Hall effect on Ferrofluids -...

CHAPTER-2

2.1 COMBINED EFFECT OF SUSPENDED PARTICLES, ROTATION

AND MAGNETIC FIELD ON THERMOSOLUTAL CONVECTION

IN RIVLIN-ERICKSEN ELASTICO-VISCOUS FLUID IN POROUS

MEDIUM

2.2 EFFECT OF SUSPENDED PARTICLES, ROTATION AND

MAGNETIC FIELD ON THERMOSOLUTAL CONVECTION IN

RIVLIN-ERICKSEN FLUID WITH VARYING GRAVITY FIELD IN

POROUS MEDIUM

37

COMBINED EFFECT OF SUSPENDED PARTICLES, ROTATION AND

MAGNETIC FIELD ON THERMOSOLUTAL CONVECTION IN RIVLIN-

ERICKSEN ELASTICO-VISCOUS FLUID IN POROUS MEDIUM

2.1.1 INTRODUCTION

The problem of thermosolutal convection has been extensively studied in the recent past

on account of its interesting complexities as a double diffusive phenomenon as well as its direct

relevance in many problems of practical interest in the field of astrophysics, chemical

engineering and oceanography etc. A detailed account of the theoretical and experimental study

of thermal instability in Newtonian fluids, under varying assumptions of hydrodynamics and

hydromagnetics, has been given by Chandrasekhar [11]. The use of Boussinesq approximation

has been made throughout, which states that the density may be treated as a constant in all the

terms in equations of motion except the external force term. Veronis [104] has investigated the

thermosolutal convection in a layer of fluid heated from below and subjected to a stable solute

gradient.

Chandra [199] observed a contradiction between the theory for the onset of convection in

fluids heated from below and his experiment. Scanlon and Segel [169] studied the effect of

suspended particles on the onset of Bénard convection and found that the critical Rayleigh

number was reduced solely because the heat capacity of the pure gas was supplemented by that of

the particles. Sharma [200] has studied the thermal instability of a layer of viscoelastic

(Oldroydian) fluid acted on by a uniform rotation and found that rotation has destabilizing as well

as stabilizing effects under certain conditions in contrast to that of a Maxwell fluid where it has a

destabilizing effect.

There are many elastico-viscous fluids that cannot be characterized by Maxwell’s

constitutive relations or Oldroyd’s constitutive relations. Two such classes of fluids are Rivlin-

Ericksen and Walter’s (model B΄) fluids. Rivlin and Ericksen [69] have proposed a theoretical

model for such one class of elastico-viscous fluids. Sharma and Kumar [201] have studied the

effect of rotation on thermal instability in Rivlin-Ericksen elastico-viscous fluid whereas the

thermal convection in electrically conducting Rivlin-Ericksen fluid in presence of magnetic field

has been studied by Sharma and Kumar [202]. Sharma and Aggarwal [203] have studied the

38

effect of compressibility and suspended particles on thermal convection in a Walters’ (model B΄)

elastico-viscous fluid in hydromagnetics. Aggarwal [204] has studied the effect of rotation on

thermosolutal convection in a Rivlin-Ericksen fluid permeated with suspended particles in porous

medium.

The medium has been considered to be non-porous in all above studies. In recent years,

the investigation of flow of fluids through porous media has become an important topic due to the

recovery of crude oil from the pores of reservoir rocks. When the fluid permeates a porous

medium, the gross effect is represented by the Darcy’s law. As a result of this macroscopic law,

the usual viscous and viscoelastic terms in the equation of motion are replaced by the resistance

term ,1

1

q

tk where and are the viscosity and viscoelasticity of the Rivlin-

Ericksen fluid, 1k is the medium permeability and q is the Darcian (filter) velocity of the fluid.

In recent years, the investigations on flow of visco-elastic fluids, such as oils, paints,

some organic liquids and polymer solutions, through porous medium play an important role in

many engineering technological fields and modern industries and their study has become

important and attracted the researchers. A great number of applications in geophysics may be

found in a book by Phillips [205].

In the present section, we have considered the effect of suspended particles, rotation and

magnetic field on thermosolutal convection in Rivlin-Ericksen elastico-viscous fluid in porous

medium. Here, we have extended the results reported by Sharma et al. [206] to include the effect

of suspended particles for Rivlin-Ericksen fluids.

2.1.2 MATHEMATICAL FORMULATION

Consider an infinite horizontal fluid particle layer of Rivlin-Ericksen elastico-viscous

fluid of thickness d bounded by the planes 0z and dz in porous medium, is acted upon by

a uniform rotation ) (0,0,Ω and a uniform magnetic field ).,0,0( HH as shown in figure 2.1.1.

This layer is heated from below and subjected to stable solute gradient such that a uniform

temperature gradients dz

dT and solute concentration gradients dz

dC are maintained

across the layer.

39

Figure 2.1.1: Geometrical Configuration

The equations of motion and continuity governing the flow are

HHΩq

qqqgqqq

)(4

2

11

111

0

'

1000

πρ

μ

ε

tυυ

kερ

KN

ρ

δρp

ρεtε

e

d

(2.1.1)

0 q (2.1.2)

where q and, p denote respectively, the density, the pressure and the filter velocity of the pure

fluid; ),(and txNdq denote the velocity and number density of the suspended particles,

respectively. ,6πμηK where is the particle radius, is the Stokes’ drag coefficient,

),,( zyxx and )1,0,0(λ . Let gk and,,, 1 stand for medium porosity, medium

permeability, viscosity of fluid, viscoelasticity of fluid and acceleration due to gravity,

respectively.

In writing equation (2.1.1) we have assumed uniform size of fluid particles, spherical

shape and small relative velocities between the fluid and particles. Then the net effect of the

suspended particles on the fluid through porous medium is equivalent to an extra body force term

40

per unit volume qq

d

KN. Since the force exerted by the fluid on the particles is equal and

opposite to that exerted by the particles on the fluid, there must be an extra force term, equal in

magnitude but opposite in sign, in the equation of motion of the particles. The distances between

particles are assumed to be so large compared with their diameter that interparticle reactions need

not be accounted for. The effects of pressure, gravity and Darcian force on the suspended

particles (assumed large distances apart) are negligibly small and therefore ignored. If mN is the

mass of particles per unit volume, then the equations of motion and continuity for the particles,

under the above assumptions, are

ddd

d KNt

mN qqqqq

1 (2.1.3)

0

dN

t

Nq (2.1.4)

Let qqCTcc ptand,,,, denote respectively the specific heat of fluid at constant pressure,

specific heat of fluid particles, temperature, solute concentration, effective thermal conductivity

of the pure fluid and an analogous effective solute conductivity. If we assume that the particles

and the fluid are in thermal and solute equilibrium, then the equation of heat and solute

conduction gives

TTt

εmNcTcρt

Tεcρεcρ dptss

2q1

qq (2.1.5)

CCt

εmNcCcρt

Cεcρεcρ dptss

2q1

qq (2.1.6)

where ss c, are the density and the specific heat of the solid material respectively, and ε is

medium porosity. The equation of state for the fluid is given by

000 1 CCαTTαρρ , (2.1.7)

where 000 and, CT are the density, temperature and solute concentration of the fluid at bottom

surface ,0z is the coefficient of thermal expansion and is the analogous coefficient of

solvent expansion.

41

2.1.3 THE PERTURBATION EQUATIONS

The initial state of the system, denoted by subscript 0, is taken to be a quiescent layer (no

setting) with a uniform particle distribution 0N i.e. 0,0,0,0,0,0 dqq and 0NN is a

constant. The character of the equilibrium of this initial static state can be determined, as usual,

by supposing that the system is slightly disturbed and then by following its further evolution.

The steady state solution to the governing equations is

zβαzαβρρzβCCzβTTd 1,,,0,0,0,0,0,0 000qq .

Let Nsrlwvup d and,,,,,,,,, qq denote respectively the small perturbations in

density , pressure ,p temperature ,T fluid velocity )0,0,0(q , fluid particle velocity

),0,0,0(dq solute concentration C and number density 0N . Then the linearized perturbation

equations of motion, continuity, heat conduction and solute conduction for Rivlin-Ericksen

elastic-viscous fluid-particle layer are

HhΩq

qqqq

o

e

o

d

oo tK

KNgp

t

4

2

111

01

λ

(2.1.8)

0 q (2.1.9)

00

dN

t

Nq (2.1.10)

qq

d

tK

m1 (2.1.11)

2

hsw

thE (2.1.12)

2

hsw

thE (2.1.13)

where, 00000

0 and,,,,,

c

q

c

qmNf

c

fch

pt

In writing equation (2.1.8), use has been made of the Boussinesq equation of state

0 .

42

2.1.4 EXACT SOLUTION AND DISPERSION RELATION

Analyzing the perturbations into normal modes by seeking solutions in the form

)exp()(),(),(),(),(),(,,,,, ntyikxikzXzZzKzzzWhw yxz (2.1.14)

where yx kk , are the wave numbers along x and y directions respectively, and 21

22 )( yx kkk is

the resultant wave number of the disturbance and n is the growth rate which is , in general , a

complex constant and y

h

x

hξ

y

u

x

vζ xy

and

stand for the z-components of vorticity and

current density, respectively. Eliminating the physical quantities using the non-dimensional

parameters ,kda

2nd

,

1p ,

κ

υp

1,

21

d

τυτ ,

0

0

mNM ,

2p , ,

2

1

d

kpl

,2d

F

hEE 1 , dDDhHhEE *

2 ,1, and dropping (*) for convenience,

the linearized dimensionless perturbation equations are

DKaDυπρ

HdμDZ

υε

d

ααυ

dgaWaD

P

σF

στ

M

ε

σ

e

l

22

0

3

2222

1

4

2

1

11

(2.1.15)

DXHd

DWd

ZP

FM e

l

01 4

21

11

(2.1.16)

WHd

pEaD

1

1

2

11

22

1 (2.1.17)

WHd

pEaD

1

1

2''

12

22

1 (2.1.18)

DWHd

KpaD

2

22 (2.1.19)

DZHd

XpaD

2

22 (2.1.20)

43

Eliminating ,,, KZ and X between the equations (2.1.15) – (2.1.20), we obtain

WDaDpEaD

pEaDQDpaDP

FMQ

WDpaDpEaDpEaDT

pEaDSpEaDR

QDpaDP

FMpaDa

H

WaDpaDpEaDpEaD

P

FMQDpaD

P

FM

l

A

l

ll

222

12

22

11

222

2

22

1

22

2

22

12

22

11

22

2

11

22

12

22

2

2

22

1

2

222

1

1

22

2

22

12

22

11

22

1

2

2

22

1

.1

11

.1

11..

1

1

11

1

11

(2.1.21)

where,

4dgR is the thermal Rayleigh number,

4dgS is the analogous solute

Rayleigh number,

222

dTA is the Taylor number and

0

22

4

dHQ e is the Chandrasekhar

number.

Consider the case where both the boundaries are free from stresses and perfect conductors

of heat and solute, the medium adjoining the fluid is perfectly conducting and temperatures at the

boundaries are kept fixed. The boundary conditions appropriate for the problem are

(Chandrasekhar [11])

,02 WDW ,0 1and0when0 zDK (2.1.22)

Proper solution of W characterizing the lowest mode is

zWW sin0 where 0W is a constant, (2.1.23)

Substituting the proper solution (2.1.23) in equation (2.1.21), we obtain the dispersion relation

44

21

111

2

11

2

111

121

2

1

2

11

1

11121

2

11

2

11

2

1112

11

2

11

2

1

2

11

11

1

'

11

1111

1

111

11

11

1)1(1

11.

11

11

1

1

1

pix

pEx

x

x

iH

iQ

QpixP

iF

i

Mi

piExpix

iH

i

x

T

x

xpiEx

iH

i

P

iF

i

MiS

piEx

piExR

A

(2.1.24)

.and,,,,,where 2

212

2

4214141 1 lA PPia

xT

TQ

QS

SR

R

Equation (2.1.24) is the required dispersion relation including the effects of magnetic field,

rotation, medium permeability, kinematic viscosity and stable solute gradient on the

thermosolutal instability of Rivlin-Ericksen fluid in porous medium in hydromagnetics. In the

absence of suspended particles, the dispersion relation (eq. (2.1.24)) reduces to the results

reported by Sharma et al. [206].

2.1.5 THE STATIONARY CONVECTION

For stationary convection, the marginal state will be characterized by 0 . Thus

equation (2.1.24) reduces to

1

2

21111

1111

QP

x

x

xH

TQ

P

x

xH

xSR

A

(2.1.25)

The above relation expresses the modified Rayleigh number 1R as a function of the

modified solute gradient parameter ,1S modified magnetic field parameter ,1Q suspended

particles parameter ,H modified rotation parameters ,1AT medium permeability parameter P

and dimensionless wave number x . The parameter F accounting for the kinematic viscoelasticity

vanishes for the stationary convection. To study the effect of stable solute gradient, magnetic

field, suspended particles, rotation and permeability, we examine the nature of

1

11

1

1

1

1 ,,,AdT

dR

dH

dR

dQ

dR

dS

dR and

dP

dR1 analytically. Equation (2.1.25) yields

45

11

1 dS

dR (2.1.26)

1

2

2

1

1

11

1 QP

x

x

HxdT

dR

A (2.1.27)

1

1

22

1

1

1111 Q

QP

x

xT

P

x

xH

x

dH

dR A

(2.1.28)

It follows from equations (2.1.26), (2.1.27) and (2.1.28) that the stable solute gradient and

rotation have a stabilizing effect whereas the suspended particles has a destabilizing effect on

thermosolutal convection in Rivlin-Ericksen elastico- viscous fluid in porous medium.

(2.1.29)

2

1

2

2

1

1

1

)1(.1

11

PQx

PxT

Hx

x

dQ

dR A

(2.1.30)

In the absence of rotation ( 01AT ),

dP

dR1 is always negative and 1

1

dQ

dRis always

positive, which means that medium permeability has a destabilizing effect, whereas magnetic

field has a stabilizing effect on the system in the absence of rotation. For a rotating system, the

medium permeability has a destabilizing (or stabilizing) effect and magnetic field has a

stabilizing (or destabilizing) effect if

.

1

1 2

2

2

1

1ε

xP

PQxorTA

(2.1.31)

The dispersion relation (2.1.24) is analyzed numerically also. In figure 2.1.2, 1R is plotted

against modified solute gradient parameter 1S for ,20H ,10P ,501AT 2.0 , ,151 Q

10and8,6,4,2x and in figure 2.1.3, 1R is plotted against wave number x for .60,40,201 S

Both the figures show the stabilizing effect of the stable solute gradient as Rayleigh number

2

1

22

2

1

1

1111

PQx

xT

PxH

x

dP

dR A

46

increases with the increase in stable solute gradient parameter. In figure 2.1.4, 1R is plotted

against rotation parameter 1AT for ,101 S ,3.0P ,15H ,6.0 101 Q and different wave

numbers .10and8,6,4,2x In figure 2.1.5, 1R is plotted against wave number x for

.60,40,201AT Here, also we find the stabilizing role of the rotation as the Rayleigh number

increases with the increase in rotation parameter. In figure 2.1.6, the graph of 1R is plotted

against magnetic field parameter 1Q for ,101 S ,9.0P ,5H ,7.0 701AT ,

.10,8,6,4,2x Figure 2.1.7 represents the plot of Rayleigh number versus wave number for

various values of 1Q . Here, it is observed that magnetic field has a stabilizing (or destabilizing)

effect under certain conditions. In figures 2.1.8 and 2.1.9, 1R is plotted against suspended

particles parameter H for ,501 S ,5.0P ,5.0 ,201AT 51 Q , 10and8,6,4,2x and

wave number x for 70,50,30H , respectively. It is clear from the figures 8 and 9 that suspended

particles have destabilizing effect as the Rayleigh number decreases with the increase in

suspended particles. In figure 2.1.10, 1R is plotted against permeability parameter P for

,101 S 201AT , ,10H ,15.0 901 Q and different wave numbers .10and8,6,4,2x

Figure 2.1.11 represents the graph of Rayleigh number versus wave number for various values of

P . Here, we find that the permeability has stabilizing (or destabilizing) effect under certain

conditions in the presence of rotation as the Rayleigh number increases (or decreases) with the

increase in permeability parameter P .

The modified Rayleigh number 1R (a function of x ), given by the equation (2.1.25) attains

its extreme value when

0113

131121

3

1

322

1

3222

1

22

2222

1

23222

1

222

1

242

1

25

1

11

xQPεHxxQPTHxxεQPH

xxPTHεPQHxεxHQPTεQPxεPQεx

A

AA

(2.1.32)

We have determined the critical Rayleigh number cR and the associated critical wave

number cx for various values of the governing parameters HSTPQ A and,,, 11 1 in accordance with

equation (2.1.25) rather than using equation (2.1.32). The critical Rayleigh numbers are obtained

numerically from figures 2.1.2 to 2.1.11 and are listed in the tables 2.1.1-2.1.2.

47

From table 2.1.1, one may find that as P increases, cR decreases for 01AT , whereas

for higher values of rotation parameter 1AT , cR first decreases for lower values of P and then

increases for higher values of P and hence showing the destabilizing effect of medium

permeability in the absence of rotation and destabilizing (stabilizing) effect in the presence of

rotation parameter. This behaviour can also be observed in figure 2.1.12. From table 2.1.2, it is

observed that magnetic field parameter has destabilizing (stabilizing) effect in the presence of

rotation as critical Rayleigh number first decreases for lower values of 1Q and then increases for

higher values of 1Q , whereas magnetic field parameter has stabilizing effect in the absence of

rotation parameter. This behaviour can also be observed in figure 2.1.13. Figures 2.1.14 and

2.1.15 indicates the stabilizing behaviour of rotation parameter1AT and stable solute gradient

parameter 1S , as the critical Rayleigh number cR increases with the increase in rotation parameter

1AT and stable solute gradient parameter 1S , respectively, which can also be observed from tables

2.1.3 and 2.1.4. Figure 2.1.16 and table 2.1.5 shows the destabilizing behaviour of suspended

particles parameter, as critical Rayleigh number decreases with the increase in suspended

particles parameter.

25

45

65

85

105

125

145

165

0 50 100

Solute Gradient

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

26

32

38

44

50

56

62

68

0 2 4 6 8 10Wave Number

Rayle

igh

Nu

mb

er

S1=20

S1=40

S1=60

Figure 2.1.2: Variation of 1R with 1S Figure 2.1.3: Variation of 1R with x

48

10

20

30

40

50

60

70

80

0 40 80 120 160 200 240

Rotation

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

15

17

19

21

23

25

0 1 2 3 4

Wave Number

Rayle

igh

Nu

mb

er

TA1=20

TA1=40

TA1=60

Figure 2.1.4: Variation of 1R with 1AT Figure 2.1.5: Variation of 1R with x

20

26

32

38

44

50

10 30 50 70 90 110

Magnetic Field

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

20

22

24

26

28

30

1 2 3 4 5Wave Number

Rayle

igh

Nu

mb

er

Q1=20

Q1=30

Q1=40

Figure 2.1.6: Variation of 1R with 1Q Figure 2.1.7: Variation of 1R with x

49

10

15

20

25

30

35

10 30 50 70 90Suspended Particles

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

11

12

13

14

15

16

0 0.9 1.8 2.7 3.6 4.5Wave Number

Rayle

igh

Nu

mb

er

H=30

H=50

H=70

Figure 2.1.8: Variation of 1R with H Figure 2.1.9: Variation of 1R with x

10

14

18

22

26

30

0 0.1 0.2 0.3 0.4 0.5Permeability

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

18

19

20

21

22

23

0.5 1.2 1.9 2.6 3.3 4Wave Number

Rayle

igh

Nu

mb

er

P=0.09

P=0.2

P=0.3

Figure 2.1.10: Variation of 1R with P Figure 2.1.11: Variation of 1R with x

50

20

30

40

50

60

70

80

0 0.1 0.2 0.3

Permeability

Cri

tical R

ayle

igh

Nu

mb

er

TA1=200

TA1=100

TA1=0

10

20

30

40

0 50 100 150 200

Magnetic FieldC

riti

cal R

ayle

igh

Nu

mb

er

TA1=0

TA1=100

TA1=150

Figure 2.1.12: Variation of cR with P Figure 2.1.13: Variation of cR with 1Q

10

15

20

25

30

0 40 80 120 160

Rotation

Cri

tical R

ayle

igh

Nu

mb

er

P=0.01

P=0.04

P=0.07

0

40

80

120

160

0 50 100Solute Gradient

Cri

tical R

ayle

igh

Nu

mb

er

P=0.01

P=0.04

P=0.07

Figure 2.1.14: Variation of cR with 1AT Figure 2.1.15: Variation of cR with 1S

51

10

19

28

37

46

55

0 50 100Suspended Particles

Cri

tical R

ayle

igh

Nu

mb

er

P=0.01

P=0.02

P=0.07

Figure 2.1.16: Variation of cR with H

Table 2.1.1: Critical Rayleigh numbers and wave numbers of the associated disturbances for the onset of instability

as stationary convection for various values of medium permeability parameter P .

P

01AT 100

1AT 200

1AT

cx cR cx cR cx cR

0.01 1.41 68.2844 1.43 73.6366 1.54 78.9064

0.05 2.45 33.7979 2.2 46.4684 1.99 58.9990

0.09 3.16 29.2495 2.1 44.9744 1.81 59.9232

0.14 3.87 26.9614 2.07 44.6797 1.70 60.9003

0.2 4.58 25.5825 1.99 44.6413 1.63 61.5931

0.25 5.10 24.8792 1.96 44.6544 1.60 61.9577

0.3 5.57 24.3785 1.94 44.6744 1.59 62.2168

52


as stationary convection for various values of magnetic field parameter 1Q .


as stationary convection for various values of rotation parameter .1AT

1AT

01.0P 04.0P 07.0P

cx cR cx cR cx cR

0 1.1 25.2727 1.5 15.1389 1.8 13.6296

20 1.2 25.7558 1.6 16.4636 1.9 15.4702

40 1.2 26.2338 1.8 17.7662 2.0 17.3029

60 1.2 26.7118 1.9 19.0534 2.1 19.1311

80 1.3 27.1882 2.0 20.3293 2.1 20.9563

100 1.3 27.6519 2.1 21.5962 2.2 22.7784

160 1.4 29.405 2.3 25.3566 2.3 28.2380

1Q

01AT 100

1AT 150

1AT

cx cR cx cR cx cR

5 2.3 11.2434 1.6 32.7117 1.7 21.9912

20 4.4 13.1909 1.6 21.1029 1.9 17.3565

30 5.3 14.3981 2.0 20.4 2.5 47.7655

40 6.1 15.5739 2.3 20.6032 3.0 67.7655

50 6.8 16.7294 2.8 21.1632 3.6 87.7655

100 9.5 22.3421 4.8 25.4950 6.1 107.7655

200 13.5 33.2119 8.4 35.5491 10.1 127.7655

53


as stationary convection for various values of solute gradient parameter 1S .

1S

01.0P 04.0P 07.0P

cx cR cx cR cx cR

0 1.7 15.3349 2.8 8.7841 3.3 7.7655

20 1.7 35.3349 2.8 28.7841 3.3 27.7655

40 1.7 55.3349 2.8 48.7841 3.3 47.7655

60 1.7 75.3349 2.8 68.7841 3.3 67.7655

80 1.7 95.3349 2.8 88.6841 3.3 87.7655

100 1.7 115.3349 2.8 108.7841 3.3 107.7655

120 1.7 135.3349 2.8 128.7841 3.3 127.7655


as stationary convection for various values of suspended particles parameter H .

H

01.0P 02.0P 07.0P

cx cR cx cR cx cR

10 1.1 51.809 1.17 33.487 1.68 21.6378

30 1.1 24.2699 1.16 17.8291 1.68 13.8792

50 1.1 18.5619 1.16 14.6974 1.68 12.3275

70 1.1 16.1157 1.16 13.3553 1.68 11.6625

90 1.1 14.7566 1.16 12.6097 1.68 11.2931

110 1.1 13.8918 1.16 11.9573 1.68 11.0579

54

2.1.6 STABILITY OF THE SYSTEM AND OSCILLATORY MODES

Multiplying equation (2.1.15) by ,*W (the complex conjugate of W ) and integrating over

the range of z

1

0

22

0

1

0

3

1

0

1

0

2222

1

0)(*4

*2

*)(*1

11

DKdzaDWHd

DzdzWd

dzWdga

WdzaDWP

FM

e

e

(2.1.33)

Integrating equation (2.1.33) and using boundary conditions (2.1.22)-(2.1.23) together with

equations (2.1.16)-(2.1.20), we obtain

0*44

*1

*11

*

***

*11

11

1029

0

827

0

2

6

1

2

5

'

1243112

1

1

2

1

1

IσpIνπρ

ημIσpI

νπρ

dημI

P

σF

στ

M

ε

σd

IpEσIβ

καIpEσI

β

ακ

στH

στ

ν

gaI

P

σF

στ

M

ε

σ

ee

e

e

(2.1.34)

where,

,

1

0

222

1 dzWaDWI ,

1

0

222

2 dzaDI

,

1

0

2

3 dzI ,

1

0

222

4 dzaDI

,

1

0

2

5 dzI ,

1

0

2

6 dzZI

,

1

0

222

7 dzXaDXI 1

0

2

8 ,dzXI

1

0

24222

2

9 ,2 dzKaDKaKDI

1

0

222

10 .dzKaDKI

and * is the complex conjugate of . The integrals 101 II are all positive definite. Putting

ii ii * in equation (2.1.34) and equating imaginary parts, we obtain

55

0

4411

1

1.

11

1

102

0

82

0

2

622

2

512311222

222

42222

1

2

122

IpIpd

IP

FMd

IpEIpEH

Hga

IIH

HgaI

P

FM

ee

lii

ii

ii

iilii

i

(2.1.35)

In the absence of rotation, magnetic field and stable solute gradient, equation (35) reduces to

01

.1

11

3222

22

11

2

2222

1

2

122

I

H

HpE

gaI

H

HgaI

P

FM

ii

ii

iilii

i

(2.1.36)

Equation (2.1.36) implies that i is zero (as the terms in the bracket are positive definite) in the

absence of rotation, magnetic field and stable solute gradient, which means that oscillatory modes

are not allowed and the principle of exchange of stabilities is satisfied. It is evident from equation

(2.1.35) that the presence of magnetic field, rotation and stable solute gradient introduces

oscillatory modes (as i may not be zero).

2.1.7 CONCLUSIONS

In this section, we have studied the effect of magnetic field, rotation and suspended particles

on thermosolutal convection in Rivlin-Ericksen elastico-viscous fluid saturating a porous

medium. The principal conclusions from the analysis are as follows:

(i) For stationary convection, the suspended particles have destabilizing effect on

thermosolutal convection in Rivlin-Ericksen elastico- viscous fluid in porous medium.

(ii) The medium permeability has a destabilizing effect in the absence of rotation whereas

in the presence of rotation it has a destabilizing effect when

2

2

2

1

1

11

εxP

PQxTA

and has stabilizing effect when

.

1

1 2

2

2

1

1ε

xP

PQxTA

56

(iii) The magnetic field has a stabilizing effect in the absence of rotation whereas in the

presence of rotation it has a stabilizing effect when

2

2

2

1

1

11

εxP

PQxTA

and has

destabilizing effect when

.

1

1 2

2

2

1

1ε

xP

PQxTA

(iv) The rotation and stable solute gradient have stabilizing effect on thermosolutal

convection of the system.

(v) The critical Rayleigh numbers and critical wave numbers for the onset of instability

are also determined and the sensitiveness of the critical Rayleigh number cR with

respect to changes in medium permeability, magnetic field, rotation, solute gradient

and suspended particles parameter is depicted graphically in figures 2.1.12-2.1.16

which are in agreement with analytical results.

(vi) The principle of exchange of stabilities is satisfied in the absence of magnetic field,

rotation and stable solute gradient. The presence of magnetic field, rotation and stable

solute gradient introduces oscillatory modes (as i may not be zero).

57

EFFECT OF SUSPENDED PARTICLES, ROTATION AND MAGNETIC

FIELD ON THERMOSOLUTAL CONVECTION IN RIVLIN-

ERICKSEN FLUID WITH VARYING GRAVITY FIELD IN POROUS

MEDIUM

2.2.1 INTRODUCTION

Thermal stability of a fluid layer under variable gravitational field heated from below or

above is investigated analytically by Pradhan and Samal [207]. Although the gravity field of the

earth is varying with height from its surface, we usually neglect this variation for laboratory

purposes and treat the field as a constant. However, this may not be the case for large-scale flows

in the ocean, the atmosphere or the mantle. It can become imperative to consider gravity as a

quantity varying with distance from the centre. Sharma and Rana [208, 209] have also studied the

instability of streaming the Rivlin–Ericksen fluid in porous medium in hydromagnetics and the

thermosolutal instability of the Rivlin–Ericksen rotating fluid in the presence of magnetic field

and variable gravity field in a porous medium. Recently, Rana and Kumar [210] have studied the

stability of Rivlin-Ericksen elastico-viscous rotating fluid permeating with suspended particles

under variable gravity field in porous medium.

In this section, we have considered the effect of suspended particles, rotation and magnetic

field on thermosolutal convection in Rivlin-Ericksen elastico-viscous fluid in porous medium.

Here, we have extended the results reported by Sharma and Rana [211] to include the effect of

magnetic field for Rivlin-Ericksen fluids.

2.2.2 MATHEMATICAL FORMULATION

Consider an infinite horizontal fluid particle layer of Rivlin-Ericksen elastico-viscous

fluid of thickness d bounded by the planes 0z and dz in porous medium, is acted upon by

a uniform rotation ) (0,0,Ω , a uniform magnetic field ),0,0( HH and variable gravity

g,0,0g ,where 0g,gg 00 is the value of g at 0z and can be positive or negative as

gravity increases or decreases upwards from its value 0g (see figure 2.2.1).This layer is heated

58

from below and subjected to stable solute gradient such that a uniform temperature gradients

dzdT and solute concentration gradients

dzdCβ are maintained across the layer.

Figure 2.2.1: Geometrical Configuration

Then the equations of motion and continuity governing the flow are

HHq

qqqgqqq

)(4

2

11

111

0

'

1000

πρ

μ

ε

tυυ

kερ

KN

ρ

δρp

ρεtε

eΩ

d

(2.2.1)

0 q (2.2.2)

where q and, pρ denote respectively, the density, the pressure and the filter velocity of the pure

fluid; ),(and txNdq denote the velocity and number density of the suspended particles,

respectively. ,6K where is the particle radius, is the Stokes’ drag coefficient,

),,( zyxx . Let ,,k, 1 stand for medium porosity, medium permeability, kinematic

viscosity of fluid and kinematic viscoelasticity of fluid, respectively.

In writing equation (2.2.1) we have assumed uniform size of fluid particles, spherical

shape and small relative velocities between the fluid and particles. Then the net effect of the

59

suspended particles on the fluid through porous medium is equivalent to an extra body force term

per unit volume qq d

ε

KN. Since the force exerted by the fluid on the particles is equal and

opposite to that exerted by the particles on the fluid, there must be an extra force term, equal in

magnitude but opposite in sign, in the equation of motion of the particles. The distances between

particles are assumed to be so large compared with their diameter that interparticle reactions need

not be accounted for. The effects of pressure, gravity and Darcian force on the suspended

particles (assumed large distances apart) are negligibly small and therefore ignored. If mN is the

mass of particles per unit volume, then the equations of motion and continuity for the particles,

under the above assumptions, are

ddd

d KNεt

mN qqqqq

1 (2.2.3)

0

dN

t

Nε q (2.2.4)

Let qqCTcc ptand,,,, denote respectively the specific heat of fluid at constant

pressure, specific heat of fluid particles, temperature, solute concentration, effective thermal

conductivity of the pure fluid and an analogous effective solute conductivity. If we assume that

the particles and the fluid are in thermal and solute equilibrium, then the equation of heat and

solute conduction gives

TTt

εmNcTcρt

Tεcρεcρ dptss

2q1

qq (2.2.5)

CCt

εmNcTcρt

Tεcρεcρ dptss

2q1

qq (2.2.6)

where ss c, are the density and the specific heat of the solid material respectively, and ε is

medium porosity. The equation of state for the fluid is given by

000 1 CCTT , (2.2.7)

where 000 and, CT are the density, temperature and solute concentration of the fluid at bottom

surface ,0z is the coefficient of thermal expansion and is the analogous coefficient of

solvent expansion.

60

2.2.3 THE PERTURBATION EQUATIONS

The initial state of the system, denoted by subscript 0, is taken to be a quiescent layer (no

setting) with a uniform particle distribution 0N i.e. 0,0,0,0,0,0 dqq and 0NN is a

constant. The character of the equilibrium of this initial static state can be determined, as usual,

by supposing that the system is slightly disturbed and then by following its further evolution.

The steady state solution to the governing equations is

zβαzαβρρzβCCzβTTd 1,,,0,0,0,0,0,0 000qq

Let Nγsrlwvuθpδδρ d and,,,,,,,,, qq denote respectively the small perturbations in

density , pressure ,p temperature ,T fluid velocity )0,0,0(q fluid particle velocity

),0,0,0(dq solute concentration C and number density 0N . Then the linearized perturbation

equations of motion, continuity, heat conduction and solute conduction for Rivlin-Ericksen

elastic-viscous fluid-particle layer are

HhΩq

qqqq

o

e

o

d

oo

πρ

μ

ερ

tμμ

ρKερ

KNγααθgpδ

ρtε

4

2

111

01

0 λ

(2.2.8)

0 q (2.2.9)

00

dN

t

Nε q (2.2.10)

qq

d

tK

m1 (2.2.11)

2

hsw

thE (2.2.12)

2

hsw

thE (2.2.13)

where,

00000

0 and,,,,,

c

q

c

qmNf

c

fch

pt

61

In writing equation (2.2.8), use has been made of the Boussinesq equation of state

0 .

2.2.4 EXACT SOLUTION AND DISPERSION RELATION

Analyzing the perturbations into normal modes by seeking solutions in the form

)exp(.)(),(),(),(),(),(,,,,, ntyikxikzXzZzKzzzWhw yxz (2.2.14)

where yx kk , are the wave numbers along x and y directions respectively, and 21

22 )( yx kkk is

the resultant wave number of the disturbance and n is the growth rate which is , in general , a

complex constant and y

h

x

h

y

u

x

v xy

and

stand for the z-components of vorticity and

current density, respectively. Eliminating the physical quantities using the non-dimensional

parameters

,kda

2nd

,

1p ,

1p ,

21d

,

0

0

mNM ,

2p , ,

2

1

d

kpl ,

2dF

hEE 1 , dDDhHhEE *

2 ,1, and dropping (*) for convenience, the

linearized dimensionless perturbation equations are

DKaDυπρ

Hdμ

DZυε

dαα

υ

dgaWaD

P

σF

στ

M

ε

σ

e

l

22

0

32222

1

4

21

11

(2.2.15)

DXHd

DWd

ZP

FM e

l

01 4

21

11

(2.2.16)

WHd

pEaD

1

1

2

11

22

1 (2.2.17)

WHd

pEaD

1

1

2''

12

22

1 (2.2.18)

DWHd

KpaD

2

22 (2.2.19)

62

DZHd

XpaD

2

22 (2.2.20)

Eliminating ,,, KZ and X between the equations (2.2.15) – (2.2.20), we obtain

WDaDpEσaDpEσaDQDσpaDP

σF

στ

M

ε

σQ

WDσpaDpEσaDpEσaDε

T

pEσaDλSpEσaDλR

QDσpaDP

σF

στ

M

ε

σσpaDa

στ

στH

WaDσpaDpEσaD

pEσaDP

σF

στ

M

ε

σQDσpaD

P

σF

στ

M

ε

σ

l

A

l

ll

222

12

22

11

222

2

22

1

22

2

22

12

22

11

22

2

11

22

12

22

2

2

22

1

2

222

1

1

22

2

22

12

22

11

22

1

2

2

22

1

.1

11

.

1

11..

1

.

1

11.

1

11

(2.2.21)

where,

4dgR is the thermal Rayleigh number,

4dgS is the analogous solute

Rayleigh number,

222

dTA is the Taylor number and

0

22

4

dHQ e is the Chandrasekhar

number.

Consider the case where both the boundaries are free from stresses and perfect conductors

of heat and solute, the medium adjoining the fluid is perfectly conducting and temperatures at the

boundaries are kept fixed. The boundary conditions appropriate for the problem are

(Chandrasekhar [11])

,02 WDW ,0 1and0when0 zDK (2.2.22)

Proper solution of W characterizing the lowest mode is

zWW sin0 where 0W is a constant. (2.2.23)

63

Substituting the proper solution (2.2.23) in equation (2.2.21), we obtain the dispersion relation

21

111

2

11

2

111

121

2

1

2

11

1

11121

2

11

2

11

2111

2

11

2

11

2

1

2

11

11

1

'

11

1111

1

11

1

11

11

1)1(.

111

.1

.1

11

1

1

1

pσix

σpExx

πστiH

πστiQ

QpσixP

πσiF

πστi

M

ε

σi

σpiExpσix

πστiH

πστi

ε

TxσpiEx

πστiH

πστi

P

πσiF

πστi

M

ε

σiλxS

σpiEx

σpiExλxR

A

(2.2.24)

.and,,,,,where 2

21

2

2

421

41

41 1 lA PPi

ax

TT

QQ

SS

RR

Equation (2.2.24) is the required dispersion relation including the effects of magnetic

field, rotation, medium permeability, kinematic viscosity, stable solute gradient and variable

gravity field on the thermosolutal instability of Rivlin-Ericksen fluid in porous medium. The

dispersion relation vanishes to the one derived by Sharma and Rana [211] if the magnetic field

parameter is vanishing.

2.2.5 THE STATIONARY CONVECTION

For stationary convection, the marginal state will be characterized by 0 . Thus

equation (2.2.24) reduces to

1

2

2111

1

1111

QP

x

x

Hx

TQ

P

x

xH

xSR

A

(2.2.25)

The above relation expresses the modified Rayleigh number 1R as a function of the

modified solute gradient parameter ,1S modified magnetic field parameter ,1Q suspended

particles parameter ,H modified rotation parameters ,1AT medium permeability parameter P

and dimensionless wave number x . The parameter F accounting for the kinematic viscoelasticity

vanishes for the stationary convection and the Rivlin-Ericksen fluid behaves like an ordinary

Newtonian fluid. To study the effect of stable solute gradient, magnetic field, suspended particles,

64

rotation and permeability, we examine the nature of

1

11

1

1

1

1 ,,,

AdT

dR

dH

dR

dQ

dR

dS

dR and

dP

dR1

analytically.

Equation (2.2.25) yields

11

1 dS

dR (2.2.26)

which is positive implying thereby that the effect of stable solute gradient is to stabilize the

system. This is in agreement with the result of figure 2.2.2 where 1R is plotted against 1S

for ,60H ,001.0P ,201AT 15.0ε , ,2001 Q ,2λ 10,8,6,4,2x and figure 2.2.3

where 1R is plotted against x for 80,60,40,201 S . This stabilizing effect of stable solute

gradient is in good agreement with the earlier works of Sharma and Rana [211].

From equation (2.2.25), we get

1

2

2

1

1

11

1 QP

x

x

HxdT

dR

A (2.2.27)

which shows that rotation has a stabilizing effect on the system in porous medium when gravity

increases upwards from its value )0i.e.(0 g and destabilizes the system when gravity decreases

upwards. Also figure 2.2.4 and 2.2.5 confirms the above result numerically for

fixed ,10H ,2.0P ,401 S 15.0ε , ,2001 Q 2 and various values of x and

1AT respectively. This stabilizing effect of rotation is in good agreement with earlier works of

Sharma and Rana [211] and Rana and Kumar [210].

Also from equation (2.2.25), we get

1

1

22

1

1

1111 Q

QP

x

xT

P

x

Hx

x

dH

dR A

(2.2.28)

which is negative, implying thereby that the effect of suspended particles is to destabilize the

system when gravity increases upwards (i.e. 0 ) and stabilizes the system when gravity

decreases upwards. Also in figures 2.2.6 and 2.2.7, 1R decreases with the increase in H which

65

confirms the above result numerically. This result is in agreement with the result of Sharma and

Aggarwal [203] in which effect of compressibility and suspended particles is investigated on

thermal convection in a Walters’ B΄ elastico-viscous fluid in hydromagnetics.

It is evident from equation (2.2.25) that

2

1

22

2

1

1

1111

PQx

x

ε

T

PHλx

x

dP

dR A (2.2.29)

2

1

2

2

1

1

1

)1(.1

11

PQx

Px

ε

T

λHx

x

dQ

dR A

(2.2.30)

In the absence of rotation ( 01AT ),

dP

dR1 is always negative and 1

1

dQ

dRis always

positive, which means that medium permeability has a destabilizing effect, whereas magnetic

field has a stabilizing effect on the system when gravity increases upwards from its value .0g For

a rotating system, when gravity increases upwards, the medium permeability has a destabilizing

(stabilizing) effect and magnetic field has a stabilizing (destabilizing) effect if

2

2

2

1

1

1or

1ε

xP

PQxTA

(2.2.31)

In figure 2.2.8, 1R is plotted against 1Q for ,10H ,13.0P ,101 S 15.0ε , 2

and .10,8,6,4,2x Figure 2.2.9 represents the graph of Rayleigh number 1R versus x for

.180,140,100,601 Q Here, we find that magnetic field has stabilizing as well as destabilizing

effect under condition (2.2.31). In figure 2.2.10, 1R is potted against P for ,1001AT ,40H

,301 Q ,101 S 15.0ε , 2 and 10,8,6,4,2x whereas figure 2.2.11 represents 1R versus

x for .25.0,09.0,05.0,01.0P These figures show that the permeability has destabilizing

(stabilizing ) effect in the presence of rotation under condition (2.2.31).

In figure 2.2.12, 1R is plotted against wave number x for ,101 S 01AT ,

,10H ,15.0 ,301 Q 2 and different values of permeability parameter

25.0,09.0,05.0,03.0P and figure 2.2.13 shows the graph of 1R versus x for

,101 S ,100H ,01AT ,15.0 2 and .180,140,100,601 Q Figure 2.2.12 shows

66

stabilizing effect of magnetic field whereas figure 2.2.13 shows the destabilizing effect of

medium permeability in the absence of rotation .

45

85

125

165

205

245

0 50 100Solute Gradient

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

50

70

90

110

130

0.4 0.8 1.2 1.6 2

Wave Number

Rayle

igh

Nu

mb

er

S1=20

S1=40

S1=60

S1=80

Figure 2.2.2: Variation of 1R with 1S Figure 2.2.3: Variation of 1R with x

50

60

70

80

0 40 80 120 160 200 240

Rotation

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

50

55

60

65

70

75

80


Rayle

igh

Nu

mb

er

TA1=20

TA1=80

TA1=140

TA1=200

Figure 2.2.4: Variation of 1R with 1AT Figure 2.2.5: Variation of 1R with x

67

10

12

14

16

18

20

22

0 20 40 60 80 100Suspended Particles

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

10

11

12

13

14

15

0 1 2 3 4 5

Wave Number R

ay

leig

h N

um

be

r

H=30

H=50

H=70

H=90

Figure 2.2.6: Variation of 1R with H Figure 2.2.7: Variation of 1R with x

25

30

35

40

45

20 60 100 140 180 220Magnetic Field

Rayle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

27

29

31

33

35

37


Rayle

igh

Nu

mb

er

Q1=60

Q1=100

Q1=140

Q1=180

Figure 2.2.8: Variation of 1R with 1Q Figure 2.2.9: Variation of 1R with x

68

12

16

20

24

28

0 0.1 0.2 0.3 0.4 0.5Permeability

Ra

yle

igh

Nu

mb

er

x=2

x=4

x=6

x=8

x=10

20

25

30

35

40

45

50

0 1 2 3Wave Number

Rayle

igh

Nu

mb

er

P=0.01

P=0.05

P=0.09

P=0.25

Figure 2.2.10: Variation of 1R with P Figure2.2.11: Variation of 1R with x

10

10.8

11.6

12.4

13.2

14

0.2 0.9 1.6 2.3 3Wave Number

Rayle

igh

Nu

mb

er

P=0.03

P=0.05

P=0.09

P=0.25

17

21

25

29

33


Rayle

igh

Nu

mb

er

Q1=60

Q1=100

Q1=140

Q1=180

Figure 2.2.12: Variation of 1R with x for 01AT Figure 2.2.13: Variation of 1R with x for 0

1AT

69

2.2.6 STABILITY OF THE SYSTEM AND OSCILLATORY MODES

Multiplying equation (2.2.15) by ,*W the complex conjugate of ,W and integrating over

the range of z

1

0

22

0

1

0

3

1

0

1

0

22

022

1

0)(*4

*2

*)(*.1

11

DKdzaDWνπρ

HdμDzdzW

εν

d

dzααWν

dagλWdzaDW

P

σF

στ

M

ε

σ

e

e

(2.2.32)

Integrating equation (2.2.32) and using boundary conditions (2.2.22) - (2.2.23) together

with equations (2.2.16) - (2.2.20), we obtain

0*44

*1

*11

*

***

*11

11

1029

0

827

0

2

6

1

2

5

'

1243112

1

1

2

0

1

1

IσpIνπρ

ημIσpI

νπρ

dημI

P

σF

στ

M

ε

σd

IpEσIβ

καIpEσI

β

ακ

στH

στ

ν

agλI

P

σF

στ

M

ε

σ

ee

e

e

(2.2.33)

where,

,

1

0

222

1 dzWaDWI ,

1

0

222

2 dzaDI

,

1

0

2

3 dzI ,

1

0

222

4 dzaDI

,

1

0

2

5 dzI ,

1

0

2

6 dzZI

,

1

0

222

7 dzXaDXI 1

0

2

8 ,dzXI

1

0

24222

2

9 ,2 dzKaDKaKDI

1

0

222

10 .dzKaDKI

and * is the complex conjugate of . The integrals 101 II are all positive definite. Putting

ii ii * in equation (2.2.33) and equating imaginary parts, we obtain

70

0

4411

1

.

1.

11

1

102

0

82

0

2

622

2

512311222

222

0

42222

1

2

0

122

Ipνπρ

ημIp

νπρ

dημI

P

F

τσ

M

εd

IpEβ

καIpE

β

ακ

τσH

τσH

ν

agλ

Iβ

καI

β

ακ

τσH

H

ν

τagλI

P

F

τσ

M

ε

σ

ee

lii

ii

ii

iilii

i (2.2.34)

In the absence of rotation, magnetic field and solute gradient, equation (2.2.34) reduces to

0

1.

11

1

3222

22

11

2

0

2222

1

2

0

122

IτσH

τσHpE

β

ακ

ν

agλ

Iβ

ακ

τσH

H

ν

τagλI

P

F

τσ

M

εσ

ii

ii

iilii

i (2.2.35)

Equation (2.2.35) implies that i is zero (as the terms in the bracket are positive definite)

in the absence of rotation, magnetic field and stable solute gradient, which means that oscillatory

modes are not allowed and the principle of exchange of stabilities is satisfied. It is evident from

equation (2.2.34) that the presence of magnetic field, rotation and stable solute gradient

introduces oscillatory modes (as i may not be zero).

2.2.7 CONCLUSIONS

In this section, we have studied the effect of magnetic field, rotation, variable gravity field

and suspended particles on thermosolutal convection in Rivlin-Ericksen elastico-viscous fluid

saturating a porous medium. The principal conclusions from the analysis are as follows:

(i) For stationary convection, the suspended particles have destabilizing effect for 0λ

and stabilizing effect for .0λ

(ii) When gravity increases upwards (i.e. 0λ ), the medium permeability has a

destabilizing effect in the absence of rotation whereas in the presence of rotation it has

71

a destabilizing effect when

2

2

2

1

1

11

xP

PQxTA and has stabilizing effect when

.1

1 2

2

2

1

1

xP

PQxTA

(iii) The magnetic field has a stabilizing effect in the absence of rotation when gravity

increases upwards whereas in the presence of rotation it has a stabilizing effect when

2

2

2

1

1

11

xP

PQxTA and has destabilizing effect when

.1

1 2

2

2

1

1

xP

PQxTA

(iv) For stationary convection, the stable solute gradient is found to have stabilizing effect

whereas effect of rotation is stabilizing for 0λ and destabilizing for .0λ

(v) The principle of exchange of stabilities is satisfied in the absence of magnetic field,

rotation and stable solute gradient. The presence of magnetic field, rotation and stable

solute gradient introduces oscillatory modes (as i may not be zero).

Hall effect on Ferrofluids -...

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