature modulation. Weobserve that as Pr increases, R

#decreases. Thus, the

increase in the value of Pr is to destabilizethe system. Increase in Pr is due to increase inviscosity and increase in viscosity is due to severalfactors as discussed earlier. This means that the#uid with suspended particles is more vulnerable to



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Fig. 4. Plot of R#

versus c for di!erent values of N1, Q and Pr.

Fig. 3. Plot of R#

versus c for di!erent values of couplingparameter N

1, Q and Pr.

Fig. 2. Plot of critical Rayleigh number R#

versus frequency ofmodulation c for di!erent values of Chandrasekhar numberQ & Prandtl number Pr.

instability by modulation than the #uid withoutsuspended particles. It is also observed that theincrease in Q increases R

#. Thus, the e!ect of mag-

netic "eld is to stabilize the system.Fig. 3 is the plot of R

#versus c for di!erent values

of coupling parameter N1, Q and Pr in respect of

in-phase temperature modulation. The increase inN

1increases the concentration of suspended par-

ticles. These particles consume the greater part ofthe energy of the system in developing the gyra-tional velocity and thereby delay the onset of con-vection. Hence, increase in N

1increases R

#. Thus,

the increase in the concentration of the suspendedparticles stabilizes the system. It is observed that fora given value of N

1, R

#decreases for small values of

c and increases for moderate values of c. Thus,small values of c destabilize and moderate values ofc stabilize. This is due to the fact that when thefrequency of thermal modulation is low, the e!ectof modulation is felt throughout the #uid layer.

Fig. 4 is the plot of R#

versus c, for di!erentvalues of N

1, Q and Pr in respect of out-of-phase

temperature modulation. We observe that asN

1increases, R

#increases. For a given value of N

1,

R#

decreases for small values of c and almost re-mains same for moderate values of c. This is due tothe fact that in the case of out-of-phase modulation,the temperature "eld is essentially a linear gradientvarying in time, so that the Rayleigh number issupercritical for half a cycle and subcritical duringthe other half cycle.

The above results on the e!ect of N1, Q and Pr

on R#

for out-of-phase modulation do not changequalitatively for the case of temperature modula-tion of just the lower boundary. This is illustratedwith the help of Fig. 5.

We now discuss the results of gravity modula-tion. Unlike temperature modulation, which is atthe boundaries, gravity modulation a!ects the en-tire bulk of the #uid between the bounding plates.Fig. 6 shows the e!ect of N

1, Q and Pr on R

#and is

qualitatively similar to that of out-of-phase temper-ature modulation.

The above illustrative "gures reiterate the quali-tative "ndings, in terms of parameters in#uence onR

#, arrived at in respect of weak electrically con-

ducting #uids.From the above "gures we can conclude that in

the temperature/gravity modulation problem in-volving a "nitely conducting #uid, the criticalRayleigh number is less than that of the weak



Fig. 6. Plot of R#


Fig. 5. Plot of R#


electrically conducting #uid. This is true for clean#uids as well as for #uids with suspended particles.

We now present a few conclusions made by ana-lyzing the obtained results against the backgroundof the results of the weak conducting #uid [1]

1. R&*/*5% #0/$6#5*7*5:#

(R8%!, #0/$6#5*7*5:#

.2. In the case of out-of-phase temperature modula-

tion and lower wall modulation, the Prandtlnumber Pr in#uences the system in the "nitelyelectrically conducting #uid for smaller values ofc compared to that of the weak electrically con-ducting #uid.

3. In the "nitely conducting #uid R#

is almostindependent of c for very small values of c whichis not the case in weak electrically conducting#uids.

4. In the case of gravity modulation (from Fig. 6),in "nitely electrically conducting #uids, for largePr and small values of c

1, R

#decreases with c

1,

thereby destabilizing the system and sub-sequently increases with c

1for higher values of

c1

. In contrast, R#increases with c

1for large Pr

and for all values of c1

in the case of weakelectrically conducting #uids.

The comparison between the results of the presentpaper with those of Siddheshwar and Pranesh [1]has been possible because computations reveal thatthe e!ect of Pm on R

#is negligibly small. We also

note that Pm always occurs in a ratio with Pr, as itsdenominator.

The observed results of the present paper havebeen analyzed bearing in mind that in the case ofweak electrically conducting #uids the resistance to#ow (i.e., linear damping (see Ref. [1]) ) is highercompared to the "nitely electrically conductingcase. The above results suggest that instead of tak-ing electrically non-conducting #uids it is better toconsider electrically conducting #uids with weakelectrical conductivity as this ensures a stabler envi-ronment in the presence of a magnetic "eld. Theresults also suggest the need for experiments to studyelectrically conducting #uids with suspended par-ticles (both weak and "nitely conducting) in an ap-plication situation and alternately use ferro#uids.This might throw up interesting possibilities andpave way for a new understanding of di!erent typesof suspension in the presence of a magnetic "eld.

Acknowledgements

The work was supported by the UGC-DSA andUGC-COSIST programmes being implemented atthe Department of Mathematics, Bangalore Uni-versity, India. The second author (S.P.) would liketo acknowledge the support of the Christ Collegeadministration in deputing him to the Ph.D. pro-gramme. The authors wish to acknowledge themost useful and educative comments of the refereewhich exerted the grey area considerably and ledto substantial improvement of the quality of thepaper after two revisions.

References

[1] P.G. Siddheshwar, S. Pranesh, J. Magn. Magn. Mater 192(1999) 159.



[2] P.G. Siddheshwar, S. Pranesh, Int. J. Eng. Sci. 36 (1998)1173.

[3] P.G. Siddheshwar, S. Pranesh, Int. J. Eng. Sci. 36 (1998)1183.

[4] Lukaszewicz, Micropolar Fluid Theory and Applications,Birkhauser, Boston, MA, USA, 1998.

[5] A.C. Eringen, Microcontinuum Field Theory, Springer,Berlin, 1999.

[6] J.K. Platten, J.C. Legros, Convection in Liquids, Springer,Berlin, 1984.

[7] S. Chandrasekhar, Hydrodynamic, Hydromagnetic Stabil-ity, Dover, New York, 1961.

[8] G.I. Venezian, J. Fluid Mech. 35 (1969) 243.[9] A. Einstein, Eine neue Bestimmung der Molekuldimen-

sionen. Ann. Phys. 19 (1906) 289.[10] V.K. Stokes, Phys. Fluids 9 (1966) 1079.







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