Effect of magnetic field dependent viscosity on Soret-driven ferrothermohaline convection saturating an anisotropic porous medium of sparse particle suspension

Effect of magnetic field dependent viscosity onSoret-driven ferrothermohaline convectionsaturating an anisotropic porous medium ofsparse particle suspension

by

R. Sekar and K. Raju

reprinted from

WORLD JOURNALOF ENGINEERING

VOLUME 11 NUMBER 3 2014

MULTI-SCIENCE PUBLISHING COMPANY LTD.

World Journal of

EngineeringWorld Journal of Engineering 11(3) (2014) 213-228

World Journal of

Engineering

1. IntroductionFerrohydrodynamics is an interdisciplinary field

having inherent interest of physical andmathematical nature with applications in medicine,separations science, tribology, information display,instrumentation, printing and other fields(Rosensweig, 1985).

Ferrofluids are stable colloidal suspensionsconsisting of single – domain magnetic particlescoated with surfactant and immersed in a carrier

Effect of magnetic field dependent viscosity onSoret-driven ferrothermohaline convectionsaturating an anisotropic porous medium of

sparse particle suspension

R. Sekar* and K. RajuDepartment of Mathematics, Pondicherry Engineering College,

Pillaichavady, Puducherry 605 014, India*E-mail: [email protected]

(Received 2 June 2013; accepted 19 March 2014)Abstract

Thermoconvective instability with Soret effect in multi-component fluids has wide rangeof applications in heat and mass transfer. This work deals with the theoretical investigationof the effect of magnetic field dependent (MFD) viscosity on Soret-driven ferrothermohalineconvection heated and salted from below in an anisotropic porous medium subjected to atransverse uniform magnetic field. The resulting eigen value problem is solved usingBrinkman model. An exact solution is obtained for the case of two free boundaries and thestationary and oscillatory instabilities are investigated by using linear stability analysis andnormal mode technique for the vertical of anisotropic porous medium. The analysis has beenmade for different parameters like porosity, anisotropy, ratio of heat transport to masstransport, buoyancy magnetization, non-buoyancy magnetization, Soret parameter andSalinity Rayleigh number. The effect of MFD viscosity is assumed to be isotropy. It is foundthat the presence of MFD viscosity has a stabilizing effect, whereas magnetization has adestabilizing effect.

Key words: Thermohaline convection, Ferromagnetic fluid, Soret effect, Brinkman model,Magnetic field dependent viscosity, Anisotropy effect

World Journal of

Engineering

fluid. Ferrofluids are being greatly used in manymagnetic fluid based scientific devices likeaccelerometer, sensors, pressure transducers,magnetostatic support, jet printers, flow control, dragreduction, transducers and medical applications.Instability of thermal convection in a horizontal fluidlayer heated from below due to buoyancy force andthe effect of a uniform vertical magnetic field onsuch instability has been studied by Chandrasekhar(1961). The thermal convection in ferromagnetic

ISSN:1708-5284

214 R. Sekar and K. Raju/World Journal of Engineering 11(3) (2014) 213-228

fluid has been studied by Finlayson (1970), Lalasand Carmi (1971) and Gotoh and Yamada (1982).Later, Vaidyanathan et al. (1991) investigated theferroconvective instability of fluids saturating aporous medium. This work has extended to verticalanisotropic porous medium heated from below bySekar et al. (1996) for single component fluid. ParasRam et al. (2010) examined the effect of magneticfield dependent viscosity on the revolvingaxi–symmetric steady flow of ferrofluid in a disc byusing Neuringer–Rosensweig model for differentKarman’s number and the calculation of velocitycomponents and pressure for various values ofmagnetic field dependent viscosity have beencalculated. The stability of a conducting fluid in aporous medium, in the presence of a uniformmagnetic field is investigated using the Brinkmanmodel by Alchaar et al. (1995).

The study of convective instability offerromagnetic fluid with magnetic field dependentviscosity and vertical anisotropic porous mediumusing Darcy model have been analyzed byRamanathan and Suresh (2004). Vaidyanathan et al.(2002; 2002a) investigated the presence and absenceof porous medium on the effect of magnetic fielddependent viscosity and rotation on ferroconvectionof sparse particle suspension for two componentfluid. The ferroconvection induced by magnetic fielddependent viscosity in an anisotropic porous mediumusing Darcy model have been considered by SureshGovindan and Vasanthakumari (2009). Furtheranalysis has been carried out for comparison oftheoretical and computational ferroconvection.

The effect magnetic field dependent viscosity onferroconvection in dusty ferrofluids with andwithout porous and rotation have been studied bySunil et al. (2004; 2006; 2005). Also, Sunil et al.(2004) examined the thermal convection in aferromagnetic fluid in a porous medium with theeffect of magnetic field dependent viscosity. Anonlinear study of stability analysis for thermalconvection in a ferromagnetic fluid with magneticfield dependent viscosity has been investigated bySunil et al. (2008).

In binary fluid mixtures, Soret-driven convectionwas first studied by Hurle and Jakeman (1971) andmany other researchers (Legros et al., 1972;Gutkowicz-Krusin et al., 1979). An experimentalstudy of linear stability results for Soret convectionin binary fluid mixtures analyzed by Knobloch andMoore (1988). Bourich et al., (2002) considered theSoret effect on thermosolutal convection in a shallow

porous enclosure. In the presence of Soret effect, thethermal instability in an electrically conducting twocomponent fluid saturated anisotropic porousmedium has been investigated by Srivastava et al.(2012) by using generalized Darcy model.

The presence and absence of porous medium onthe effect of magnetic field dependent viscosity onthermosolutalconvection in ferromagnetic fluid hasbeen studied by Sunil et al., (2005a, 2005b).Vaidyanathan et al., (2005) examined the Soreteffect due to thermohaline convection in ferrofluid.Further, the presence and absence of anisotropyeffect on Soret-driven ferrothermohaline convectionin a porous medium of sparse particle suspensionhave analyzed by Sekar et al. (2006; 2013) formulti-component fluid and Darcy porous medium isstudied on ferrothermohaline convection with Soreteffect by Sekar et al. (2013a). The effect of coriolisforce on Soret-driven thermoconvective instabilityof ferromagnetic fluid in an anisotropic porousmedium of sparse particle suspension has analyzedby Sekar et al., (2013b) and this analysis has beenextended to the effect of magnetic field dependentviscosity by Sekar and Raju (2013).

Presently, the effect of magnetic field dependentviscosity on Soret-driven ferrothermohalineconvection in an anisotropic porous medium heatedand salted from below for multi-component fluid isconsidered. The conditions for the stationary andoscillatory instabilities are obtained by Brinkmanmodel. The distribution is assumed to be isotropicalong the horizontal plane and anisotropy along thevertical plane. The study is made for various cases.

2. Formulation of mathematical analysisA horizontal spread layer of an incompressible

ferromagnetic fluid of thickness ‘d’ in the presenceof transverse applied magnetic field heated andsalted from below is considered. The temperatureand salinity at the bottom and top surfaces z = +− d/2are T0 +− ∆T / 2 and S0 ± ∆S / 2, respectively (Fig. 1).Further the system is assumed to be anisotropy alongthe vertical direction with magnetic field dependentviscosity. Both boundaries are taken to be free andperfect conductors of heat and salt. The Soret effectis considered on the temperature gradient.

The fluid viscosity is assumed to be magneticdependent in the form (Vaidyanathan et al., 2002a;Ramanathan and Suresh, 2004)

(1)µ µ δ= +1

1( )B

R. Sekar and K. Raju/World Journal of Engineering 11(3) (2014) 213-228 215

where µ1 is viscosity of the fluid when the appliedmagnetic field is absent and δ is positive quantity.The coefficient of magnetic field dependentviscosity δ (i.e, δ1, δ2, δ3) has been taken to beisotropic. Hence the component wise µ can bewritten as: µx,y,z = µi (1 + δBi), (i = 1,2,3). As a firstapproximation for small field variation, linearvariation of magnetic viscosity has been used.Considering the mathematical equations governingthe above investigation are given below.

The continuity equation for an incompressibleBoussinesq fluid is

(2)

The corresponding momentum equation(Vaidyanathan et al. 2012a) is

(3)

The temperature equation for an incompressibleferrofluid is

(4)

The mass flux equation is given by

(5)ρ0

2 2( / )∂ ∂ + = ∇ + ∇t S K S S TS T

q

ρ µ

µ

0

0

C T

dT dt T T

v H o v H

v

, ,/

( / ) /

− ∂ ∂( )

+ ∂ ∂( )H M

M,,

. ( / )H

d dt

K T

H =

∇ +1

2 ϕ

ρ ρ

µµ δ

0

2 11

( / . ) .

( )

∂ ∂ + ∇ = −∇ + + ∇ ( ) +

∇ −+

t p

k

q q g HB

qB

q

∇ =. q 0

The Maxwell’s equations for non-conductingfluids are

(6)

Further B and H are related by

(7)

The density equation of state for Boussinesqmagnetic fluid is

(8)

Using Eq. (6), one can assume that themagnetization depends on the magnetic field,temperature and salinity, so that

(9)

In order to evaluate the partial derivatives ofmagnetization M, the linearized magnetic equationof state is

(10)

The basic state is assumed to be quiescent state.The basic state quantities are obtained bysubstituting velocity of quiescent state in Eqs.(2)–(5). The basic state quantities obtained are

M M H K T T K S S= + − − − + −0 0 0 2 0χ( ) ( ) ( ) ,H

MH

( , , ).= H

M H T S

ρ ρ α α= − − + −0 0 0

1[ ( ) ( )]t T T S SS

B H M= +( )µ0

∇ = ∇ × =. , B H0 0

Heated and salted from below

z-axis

Incompressible ferromagneticfluid saturating an anisotropicporous medium with magneticfield dependent viscosity

x-axisH = (0, 0, H0)

g = (0, 0, −g)

y-axis

z = +d/2

z = −d/2

Oz = 0

Fig. 1. Geometrical configuration.


where k is the vertical direction unit vector, βt and βs

are non-negative constants for temperature andsolute respectively. In the case of Finlayson (1970)and Sekar et al., (2013), only the spatial variation ofH0 and M0 are taken into account for the analysis.The perturbation equations can be obtained byimposing a small thermal perturbation on all thedynamical quantities.

The modified component wise Navier-Stoke’sequation can be obtained on linearization as

(12)

(13)

(14)

ρ µ

µ βµ β

0 0 0 03

0 30

2

∂∂

= − ∂∂

+ +∂∂

−

+

wt

pz

M HH

z

K HK

t

( )'

' tt

t

S T

KK S T KK S

T

S T

( )

( )

'

' '

1

1

1

1 10 2 0 2

−+

−−

+−

χ

µ βχ

µ β++

++

−

+ − +

χ

µ βχ

µ β

ρ α ρ α

0 22

0 2 3

0 0

1

K SK H

g T g S

SS

St

''

' ' µµ

µ δµµ

02

1 0 0 02

2

1

2

1

∇ +

+ ∇ −

−

w

M H wk

wk

w( )

ρ µ

µµ

0 0 0 02

02 1

1

∂∂

= − ∂∂

+ +∂∂

+

∇ −

vt

py

M HH

z

vk

v

( )'

ρ µ

µµ

0 0 0 01

02 1

1

∂∂

= − ∂∂

+ +∂∂

+

∇ −

ut

px

M HH

z

uk

u

( )'

(11)

qb b b ST T z S S z

z zt

t t

= = − = −

= + −

0

10 0

0

, , ,

( ) [

β βρ ρ α β αSS S b

bS

z p p z

H z HK z K zt

β

βχ

βχ

], ( )

( )

=

= −+

++

0

2

1 1

= ++

−+

k

k

,

( ) .M z MK z K z

bSt

02

1 1

βχ

βχ

The vertical component of Eq. (3) can becalculated as:

The perturbed linear form of Eq. (4) is

(16)

where

The salinity equation is

(17)

Using an analysis similar to Sekar et al., (2013)on gets

(18)

where

(19)

∇ = ∂ ∂ + ∂ ∂

∇ = ∇ + ∂ ∂

12 2 2 2 2

212 2 2

( / ) ( / )

( / ).

x y

z

and

( )1 12

20

012

2

+ ∂∂

+ +

∇ − ∂

∂+

∂∂

+

χ φ φ θz

M

HK

z

KSz

STT Kz

∂∂

=θ0.

∂∂

+ = ∇ + ∇St

w K S SS S Tβ θ( ) ( ).2 2

ρ ρ ρ0 0 0 0C C K Hv H= +, .

ρ θ µ φ θ

ρ β

0 0 0 12

0

Ct

K Tt z

K

c

v H

t

, ( )∂∂

− ∂∂

∂∂

= ∇ +

−−+

+

+

µ βχ

µ βχ

02

02

0 2 0

1 1

K T K K Tt S

w,

(15)

ρ ρ α θ ρ α

µ βχ

02

0 12

0 12

0

1

∂∂

∇ = ∇ − ∇ +

+

tw g g S

K

t

t

S( ) '

− ∇ − + ∂

∂∇

−

K Sz

K

T( ) ( ) ( ')1 112

12

0 2

θ χ φ

µ βSS

zK S

11 1

22 1

2

0

+

+ ∂

∂∇ + ∇

+

χχ φ

µ

( ) ( ') '

(( ( ))

( )

∇ ∇

++

− ∇ − ∇

2 2

0 212

12

11

w

KKSS T t

µχ

β θ β SS

kw

z kw

M H

'

( ) (

− ∂∂

− ∇

+ + ∇

µ µ

µ δµ

1

2

21

212

1 0 0 0 12 ∇∇ − ∇

2

2121

wk

w) ( )


(25)

The following non-dimensional numbers havebeen used

Then Eqs. (22)–(25) become

(28)

(29)Pr

∂∂

= − − +

−

St

D a S aR M w

S M M R

S

T

**

( ) * *

( /

/τ 2 2 1 26

5 61 RR D a TS ) ( ) *,/1 2 2 2−

PTt

Mt

D D a

T aR

r

∂∂

− ∂∂

= −

+

** *

( *) ( )

* (/

22 2

1 2

φ

11 2 2 5− −M M M w) *,

(27)

∂∂

− =

− + −

tD a w

a R M D M S TT

*( ) *

* ( ( ) *)/

2 2

1 21 11 1φ ++

− −

+ −

a R M M D a R M M S T

D a w

T1 2

1 51 2

1 5

2 2 2

1/ /* ( ) *

( ) *

φ

++ + +

− + −

−a R M M M

Sk

D wa

kw

S1 2

4 4 51

1

22

2

1

1

/

* ** * * aa M

D ak

w

23

2 2

2

1

δ *

**

− −

(26)

wwd

tt

dT

K aR

C dv H t* , * , * ,

/

,

= = =

ν

νρ β ν

θ2

11 2

0

φφχ

ρ β νφ*

( ), *

/

,

=+

=1 1

1 2

02

K aR

C K dz

zd

v H t

,,

,*

, */

,

a k d Dz

SK aR

C dSS S

v H S

= = ∂∂

=

0

1 2

0ρ β ν,,

, , .

* ( )

* *ν µρδ δµ χ

= = =

= +0

112 2

22

0 0 1

kk

dk

k

d

Hand

( )1 12

20

002

2

+ ∂∂

− +

−

∂∂

+ ∂∂

+

χ φ φ

θz

M

Hk

Kz

KSz

STT Kz

∂∂

=θ0.

Further analysis has been carried out as given inFinlayson (1970) and Sekar et al., (2005, 2013). Thenormal mode solutions of all dynamical variablescan be written as:

(20)

The wave number k0 is given by

(21)

With the help of Eqs. (20) and (21), Eqs. (15)-(18) become

(23)

(24)∂∂

+ = ∂∂

−

+ ∂

∂−

S

tw K

zk S S

zkS S Tβ

2

2 02

2

2 02

θ.

ρ θ µ φ0 0 0 1

2

2 02C

tK T

t zK

zkv H,

∂∂

− ∂∂

∂∂

= ∂∂

−

+ −+

+

+θ ρ β

µ βχ

µ β0

02

02

0 2 0

1 1c

K T K K Tt

t S

χχ

w

(22)

ρ ρ α ρ α θ

µ

0

2

2 02

0 02

0 02∂

∂∂∂

−

= − +

t zk w g k S g kS t

00 22 0

2

0

11

K

zK S k

K

S

t

βχ

χ φ

µ β

+

+ ∂

∂+

+

( )

111 1

02 0 2

+

+ ∂

∂− −

−

χχ φ θ

µ

( ) ( )z

K S

kKK

T

111 0

2

2

2 02

+

− −

+ ∂∂

−

χβ θ β

µ

S TS S k

zk

t( )

− ∂

∂+

+ +

2

1

2

21

2

02

1 0 0 0 02

1

wk

w

z k

k w M H k

µ µ

µ δµ ( )

kk zk w

2

2

2 02− ∂

∂−

k k kx y02 2= + .

f x y z t f z t i k x k yx y( , , , ) ( , )exp ( )= +( )


(30)

3. Free boundary solutionsThe boundary conditions for stress free non-

conducting boundaries are:

(31)

The exact solution satisfying Eq. (31) are

where A1, A2, A3 and A4 are constants and σ is thegrowth rate which is, in general, a complexconstant. Substituting of Eq. (32) in Eqs. (27) to(30), one gets

(34)

(35)

(36)− − +

+ +−

R S A R M

M A R aS T

S

1 2 22

1 2 25

61

31 2 2 2

1/ /

/

( )

(

π π

π MM A3 4

0) ,=

aR M A S M M R R a AS T S1 2

6 1 5 61 1 2 2 2

2

2

/ // ( )

(

+ ( ) +

+ +

− π

τ π aa P Ar

23

0) ,+ =σ

aR M M M A

P a A P M Ar r

1 22 2 5 1

2 22 2 4

1

0

/ ( )

( ) ,

− − −

+ + + =σ π σ

(33)

σ π π π

δ π

( ) ( )2 2 2 2 22

1

2

2

23

2 2

2

1

+ + + + + +

+ +

a ak

a

k

a M ak

− + − +

A

aR M M S MT

1

1 21 5 1

1 1/ ( ) (( )

( )/ /

1

12

1 24 4 5

13

1 2

1

− + + + +−

S

A aR M M M A aR

M

T

S

(( )1 05 4

+ =M A

(32)

w A e z

T A e z

S A e

t

t

t

* cos *,

* cos *,

* co

*

*

*

=

=

=

1

2

3

σ

σ

σ

π

π

ss *

* cos *,

* sin *

*

*

π

φ π

φπ

π

σ

σ

z

D A e z

Ae z

t

t

=

=

4

4

w D w T D S

z z

* * * * *

* / * / .

= = = = == − = +

2 0

1 2 1 2

φat and

D M a S DT

M M R R DS

T

S

23

2

5 61 1 2

1

0

φ φ* * ( ) *

/ * ,/

− − − +

( ) =−

The existence of non-trivial solutions of Eqs.(33)–(36) (Sekar et al., 2006; 2013) is given by

(37)

where

Making use of σ = 0 in Eq. (37), condition forsteady convection is obtained and substitution of σ

T a a M P

a P M

r

r

12 2 2 2

32

2 2 2

= − + + +

+

( )( )

( )

π π

π 222

22 2 2

1 5

1

1

1

π

π

τ

( )

( ) ( )

(r

−

= + +

+

S

T a M M

P

T

)) ( )

( )

( ( / )r

+ +

+

+

P a M A

P M

a

S M M

r

T

2 2 23

22

2 2 2

5 62

π

π

π

−−

−

− −

( )

( )r

1

1

3

S

AP S

T

T

T

τ

== + + + +( )+ +

( ) ( )( )

( )

π τ π τ

π

2 2 3 2 2

2 23

2

1a A P a

a M a R

r

SS

r

M M

M M a M P

a R a M

( /

) ( )

( )(

1 4 4

5 62 2

3

2 2 23

+ +

+

+ +

π

π 11

1 1 12 2 5

1 5

22

− −

+ − + −

M M M

S M M P

P M

T r

r

)

( ) ( )

(π π 22 2

5 62

21 5

2

1

1

+

− −( )+ +

a A

S M M S

a RM M P

T T

)

( / ) ( )

( )

τ

π rr

T

r

S M M M M

a P M MR

( ) ( )

(

1 1

1 1

5 2 2 5

2 22 5

− − − −

−

+

π−−

+

+

+ +

S

M M

R M M M

T

S

)

( )

( / )1 5

4 4 5

1

1

= +

+ − +( )T a R a M

S M MT

42 2 2

32

1 51 1 1

( )

( ) ( ) (

π

τ π 22 22 2 5

2 5 62

1

1

+ − −

−+ −( )

a M M M

a RS M M ST T

)( )

( / ) ( )τ

(( )

( ) ( )

1

1

2 2 5 5

2 2 21 5

− − +

+ +

+

M M M M

a M M

a

π22 2 2

4 4 5

5 6 2 2

1

1

R a M M M

S M M M M M

S

T

( )( / )

( / )(

π + + +

− − 55 62 2

3

2 2 2 2 23

) ( )

( ) ( )

+ +

− + +

M a M

a a M A

π

π τ π

and AA ak

ak

a M a M= + + + + +( ) ( )π π δ π2 2 22

1

2

2

23

2 23

− + + + =T T T T13

22

3 4 0σ σ σ .


= σ1 in Eq. (37) leads to marginal state ofconvection. Real and non-zero σ1 gives thecondition for oscillatory convection.

From Eq. (37), the Rayleigh number foroverstability can be easily obtained as (Sekar et al.,(2005; 2006; 2013))

(38)

where

The Eq. (37) helps to obtain the eigen value RSC

for which solution exists. If oscillatory instabilityexists, the time factor σ = iσ1. Since T1, T2, T3 andT4 are real, Eq. (37) could be satisfied for σ = iσ1 ifand only if σ1 = 0. The following analyses (Sekar etal. 2005; 2013), the eigen value equation for criticalRayleigh number R for stationary convection with k2

= εk1(ε is anisotropy parameter) is obtained bytaking σ1 = 0 and is given by:

(39)RXYSC =

c P a P A T

c a a M

r r12 2 2

2

22 2 2 2

1= + + + +

= + +

( )( ) ,

( )(

π τ

π π 332

2 22

2

32

1

)

( ) ( ),

(

P

a P M S

c

r

r T

−

+ −

= − +

π π

π aa AP a

a M

r2 32 2

2 23 1

)( )

( )( )

τπ

π τ+

+

+ +

( )

( ) ( )

+ +

+ − +

a R a M

S M MS

T

2 2 23

1 51 1 1

π

(( )+ + −

=

( ) ,

(

P a R M a M P T

c a

r S r2

62 2

3 3

42 2

π

π ++ + +

− −

a a a M

M M M

2 2 2 2 23

2 2 51 1

){ ( )( )

( )

τ π π

++ − +( )− +

( ) ( )

( ) ( / )

1 1

1

1 5

21 5 5 6

2

S M M

M M S M M

T

Tπ

( ) ( / ) ( )

1 12 2 52

5 6 1 5− − − + ×M M M M M M Mτπ

( )( ) ( ) },1 1 12 2 52

1 5 5

52 2

− − − − +

=

M M M S M M M

c a P

T

r

π

π MM M S M

M M S

T1 5 5

1 5

1 1

1 1 1

( )( )

( )(

+ − −( ) +

+ + − TT M M

c a a M

) ,

( ) ( ) .

( )= + +

2 5

62 2 2 2 2

3π π τ

R

c c c c c c

i cOC =

+ + +{ } +4 12

1 6 1 5 13

2 1 6

4 13

( ) ( )

(

σ σ σ σ

σ cc c c c c

c c

2 1 3 1 5 12

1 6

42

52

12

+ − +{ }+

σ σ σ

σ

) ( )

where

which express the critical Rayleigh number RSC

for stationary convection as a function of theanisotropy parameter ε permeability of the porousmedium k1, coefficient of magnetic fielddependent viscosity δ critical wave number a,Salinity Rayleigh number Rs , Soret coefficientST, ratio of mass transport to heat transport, ratioof magnetic force due to temperature fluctuationto the gravitational force M1, ratio of thermal fluxdue to magnetization to magnetic flux M2,measure of non–linearity in the magnetization M3,ratio of magnetic forces due to salinity fluctuationto gravitational force M4, ratio of salinity effecton magnetic field to pyromagnetic coefficient M5

and ratio of mass diffusivity to thermalconductivity M6.

When δ = 0 and ε = 1 one gets the Rayleighnumber identical to that of Sekar et al., (2006).When δ = 0, one gets the critical Rayleigh numbercalculated in Sekar et al., (2013). When k1 → ∞ andδ = 0 this tends to the critical Rayleigh numberobtained by Vaidyanathan et al., (2005) for multi-component fluid. When all the magnetic numbersM1 to M6 vanish, this reduces to double diffusiveconvection (Baines and Gill, 1969).

When M1 is very large, the critical magneticthermal Rayleigh number Nsc = (RM1)sc forstationary mode can be obtained using:

X a

aa

k

a M a

= +

+ + +

+

+

( )

( )

ππ επ

ε

δ π

2 2

2 2 22 2

1

23

2 22

1

2 14 4

1

1

+

− + +−

k

a R M MS

ε

τ ( MM

S M M M M M M

Y a S

T

51

5 61

2 2 5 6

2

1

1 1

−

− − − +

= + −

)

( )

[ ( TT M M

M M M

a M M

a M

) ( )

( )]

( )

1 5

2 2 5

21 5

2

2 23

1

1

1

+

− −

−+

+

π

π

− −

+

−

−

( )

/

( )

1

1

2 2 5

5

6

2

1

M M M

SM

M

S

T

T

τ

+

−M51τ


where

For physical consideration of thermal convection,ROC has to be real. Hence from Eq. (38), one gets

σ1 is to be real for overstability to occur. Using thisvalue of σ 1

2 the Rayleigh number for the onset offerroconvection oscillatory mode is obtained,which is

4. Discussion of resultsThe role of Soret effect on thermoconvective

instability of ferrofluid in an anisotropic porousmedium with magnetic field dependent viscosityis studied, for different values of MFD (magneticfield dependent) viscosity δ and δ is allowed tovary from 0.01 to 0.09 (Vaidyanathan et al., 2002;2002a). The linear stability analysis is applied forBrinkman model and small thermal perturbationmethod.

The Prandtl number Pr is taken as 0.01. Thepermeability of the porous medium k1 is taken as0.01 and 0.1. The magnetization parameter M1 isassumed to be 1000 (Sekar et al., 2013). For thesetypes of fluids M2 is insignificant and hence taken tobe zero. The measure of the non-linearity in themagnetization M3 is considered as 1(2)7 (Sekar etal., 1996). The ratio of magnetic forces due tosalinity fluctuation to gravitational force M4 and

Rc c c c c c

c cOC =+ + +{ }

+4 1

21 6 1 5 1

32 1 6

42

52

( ) ( )σ σ σ σ

σ112

.

σ12

5 6 3 4 2 4 1 5= − −( ) / ( ).c c c c c c c c

Y a M M M M

S aT

12

5 2 2 5

2

1 1

1 1

= + − −

− − +

( )( )

( ) ( MM

M M M

S M M

S

T

52

2 2 5

5 6

2

1

1

1

)

( )

/

(

π

τ

− −

( )+ −−

TT

M

a

)

/

(

+

+

−5

1

2

τ

π 223M )

NXYS C =

1

other magnetization parameters M5 and M6 areassumed to be 0.1 (Sekar et al., 2005; 2006; 2013).The range of salinity Rayleigh number RS is studiedfrom –500 to 500 and Soret coefficient ST is variedfrom –0.002 to 0.002 and τ, the ratio of masstransport to heat transport from 0.03 to 0.11 (Sekaret al., 2013). The discussions from the aboveanalysis are as under:

Figures 2 (a) and (b) plotted the critical thermalmagnetic Rayleigh number Nc versus the coefficientof MFD viscosity δ for various values of anisotropyparameter ε, permeability of the porous medium k1

= 0.1 and 0.01 respectively. Both figures exhibit astabilizing trend. This is indicated, as Nc increases, δand ε increases. From Figure 2 (b), it is found that

24.0

23.5

23.0

22.5

NC (

thou

sand

s)

22.0

21.5

21.0

20.5

0.00 0.02 0.04 0.06

δ0.08

K1 = 0.01

K1 = 0.1

0.10

ε = 0.3 ε = 2.1ε = 3.1ε = 0.7

ε = 1.3

45

40

35

NC (

thou

sand

s)

30

25

200.00 0.02 0.04 0.06

δ0.08 0.10

ε = 0.3 ε = 2.1ε = 3.1ε = 0.7

ε = 1.3

Figures 2 (a) The effect of magnetic field on the variation ofNc Vs δ for different ε, Rs = −500, ST = −0.002, τ = 0.03 and

k1 = 0.1. (b) The effect of magnetic field on the variation of Nc

Vs δ for different ε, Rs = −500, ST = −0.002, τ = 0.03and k1 = 0.01.


the early convection takes place because of theincrease in pore size due to Brinkman model.

Figure 3 (a) and (b) represent the plots of criticalmagnetic thermal Rayleigh number Nc versus RS forvarious values of δ, k1 = 0.1 and 0.01, respectively.Figure 3 (a) indicates a decrease in cell shape assalinity Rayleigh number RS is increased from -500to 0, and then it is increase phenomenonally. WhenRS increases from 0 to 500, the system has astabilizing behavior and which is not muchpronounced. In Figure 3(b), when the increasingvalue of RS from –500 to –100, Nc gets drasticdecreasing values and it then increasesphenomenonally. It is very clear that the system has

a stabilizing effect and which is much pronouncedwhen the increasing value of RS from –100 to 500.From both Figures 3 (a) and (b), the system gets thestabilizing behavior at the positive range of salinityRayleigh number RS. This is because when salteringis increasing on the convective system, the systemgets stabilizing behavior.

Figure 4 (a) shows the variation of Nc versus theratio of mass transport to heat transport τ for variousδ and for k1 = 0.1. When τ increases from 0.03 to0.11, there is a decrease in Nc promoting instabilityonly at δ = 0.01 Whereas increasing value of δ from0.03 to 0.09, Nc gets decreasing value up to τ = 0.05,it then increases very slowly and the system gets the

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

6004002000

Rs

−200−400−600

600400200

15

20

25

30

35

40

45

50

0

Rs

−200−400−600

2

3

4

5

6

7

8

9

10

NC

(tho

usan

ds)

NC

(tho

usan

ds)

k = 0.1

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

k = 0.1

Figures 3 (a) The effect of magnetic field on the variation of Nc

Vs RS for different δ, ST = −0.002, τ = 0.03, M3 = 1, ε = 0.3and k1 = 0.1. (b) The effect of magnetic field on the variation of

Nc Vs RS for different δ, ST = −0.002, τ = 0.03, M3 = 1, ε =0.3 and k1 = 0.01.

k = 0.1

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

20

10

00.02 0.04 0.06 0.08 0.10 0.12

τ

NC

(tho

usan

d)

k = 0.1

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

45

40

35

30

25

20

15

10

5

0.02 0.04 0.06τ

0.08 0.10 0.12

NC

(tho

usan

d)

Figures 4 (a) The effect of magnetic field on the variation ofNc Vs τ for different δ, RS = –500, ST = –0.002, M3 = 1, ε =

0.3 and k1 = 0.1. (b) The effect of magnetic field on thevariation of Nc Vs τ for different δ, RS = –500, ST = –0.002,

M3 = 1, ε = 0.3 and k1 = 0.01.


commencement of stability. It is observed fromFigure 4 (b) that the increase in τ, the ratio of heattransport to mass transport stabilizes the system forincreasing value of δ. This is because the increase inmass transport to heat transport adds up to the systemto be top heavy. Therefore, from the Figures 4 (a) and(b), there is a competition between stabilizing anddestabilizing behaviors for various porous medium k1 = 0.1 and 0.01. Obviously, when increasing valuesof k1, the system gets stabilizing effect.

In Figures 5 (a), (b) and (c) represent the variationof Nc versus δ for different coefficient of Soretparameter ST, k1 and ε. It is obvious from thesefigures that coefficient of MFD viscosity δ has astabilizing behavior in the system. The stabilizingbehavior of δ is much pronounced. This means that,

as δ, ST, k1 and ε increases from 0.01 to 0.09, -0.002to 0.002, 0.01 to 0.1 and 0.3 to 3.1, respectively, NC

increases. This is because the effect of Soretparameter ST, which provides additionaltemperature gradient by cross diffusion of salinityon temperature.

In the Figures 6 (a), (b) and (c), the variation ofNc versus M3 is analyzed, for different value of δ isincreased from 0.01 to 0.09, ε is increased from 0.3to 3.1 and k1 is increased from 0.01 to 0.1. It isobserved that the system gets the destabilizing effectas Nc decreases. It is also seen from the figures thecritical thermal magnetic Rayleigh number Nc isreduced due to M3. This is because magnetizationreleases extra energy which adds up to the thermalenergy to promote convection.

ST = −0.002

ST = 0.002

ST = −0.001

ST = 0.001ST = 0

= 0.3ε k1 = 0.1

23.0

22.5

22.0

21.5

21.0

20.5

0.100.080.060.04

δ0.020.00

NC (

thou

sand

)

ST = −0.002

ST = 0.002

ST = −0.001

ST = 0.001ST = 0

= 1.3εk1 = 0.0129

28

27

26

25

240.100.080.060.04

δ0.020.00

NC (

thou

sand

)

ST = −0.002

ST = 0.002

ST = −0.001

ST = 0.001ST = 0

= 3.1εk1 = 0.01

0.100.080.060.04

δ0.020.00

47

46

45

44

43

42

41

40

NC (

thou

sand

)

Figures 5 (a) The effect of magnetic field on the variation of Nc Vs δ for different ST, Rs = −500, τ = 0.03, ε = 0.3 and k1 = 0.1. (b) The effect of magnetic field on the variation of Nc Vs δ for different ST, Rs = −500, τ = 0.03, ε = 1.3 and k1 = 0.01.(c) The effect of magnetic field on the variation of Nc Vs δ for different ST, Rs = −500, τ = 0.03, ε = 3.1 and k1 = 0.01.


5. ConclusionThe effect of magnetic field dependent viscosity

on convective instability of a multi-componentferrofluid with Soret effect saturating a sparselydistributed anisotropic porous medium has beenanalysed using the Brinkman model. In thisinvestigation, we have studied the effect of variousparameters like permeability of the porous mediumk1, anisotropic parameter ε coefficient of magneticfield dependent viscosity δ, ratio of mass transportto heat transport τ , buoyancy magnetization M1,non–buoyancy magnetization M3, thermal Rayleighnumber R, salinity Rayleigh number RS andmagnetic numbers M2, M4, M5 and M6 on the onsetof convection. The thermal critical magneticRayleigh numbers for the onset of instability arealso determined numerically for sufficient large

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

16

14

12

10

8

6

4

2

1 2 3 4 5 6 7

NC (

thou

sand

s)

NC (

thou

sand

s)

M3

1

45

40

35

30

25

20

2 3 4 5 6 7

M3

k1 = 0.1 = 1.3ε

k1 = 0.01 = 1.3ε

δ = 0.01δ = 0.03δ = 0.05δ = 0.07δ = 0.09

NC (

thou

sand

s)

1

282624222018161412108

2 3 4 5 6 7M3

k1 = 0.01 = 1.3ε

Figures 6 (a) The effect of magnetic field on the variation of NC Vs M3 for different δ, RS = −500, ST = −0.002, τ = 0.03, ε = 0.3 and k1 =0.1. (b) The effect of magnetic field on the variation of NC Vs M3 for different δ, RS = −500, ST = −0.002, τ = 0.03, ε = 0.3 and k1 =

0.01 (c) The effect of magnetic field on the variation of NC Vs M3 for different δ, RS = -500, ST =-0.002, τ = 0.03, ε = 0.3and k1 = 0.01.

values of buoyancy magnetization parameter M1 andresults depicted graphically and numerically. Theprinciple of exchange of instability is applied to findout the mode of attaining instability. Both stationaryand oscillatory instabilities values have beencalculated and it has been found that the oscillatoryinstability cannot occur and the system was found tostabilize through stationary mode for variousparameters.

For the case of stationary convection, MFDviscosity has a stabilizing effect. In the absence ofMFD viscosity, the non-buoyancy magnetizationalways has a destabilizing effect, whereas in thepresence of MFD viscosity, there is a competitionbetween the stabilizing role of MFD viscosity andthe destabilizing role of non-buoyancymagnetization.


It is observed from the Tables 1–2 and Figures 2–5that increase in RS, ST, τ, δ and ε, the critical thermalmagnetic Rayleigh number NC gets increasing trend.Therefore, the system has a stabilizing behavior fork1 = 0.1 and 0.01. Whereas the non-buoyancy

magnetization M3 is found to destabilize the systemas it works in unison with thermal gradient. This isobserved as the decrease in NC for consequentincrease in M3. This can be seen in Tables 1–2 andFigures 6 (a), (b) and (c).

Table 1.Stationary mode of instability of multi-component ferrofluids for sparsely packed anisotropic porous medium having k = 0.1, M1 =1000, M2 = 0, M4 = M5 = M6 = 0.1.


AcknowledgementsThe authors are grateful to Prof. D.

Govindarajulu, Principal, Pondicherry EngineeringCollege, Puducherry for his constantencouragement. The author K. Raju is thankful toUGC for grant of Rajiv Gandhi National Fellowship2010–2011 (Award letter number: F. 14-2(SC)/2010(SA-III), Dated: May 2011).

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Nomenclature:B magnetic induction T

Cv, H effective heat capacity at constant volume and magnetic

field (kJ/m3K)

d thickness of the fluid layer m

D/Dt convective derivative s−1[D/Dt = ∂/∂t + q→

.∇]

g gravitational acceleration (0, 0, −g) ms−2

H magnetic field amp/m

k permeability of the porous medium (k1, k1, k2)

K1 thermal diffusivity W/mK

Ks concentration diffusivity W/mkg

K permeability of the porous medium

K0 resultant wave number k k k mx y02 2 1= +

−


kx, ky wave number in the x and y direction m−1

K pyromagnetic coefficient [≡ –(∂M / ∂T)H0,T0]

K2 salinity magnetic coefficient [≡ (∂M / ∂S)H0,T0]

M magnetization Ampm−1

M0 mean value of the magnetization at H = H0 and T = T0.

P hydrodynamic pressure (N/m2)

q velocity of the ferrofluid (u, v, w) ms−1

S Solute concentration kg

ST Soret coefficient

T temperature K

T time s

αt coefficient of thermal expansion K −1

αs analogous solvent coefficient of expansion K−1

βt uniform temperature gradient Km−1

βs uniform concentration gradient kgm−1

µ0 magnetic permeability of vacuum

µ dynamic viscosity kgm−1s−2

ρ0 mean density of the clean fluid kgm−3

ρ density of the fluid kgm−3

σ growth rate s−1

ϕ viscous dissipation factor containing second order

terms in velocity

φ magnetic scalar potential Amp

θ perturbation in temperature (K)

χ magnetic susceptibility [≡(∂M / ∂H)H0,T0]

vector different operator

Non – dimensional parameters M1 = [µ0K2βt / (1+χ) ρ0gα t] ratio of magnetic force due to

temperature fluctuation to the gravitational force

M2 = [µ0K2T/(1+χ)ρ0Cv,H] ratio of thermal flux due to

magnetization to magnetic flux

M3 = [(1+M0 / H0) / (1+χ)] measure of non – linearity in the

magnetization

M4 = [µ0K2βs / (1+χ) ρ0gαs] ratio of magnetic forces due to

salinity fluctuation to gravitational force

M5 = [K2βs / Kβt] ratio of salinity effect on magnetic field to

pyromagnetic coefficient

M6 = [Ks / K1] ratio of mass diffusivity to thermal

conductivity

Pr = [µCv,H/ K1] Prandtl number

R = [ρ0Cv,Hβtαtgd4 / vK1] thermal Rayleigh number

Rs = [ρ0Cv,Hβsαsgd4 / vKs] salinity Rayleigh number

τ = ρ0Cv,HKs / K1 ratio of mass transport to heat transport

Effect of magnetic field dependent viscosity on Soret-driven ferrothermohaline convection saturating an anisotropic porous medium of sparse particle suspension

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