RESEARCH ARTICLE OPEN ACCESS Effect of Rotation on … - 5/B0603050428.pdf · enough that the surfactants Vander Waals force is sufficient to prevent magnetic clumping or ... c c

Bhupander Singh. Int. Journal of Engineering Research and Application www.ijera.com

ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -5) March 2016, pp.04-28

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Effect of Rotation on a Layer of Micro-Polar Ferromagnetic

Dusty Fluid Heated from Below Saturating a Porous Medium Bhupander Singh, Department of Mathematics, Meerut College, Meerut, (U.P.) INDIA-250001

ABSTRACT This paper deals with the theoretical investigation of effect of rotation on micro-polar ferromagnetic dusty fluid

layer heated from below in a porous medium. Linear stability analysis and normal mode analysis methods are

used to find an exact solution for a flat micro-polar ferromagnetic fluid layer contained between two free

boundaries . In case of stationary convection, the effect of various parameters like medium permeability

parameter, non-buoyancy magnetization parameter, micro-polar coupling parameter, spin-diffusion parameter,

micro-polar heat conduction parameter, dust particles parameter and rotation parameter has been analyzed and

results are depicted graphically. In the absence of dust particles, rotation, micro-viscous effect and micro-inertia,

the sufficient condition is obtained for non-oscillatory modes

I. INTRODUCTION Magnetic fluids or Ferro-fluids are

colloidal liquids made of neno-scale ferromagnetic,

or ferri-magnetic particles suspended in a carrier

fluid (usually an organic solvent or water). Each

tiny particle is thoroughly coated with a surfactant

to inhibit clumping. Large ferromagnetic particles

can be ripped out of the homogeneous colloidal

mixture, forming a separate clump of magnetic dust

when exposed to strong magnetic fields. The

magnetic attraction of nano-particles is weak

enough that the surfactants Vander Waals force is

sufficient to prevent magnetic clumping or

agglomeration. Ferro-fluids usually do not retain

magnetization in the absence of an externally

applied field and thus are often classified as

superparamagnets rather than ferromagnets.

Experimental and theoretical physicists and

engineers gave significant contributions to

ferrohydrodynamics and its application. An

authoritative introduction to this subject has been

discussed in detail in the celebrated monograph by

Rosensweig[8]. Many authors [2, 6, 7, 11, 19]

discussed the thermal convection in ferromagnetic

fluids. Scanlon and Segel[4] have considered the

effects of suspended particles on the onset of

Bénard convection, whereas Sunil et al.[20] have

studied the effect of dust particles on thermal

convection in ferromagnetic fluid saturating a

porous medium by assuming isotropic properties.

During the last half century, research on magnetic

liquids has been very productive in many areas

(earthquake protection, air bags, sealing of rotating

shafts etc.) and the mechanism of controlling

convection in a ferromagnetic fluids is important in

material processing in space because of its

applications to the possibility of producing various

materials. Now-a-days micropolar ferromagnetic

fluid stabilities have become an important field of

research [1, 5]. Thermal convection in porous

medium is also of great interest in chemical

engineering, metallurgy and geophysics,

electrochemistry and biomechanics[3]. Sunil et

al.[15, 16, 18, 22, 23] have discussed the marginal

stability of micropolar ferromagnetic fluid

saturating a porous medium and thermal

convection in micropolar ferrofluid in the presence

of rotation saturating a porous and non-porous

medium. In the study of micropolar ferromagnetic

fluid layers done by Sunil et. al[15, 16, 18, 22, 23],

the fluid has been assumed to be clean (free from

dust particles). However, in many geophysical

situations, the fluid is often not pure but contains

suspended/dust particles. The effect of dust

particles on stability problems of micropolar

ferromagnetic fluid through porous medium

reflects its usefulness is several geophysical

situations, chemical engineering, biomechanics and

industry.

The effect of dust particles on

ferromagnetic fluid for porous and non-porous

medium has been studied by several anothers [12-

14, 17, 20, 21]. Reena and Rana[10] have studied

the effect of dust particles on rotating micropolar

fluids heated from below saturating a porous

medium. Reena and Rana[9] have studied the

effect of dust particles on a layer of mircopolar

ferromagnetic fluid heated from below saturating a

porous medium.

Keeping in mind the usefulness of

micropolar ferromagnetic fluids and their various

applications in several fields given above, I have

made an attempt to examine the effect of rotation

on a layer of micropolar ferromagnetic dusty fluid

heated from below saturating a porous medium and

RESEARCH ARTICLE OPEN ACCESS




to the best my knowledge this problem is

uninvestigated so far.

II. MATHEMATICAL

FORMULATION

Consider an infinite, horizontal,

electrically non-conducting incompressible thin

micro-polar ferromagnetic fluid layer of thickness

d, embeded in dust particles, heated from below as

shown in figure below.

Fig. 1

The physical structure is one of infinite

extent in the x and y directions bounded by the

planes 2

dz and

2

dz . The upper boundary is

held at fixed temperature 1T T and the lower

boundary is held at constant temperature 0T T

such that a steady adverse temperature gradient

d T

d z is maintained. A strong but uniform

magnetic field 0(0 , 0 , )e x t

HH

is applied along z-

direction. The whole system is acted upon by a

gravity (0 , 0 , )g g

and is assumed to be

rotating with uniform angular velocity

0(0 , 0 , ) Ω

about z-axis.

This micro-polar ferromagnetic fluid layer is

assumed to be flowing through an isotropic and

homogeneous porous medium of porosity and

the medium permeability 1 , where the porosity is

defined as the fraction of the total volume of the

medium that is occupied by void space. Thus 1

is the fraction that is occupied by solid. For an

isotropic medium the surface porosity (the fraction

of void area to total area of a typical cross section)

will normally be equal to . Here both the

boundaries are taken to be free and perfect

conductor of heat.

Within Boussinesq approximation, the

mathematical equations governing the motion of a

micro-polar ferromagnetic fluid saturating a porous

medium following Darcy’s law for the above

model are as follows :

The equation of continuity for an incompressible micro-polar ferromagnetic fluid is

. 0 q

...(1)

The equation of momentum for generalized Darcy model including the inertial forces is given by

0

1

1 ( )ˆ( . ) ( )z d

K Np g

t

qq q e q q q

02

( . ) ( ) ( )

H B q Ω N

...(2)

The equation of internal angular momentum is given by

2

0

1( . ) ( ) ( . )j

t

Nq N N N

0

12 ( )

q N M H

...(3)

The equation of energy is given by

0 , 0

,

. ( . ) (1 )V H s s

V H

T TC T C

T t t

NH q




+0

,

. ( . ) .p t d

V H

T m N C TT t t

M Hq H q

2

( ) .T T T N

...(4)

The equation of state is given by

0 1 ( )aT T ...(5)

The equation of motion and continuity of the dust particles are given by

1

( . ) ( )d d dm N K Nt

q q q q

...(6)

and . ( ) 0d

NN

t

q

...(7)

Where

0 1 0 ,, , , , , , , , , , , , , , , , , , , , ,d s V Hp t j T C q N q H B M

, , 6 ,p tm N C K r ,T , , aT and ˆze denote

respectively filter velocity, micro-rotation, velocity

of dust particles, pressure, time, density, reference

density, viscosity, coupling viscosity coefficient,

medium permeability, micro-polar viscosity

coefficients, magnetic permeability for free space 7

0( 4 1 0

Henry m-1

), micro-inertia

coefficient, magnetic field, magnetic induction,

magnetization, temperature, density of solid matrix,

specific heat at constant volume and magnetic

field, specific heat of solid matrix, mass of the

particles per unit volume, specific heat of dust

particles, stokes drag coefficient, thermal

conductivity, micro-polar heat conduction

coefficient, coefficient of thermal expansion,

average temperature, which is defined as

0 1

2a

T TT

, and unit vector along z-axis.

Also ( , )N N x t where ( , , )x x y z , denotes

the number density of the dust particles. 0T and 1T

are constant temperatures at lower and upper

boundaries respectively. The partial derivatives are

material properties that can be evaluated once the

magnetic equation of state is known.

In ferro-hydro-dynamics the free charge and the electric displacement are assumed to be absent, therefore the

Maxwell’s equation yield

. 0 , B H 0

...(8)

In Chu formulation of electrodynamics, the magnetic field, magnetization and magnetic induction are related by

0 ( ) B H M

...(9)

We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude

of the magnetic field as well as the temperature, so that

( , )M H TH

H

M

...(10)

The magnetic equation of state is linearlized about oH and aT , which is given by

2( ) ( )o o aM M H H K T T ...(11)

Where (0 , 0 , )e x t

oHH

i.e., ˆ ˆ,

e x t

o z zHH e e

is the unit vector along z-axis and oH is the uniform magnetic

field of the fluid layer when placed in an external magnetic field H

, and B

is the magnetic induction.

Now ,H To a

M

H

denotes the magnetic susceptibility.

2

,H To a

MK

T

denotes the pyromagnetic coefficient, and | |, | |H M H M

and ( , )o o aM M H T

III. BASIC STATE OF THE PROBLEM The basic state is assumed to be quiescent state

which is given by




(0 , 0 , 0 ), (0 , 0 , 0 ), ( ) (0 , 0 , 0 )b b d d b q q N N q q

, ( ), ( ),b b bz p p z N N

( ) , ( ) ,b b b b bz z H H H M M M B B

...(12)

Using this basic state equations (1) to (11) yield

( . ) 0b

b b b

d pg

d z H B

...(13)

1 0

( ) ,b a

T TT T z z T

d

...(14)

0 (1 )b z ...(15)

b oN N N ...(16)

From (11), we have

2( ) ( )b o b o b aM M H H K T T ...(17)

From (8) and (9), we have

. 0B

0 . ( ) 0 H M

0ˆ( )o zH M H M e

...(18)

For basic state, we have

ˆ( )b b o o zH M H M e

or b o bM M h ...(19)

From (17) and (19), we get

2 ( )

1

b a

b o

K T TM M

or 2 ( )ˆ

1

b a

b o z

K T TM

M e

...(20)

and 2 ( )

1

b a

b o

K T TH H

or 2 ( )ˆ

1

b a

b o z

K T TH

H e

...(21)

and e x t

o o oH M H and 0ˆ( )b o o zH M B e

...(22)

IV. PERTURBATION EQUATIONS

Let , , , , , , ,p 1q N q H M

and N denote

respectively the small perturbations in

, , , , , , ,d p Tq N q H M

and N , therefore the

new variable become

, ',b b q q q q N N N N

1 1( ) , , ,d d b b b bp p p T T q q q q

, ,b b b H H H M M M B B B

and

bN N N

Using above perturbations and equations (13), (14), (15), (16), (20), (21), (22), equations (1) to (11) become

. 0 q

...(23)

0

1

1

( )1 ( )ˆ( . ) ( )

o

z

K Np g

t

Nqq q e q q q

( . ) ( . ) ( ' . ) ( )b b b b H B B H B H H B

2( . ) ( . ) ( ) ( ) ( )

o

H B B H H B q Ω N

...(24)

21 1( . ) ( ) ( . ) 2o j

t

Nq N N N q N

0 ( ) ( )b b

M M H H

...(25)

, 0

,

( ) ( )( ) . ( . ) ( )

( )

b b

o V H b b

b V H

TC T

T t

M MH H q

0

,

( ) ( ) ( )(1 ) ( ) . ( . ) ( )

( )

b b b

s s b b

b V H

TC T

t T t

M M H Hq H H

1( ) ( . ) ( )b p t bm N N C Tt

q 2

( ) ( ) . ( )T b bT T N

...(26)




o ...(27)

1

0

1 1

1m L

K t

q q q

...(28)

N a constant (not a function of time) ...(29)

From equations (10) and (11), we have

2 2( ) ( ) ( ) ( ) ( )b b b o b o b aH H M H H H K K T T M M H H

Taking components along x, y, and z-directions respectively, we have

Along x-direction :

0

1 1 1

0

1M

H M HH

...(30)

Along y-direction :

0

2 2 2

0

1M

H M HH

...(31)

Along z-direction :

3 3 3 2(1 )H M H K ...(32)

Using (28), equation (26) can be rewritten as

, 0 0

,,

( ) . ( ) .( ) ( )

b

o V H b b

b b V HV H

CT T

M MH H H H

( . ) ( . ) (1 )b s sT Ct t

q q

0 0

,,

( ) ( ) .( ) ( )

b

b b

b b V HV H

T TT T

M M

( . ) ( . )bt

Hq H q H

0

1( ) ( . ) ( )b p t bm N N C T

t L

q

2( ) . ( ) .T bT N N

...(33)

In order to make linear, we ignore the terms 1( . ) , ( ) , ( . ) , ( . ) ,N q q q q M H q N

, ,

( . ) , , . , ( . ) , , ( . ) , ( , ) , ,( ) ( )b bV H V H

N NT T t

M MH B M H H q q H q

( ) . , ( . ) , ( ) N B H H B

, we obtain

. 0 q

...(34)

0 0 0 0 0

0

1

2ˆ ˆ( )z z

L m NL p g

t

qe q q e

0 2

0 0 0 3 2

ˆ( ) (1 )

(1 )

zKH M H K

z

eHN

...(35)

0

1( ) ( . ) ( ) 2j

t

NN N q N

+

0 ( )b b M H M H

...(36)

0 1 0 0 0 2( )p tL C m N C T Kt t z

2

2 0 0 2

0 2 01

T p t

T KL C w m N C w

...(37)

Where 1 0 , 0 2 0 2 0 , 0 2 0(1 ) ,V H s s V HC C C K H C C K H ,

1 2 3 1 2 3

ˆ ˆ( , , ) , ( , , ) , ( ) . , ( . )z zB B B H H H W B H N e q e

V. DISPERSION RELATIONS Taking curl twice on both sides of equation (35), and taking z-component, we get




2 2 20 0 0 0

0 0 0 1

1

2z

m NL w L g

t t z

2

2 20 2

0 2 1 11

KK

z

...(38)

Taking curl once on both sides of (36), and taking z-component, we get

2 2

0

12j w

t

...(39)

Taking curl once on both sides of equation (35), and taking z-component, we get

0 0 0 0 0

0 0

1

2ˆ( ) .z z

m N L wL L

t t z

N e

...(40)

Taking curl twice on both sides of equation (36) and taking z-component, we get

2 2

0ˆ2 ( ) . z zj

t

N e

...(41)

Eliminating ˆ( ) . z N e

between (40) and (41), we get

20 0

0 0

1

2 z

m NL j

t t t

2

2 20 0 0

0 0

22 z

L wj L

t z

...(42)

Where ˆ( ) .z z q e

From equations (30), (31) and (32), we get 2

20

1 220

1 (1 ) 0M

KH zz

...(43)

Second equation of equation (8) implies that in the absence of electric current and electromagnetic induction, the

field H

can be regarded as conservative, so we may take H

, where is the perturbed magnetic

potential.

VI. NORMAL MODE ANALYSIS

Now we analyze the perturbations , , , zw and into two-dimensional periodic waves by considering the

following form

[ , , , , ] ( , ) , ( , ) , ( , ) , ( , ) , ( , ) e x p ( )z x yw W z t z t X z t G z t z t ix k iy k ...(44)

Where 2 2

x ya k k is the wave number,

22 2

2a

z

and 2 2

1 a .

Using above normal mode analysis, equations (37), (38), (39), (42), and (43) become 2 2

2 2 20 0

0 0 02 21

m NL a W L g a a X

t t z z

2 2

20 0 0 2

0 2

2

1

K aGK a

z z

...(45)

2 22 2

0 2 22j a X a W

t z z

...(46)

220 0

0 0 21

2m N

L j a Gt t t z

22 22 20 0 0 0

0 2 2

22

L LWj a a G

t zz z

...(47)

220

220

1 ( ) (1 ) 0M

a KH zz

...(48)

and

0 1 0 0 0 2( p tL C m N C T Kt t z

222 0 0 2

0 2 02 1T p t

T KL a C W X m N C W

z

..(49)

Now converting the equations (45), (46), (47), (48) and (49) into non-dimensional form by using the following

non-dimensional parameters and non-dimensional quantities

3 2

* * * * * * *112 * 2

, , , , , , ,Kv t d z X d d

t W W Z D K a a d X G Gv d v vd z d

* 1 / 2 * 1 / 2 4

* * 2 2 112

2 2 2

(1 ), , , , ,

T Tr r

T T T

a R a R g d C v C v CR P P N

C v d vK C v d

0

2 200 2 0 0 2

2 3 1 2 32 2 20 22

1

, , , , ,(1 ) (1 ) (1 )

M

HK T KjN N j M M M

g Cd C d d




22

0* 0 0

02 *0 2

2, 1 , , ,

p t

A

m N Cm N dm vL f h T

C vK d t

We have

* 2 * 2 *10 * * *

1

11( )

N fL D a W

t K t

1/ 2

* * * * 1/ 2 2 *2 * *

0 1 1 1(1 ) ( )AT

L M D M a R N D a X D G

...(50)

2 * 2 * 2 * 2 *12 1*

( ) 2 ( )N

j N D a N X D a W

t

...(51)

* 2 * 2 *10 2 1* * * *

1

11( ) 2

NL j N D a N G

t K t t

* 1 / 2 * 2

2 * 2 * 2 * 2 *0 0 1

2 1*( ) 2 ( )

AL T L Nj N D a N D W D a G

t

...(52)

2 * * 2 * *

3 0D a M D ...(53)

*

* *

0 2* *( )r r rL P h P M P D

t t

* 2 *2 * * * 1/ 2 * * 1/ 2 * *

0 0 2 0 3( ) (1 )L D a L M h a R W a R L N X ...(54)

In equations (50) to (54) 0

v

denotes the

kinematic viscosity of the fluid, 1 3,M M and h

denote respectively buoyancy magnetization, non-

buoyancy magnetization and dust particle

parameters. 1 3, , , , ,A r rR T P P N N and j denote

respectively thermal Rayleigh number, Taylor

number, Prandtl number, coupling parameter, heat

conduction parameter and microinertia parameter.

VII. EXACT SOLUTION FOR FREE

BOUNDARIES In this problem, we consider that both the

boundaries are free as well as perfect conductor of

heat. The case of two free boundaries is of little

physical interest but mathematically, it is important

because in case of free boundaries one can derive

an exact solution. Here we consider the case of an

infinite magnetic susceptibility (i.e., ) and

we neglect the deformability of horizontal surfaces.

The exact solution of equations (50)-(54) subject to

the boundary conditions

* 2 * * * * *0W D W D X D G at

* 1

2Z and

1

2 ...(55)

is written in the form

** * * * *1 2

* * * * * * * *3

3 4

* * * * * *55

c o s c o s

c o s o r s in a n d c o s *

a n d c o s o r s in

t t

t t t

t t

W P e z P e z

PD P e z e z X P e z

PD G P e z G e z

...(56)

Where 1 2 3 4 5, , , ,P P P P P are constants and is the growth rate, which is in general, a complex number.

Substituting eq. (56) in eqs. (50)-(54) and dropping asterisks for convenience, we get

2 2 1 / 211 1 2

1

1(1 ) ( ) (1 ) (1 )

N fa P M P a R

K

1/ 2

1/ 2 2 2

1 3 1 4 5(1 ) ( ) (1 ) (1 ) 0AT

M P a R N a P P

...(57)

2 2 2 211 2 1 4( ) 2 0

Na P j N a N P

...(58)

2 1 / 22 2 1

2 1 1

1

(1 ) 1( ) 2 1

AT N fj N a N P

K

22 2 2 21

2 1 5

(1 )( ) 2 ( ) 0

Nj N a N a P

...(59)

2 2 2

2 3 3( ) 0P a M P ...(60)

1/ 2 2 2

2 1 2(1 ) (1 ) (1 ) ( ) ( )r ra R M h P P h P a P 1/ 2

2 3 3 4(1 ) (1 ) 0rM P P a R N P ...(61)

For the existence of non-trivial solutions of equations (57)-(61), the determinant of the coefficients of

1 2 3 4, , ,P P P P and 5P must vanish. Thus we have




6 2

1 1 211 1 2 1 12

(1 )

(1 ) 2AT i

j i N x N

4 4

2 1 1 1 1 3 1 1 1 1(1 ) (1 ) (1 ) ( ) 1r r rM P i i x M i i P h P x

4 22 2

1 1 11 1 1 1 1 1 2 1 1

1

(1 ) (1 )(1 ) (1 ) 2

N x N fi i i j i N x N

K

2 2

4 2 4 21 11 1 1 1 1 3 1 1 2 1 1 1 1 1

(1 )(1 ) . (1 ) (1 ) (1 )r

N xi x R M N i M P i N x i

22 2

4 1 1 11 3 1 1 1 1 1 2 1 1

1

(1 ) (1 )(1 ) (1 (1 ) 2

N x N fx M i i i j i N x N

K

2 2

1 11 1

(1 )(1 )

N xi

4 2 4 2

1 1 3 1 1 1 1 1 1 1 1 1(1 ) (1 ) (1 ) (1 ) ( ) 1r rx R N M i N x i i P h P x

22 22 1

1 2 1 1 1 1 1 1

1

(1 )(1 ) 2 1

N fj i N x N i i i

K

2 2

1 11 1 2 1 1 1 1

(1 )(1 ) 2 (1 )

N xj i N x N i

22 2

4 11 2 1 1 1 1 1 1 1

1

(1 )(1 ) (1 ) (1 )r

N fx M P i i i i i

K

6

1 1 1 1 1 1 1 2(1 ) (1 ) (1 )x R M i i M h

22 2

2 11 3 1 1 2 1 1 1 1 1 1

1

(1 )(1 ) (1 ) 2 (1 )

N fx M j i N x N i i i

K

2 2

1 11 1 2 1 1 1 1

(1 )(1 ) 2 (1 )

N xj i N x N i

22 2

2 11 1 1 1 1 1 1

1

(1 )(1 ) (1 ) (1 )

N fx i i i i

K

2 6

1 1 1 1 1 1 1 1 1 2( ) 1 (1 ) (1 ) (1 ) (1 ) 0r ri P h P x x R M i i M h

. ..(62)

Where 2

2 2 2 2 2

1 1 1 1 2 2 3 3 1 1 112 2 4 4, , , , , , , ,

AA

Ta Ri x R j j N N N N K K T

Here we assume that

1 11 1 2 1 1 2 1 2 1 3 3

1

(1 ) 11 , (1 ) 2 2 , (1 ) ,

N fb x L N x N N b N L x M L

K

, 14

1

1 NL

K

,

5 r rL P hP , we have

8 7 6 5 4 3 2

0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 0A iA A iA A iA A iA A ...(63)

Where

2 3 2

2 1 1 5 1 2 10 2

rL j b L j b M P

A




3 22 2 2

1 2 1 1 5 1 5 2 1 1 52 1 1 1 1 11 1 3 5 1 32

L L j b L b L L j b LL j b LA b L L j L

2 2 2

1 1 2 1 1 2 1 1 2 1 1 12 3 1 1 3

r r rr

j b M P L b M P j b M P Lb M P L j L

2 3 2 23 2 2 2 21 1 2 1 1 1 5 1 51 1 1 2 1 1 2 1 1 1

2 2 1 2 5 1 1 3 52 2 2

A rr

T j L j N b L b Lj M P N b L L j bA M P L L b L L

2 22 2 2

1 2 1 5 2 1 1 51 1 2 1 1 1 1 11 3 1 4 1 3 1 4 1 3

rL L b L L j b LL b M P N b N b

j L j L L L j L L L

2

2 2 213 5 1 4 5 1 52 1 1 1 1 1

2 1 1 3 1 3 5

2 2

1 3 1 1 1 1 2(1 ) (1 )

bb L L b L L b LL j L b

L j j L b L L

b L x R M M

2 2

2 3 2 1 42 1 1 1 1 1 1 2 1 1 11 3 2 3 1 2

1 1 1 1 2(1 )

r rr rr

b M P L b M P Lb M P L L j b M P jj L b M P L

x R M M

1 1 2 11 1 3 2 3 1

rr

L b M Pj j L b M P L

2 2 22 21 2 1 5 1 1 1 2 1 1 1

3 1 4 1 3 3 5 1 4 5 1 3 1 1 1 1 2(1 )(1 )L L b L N b L L j b

A j L L L b L L b L L b L x R M M

2 2 2 2 2

1 5 1 51 1 1 1 1 11 2 3 1 3 5 2 1 1 4 1 3 1 3 5

b L b LL b N b bL L jL b L L L j j L L L b L L

22 22 2

2 1 1 52 1 1 14 5 3 1 41 4

1 1 1 1 2 1 1 1 12 (1 )(1 ) (1 )

L j b LL j N bb L L b L b LL L

x R M M x R h M

2

2 21 1 12 1 1 3 3 5 1 4 5 1 3 1 1 1 1 2(1 )(1 )

L bL j j L b L L b L L b L x R M M

2 2

21 1 1 1 2 1 1 12 3 2 1 4 1 1 1 1 2 1 4 1 3(1 )

rr r

L j L b M P N bb M P L b M P L x R M M j L L L

2

1 1 2 1 1 1 2 11 1 3 2 3 1 1 1 4 1 3 2 3 1

r rr r

L b M P N b b M PL j L b M P L j j L L L b M P L

2 2 2

1 1 1 2 1 12 4 1 1 1 1 1 1 1 1 1 1 42 (1 )

rr

j j b M P N bb M P L x R M M x R M h L L

21 11 1 3 2 3 2 1 4 1 1 1 1 2(1 )r r

Lj j L b M P L b M P L x R M M

2 2 2

2 2 2 1 5 12 1 1 1 1 11 1 5 1 1 1 1 1 3 1 1 32

2 (1 )L N b LL N b j L

N b L N b x R N M j L

2 2 2

21 1 1 2 1 1 1 11 1 1 3 1 1 2 1 1 32

2r

r

N b j M P N b Lx R M N M P N b j L

2 21 1

1 1 1 1 1 2 1 2 5 1 1 1 2 5 2 1 22 22 2

A A

r r

T T

j j L M P L L j L L L b M P




2 22 21 1 1

4 1 2 1 4 1 3 3 5 1 4 5 1 3 1 1 1 1 2(1 ) (1 )N b b

A j L j L L L b L L b L L b L x R M M

2 2 2 2

2 1 52 1 1 1 11 1 1 2 1 1 1 4 2 1 1 4 1 3 5(1 ) (1 ) (1 )

b LL j N b bx R M M x R h M b L L j L L b L L

2 21 12 1 1 3 4 5 3 1 4 1 1 1 1 2 1 1 1 12 (1 )(1 ) (1 )

LL j j L b L L b L b L x R M M x R h M

2 2 2 2 21 51 1 1 1 2 1 1 4 5 3 1 4

1 2 1 4 1 3 1 3 5

1 1 1 1 2 1 1 1 12 (1 ) (1 ) (1 )

b LN b b L L j b L L b L b LL L j L L L b L L

x R M M x R h M

222 2 1

3 5 1 4 5 1 31 2 1 5 1 1 1

1 4 1 2 1 3

2

1 1 1 1 2(1 ) (1 )

bb L L b L L b LL L b L N b L

L L L L j L

x R M M

2

21 11 1 4 1 3 2 3 2 1 4 1 1 1 1 2(1 )r r

N bj j L L L b M P L b M P L x R M M

2 2

1 1 1 2 11 1 1 1 1 1 2 1 1 4 2 3 1(1 )

rr

j N b b M Px R M h x R M M j L L b M P L

2

1 1 1 11 1 3 2 4 1 1 1 1 2 1 1 1 1 1 1 4 1 32 (1 )r

L N bj j L b M P L x R M M x R M h L j L L L

2 1 1 1 1

2 3 1 2 4 1 1 1 1 2 1 1 1 12 (1 )r

r r

b M P L jb M P L b M P L x R M M x R M h

1 1 1 3 12 3 2 1 41 1 2 1 1

1 1 3 22 21 5 1 11 1 1 1 2

2 (1 )

2(1 )

r r x R N Mb M P L b M P LL L N b jL j L

N b L N bx R M M

2 22 2 2 2

1 1 5 1 12 1 5 1 1 1 1 11 4 1 3 2 1 1 3

2

1 1 1 3 1

2

(1 )

N b L N bL N b L N b Lj L L L L N b j L

x R N M

2 2 2 22 2 2 2 2 2 2

1 1 1 1 1 1 3 1 3 2 1 11 1 2 1 1 4 2 1 1 1 2 1 11 42 2

2 rr r rN b j x R M N L L M P N bj M P N b j L M P N b L b M P N b

L L

3 4 3

21 1 2 1 1 11 3 1 1 1 3 1 1 12

2r

r

N b M P N b Lj L x R M N P N b

2

2 1 12 2 21 1 12 5 2 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 2 5 12 2 2

2 2 4A A A

r r

T T L b j T

L L L b M P j j L j L L j M P L L

2 22 21 1 1

5 1 2 1 4 1 3 3 5 1 4 5 1 3 1 1 1 1 2(1 ) (1 )N b b

A L L j L L L b L L b L L b L x R M M

2 2 2

1 1 1 2 1 51 2 1 1 1 11 2 1 4 1 3 52

1 1 1 4

(1 ) (1 )

(1 )

x R M M b LL L j N b bL L L L b L L

x R h M b L

2 21 11 2 1 3 4 5 3 1 4 1 1 1 1 2 1 1 1 12 (1 ) (1 ) (1 )

LL L j L b L L b L b L x R M M x R h M

2

2 21 12 1 1 4 1 3 4 5 3 1 4 1 1 1 1 2 1 1 1 12 (1 ) (1 ) (1 )

N bL j j L L L b L L b L b L x R M M x R h M




2 2

2 21 12 1 1 4 3 5 1 4 5 1 3 1 1 1 1 2(1 )(1 )

N b bL j L L b L L b L L b L x R M M

21 12 1 1 3 1 1 1 2 1 1 1 4(1 ) (1 ) (1 )

LL j j L x R M M x R h M b L

2

21 11 1 4 1 3 2 3 2 1 4 1 1 1 1 2(1 )r r

N bL j L L L b M P L b M P L x R M M

2

1 1 1 1 2 11 1 1 1 1 1 2 1 1 4 2 3 1(1 )

rr

L j N b b M Px R M h x R M M L L L b M P L

1 1

1 1 3 2 4 1 1 1 1 2 1 1 1 12 (1 )r

LL j L b M P L x R M M x R M h

22 4 1 1 1 1 2 1 1 11 1 1 1

1 1 4 1 3 1 1 3

1 1 1 1 1 1 1 2

2 (1

) (1 )

rb M P L x R M M x R M hN b Lj j L L L j j L

x R M h x R M M

2 2

21 2 1 1 11 1 4 2 3 2 1 4 1 1 1 1 2 1 4 1 3(1 )r r

N b L N b N bj L L b M P L b M P L x R M M j L L L

2 2 22 22 2 2 2 1 5 11 1 1 1 1 1

1 1 5 1 1 1 1 1 3 1 1 1 3 1 1 42 (1 ) (1 )L N b Lj j N b N b

N b L N b x R N M x R N M L L

22 1 1 1 1 1 11 3 1 1 1 3 1 1 5 1 1 1 1 1 32

2 (1 ) 2L N b L N b j

j L x R N M N b L N b x R M N

2 2 2 2 2

21 1 1 2 1 1 11 4 1 3 1 1 1 3 1 1 2 1 1 42

rr

N b N b M P N b N bj L L L x R M N M P N b L L

21 1 1 1

1 3 2 1 1 1 1 1 3 1 1 1 1 2 1 2 5 122 2 2

A

r r

TN b Lj L M P N b x R M N L j L M P L L

2 2 21 12 1 1 1 1 1 2 2 5 2 1 1 1 1 1 1 12 2

2 2 4A A

r

T T

L b j j L M P L L L b j L L j

2

21 16 2 1 1 4 1 3 1 1 1 2 1 1 1 4(1 )(1 ) (1 )

N bA L j j L L L x R M M x R h M b L

2

2 212 1 1 4 1 1 1 1 2 1 1 1 1 4 5 3 1 42 (1 )(1 ) (1 )

N bL j L L x R M M x R h M b L L b L b L

2

2 21 11 2 1 4 1 3 1 1 1 1 2 1 1 1 1 4 5 3 1 42 (1 ) (1 ) (1 )

N bL L j L L L x R M M x R h M b L L b L b L

2 2

2 21 11 2 1 4 3 5 1 4 5 1 3 1 1 1 1 2(1 )(1 )

N b bL L L L b L L b L L b L x R M M

21 11 2 1 3 1 1 1 2 1 1 1 4(1 ) (1 ) (1 )

LL L j L x R M M x R h M b L

2 22 4 1 1 1 1 21 1 1

1 1 4 1 3 1 1 1 1 1 1 2 1 1 4

1 1 1 1

2 (1 )(1 )

rb M P L x R M MN b N bj j L L L x R M h x R M M j L L

x R M h

2

1 11 1 4 1 3 2 4 1 1 1 1 2 1 1 1 12 (1 )r

N bL j L L L b M P L x R M M x R M h




22 3 2 1 41 1 1

1 1 3 1 1 1 1 1 1 2 1 1 4 2

1 1 1 1 2

(1 )

(1 )

r rb M P L b M P LL N bL j L x R M h x R M M L L L

x R M M

2

22 1 1 11 4 1 3 1 1 1 3 1 1 5 1 12 (1 ) 2

L N b N bj L L L x R N M N b L N b

2

2 2 22 1 11 4 1 1 5 1 1 1 1 1 3 12 (1 )

L N b N bL L N b L N b x R N M

2

22 1 1 1 1 1 11 3 1 1 3 1 1 1 4 1 3 2 1 1 1 1 1 3(1 ) 2r

L N b L N b N bj L x R N M N b j L L L M P N b x R M N

2

21 1 1 1 11 1 1 3 1 3 1 4 1 1 1 3 1 1 12 r

N b L N b N bx R M N j L L L x R M N P N b

2 2 2 21 1 1

1 1 1 1 2 2 5 2 1 2 1 1 1 1 1 1 1 2 1 2 5 12 2 22 2 4

A A A

r r

T T T

L j L M P L L L b L b j L L j L M P L L

2

21 17 1 2 1 4 1 3 1 1 1 2 1 1 1 4(1 ) (1 ) (1 )

N bA L L j L L L x R M M x R h M b L

2

2 211 2 1 4 4 5 3 1 4 1 1 1 1 2 1 1 1 12 (1 ) (1 ) (1 )

N bL L L L b L L b L b L x R M M x R h M

2

212 1 1 4 1 1 1 2 1 1 1 4(1 ) (1 ) (1 )

N bL j L L x R M M x R h M b L

2 21 1 1 1 21 1 1

1 1 4 1 3 1 1 1 1 1 1 2 1 1 4

1 1 1 1 2 4

2 (1 )(1 )

r

x R M MN b N bL j L L L x R M h x R M M L L L

x R M h b M P L

2 2

1 1 1 21 2 1 1 11 1 4 1 4 1 3 1 1 3 1 1

1 1 1 2

(1(1 )

x R M hN b L N b N bj L L j L L L x R N M N b

x R M M

2 2

22 1 1 1 1 11 4 1 1 1 3 1 1 5 1 1 1 1 1 3 1 4 1 32 (1 ) 2

L N b N b N b N bL L x R N M N b L N b x R M N j L L L

2

2 21 1 1 11 4 2 1 1 1 1 1 3 1 2 5 1 2 2 2 1 1 1 12 2

2 2 2A A

r r

T TN b N bL L M P N b x R M N L L L L b M P L b L j L

2

218 1 2 1 4 1 1 1 2 1 1 1 4(1 )(1 ) (1 )

N bA L L L L x R M M x R h M b L

2 2

21 2 1 11 1 4 1 1 1 1 1 1 2 1 4 1 1 3 1 1(1 ) (1 )

N b L N b N bL L L x R M h x R M M L L x R N M N b

2

21 1 11 1 1 3 1 4 1 22

ATN b N bx R M N L L L L b

VIII. STATIONARY CONVECTION

Here we consider the stationary convection in case 2 0M , therefore in stationary convection, the

marginal state will be characterized by putting 1 0 in equation (62) which reduces to




22

2 21 12 1 4 1 22

12

1 311 1 4 2 1 1 1 1(1 )

ATN bL b L L L L b

R

N b NN bx L L L M M L h

where 1 1h h

On putting the values of 1 2,L L and 4L , we obtain

22

22 1 1 11 3 1 2 1 3 1 22

1

12

1 31 11 1 2 1 3 1 1 1 2

1

11 2 1 2

12 1 (1 ) ( 2 )

ATN N bb x M N b N b x M N b N

K

R

N b NN N bx N b N x M M h N b N

K

...(64)

In the absence of rotation (i.e., 1

0AT ), equation (64) reduces to

22 1 1

1 3 1 2

1

1

1 31 1 3 1 1 1 2

11 2

1 (1 ) 2

N N bb x M N b N

K

RN N b

x x M M h N b N

...(65)

Which is similar to the equation derived by R. Mittal and U.S. Rana[9].

In case of ferromagnetic fluid, we set 1 0N and keeping 2N arbitrary in (65), we get

2 2

1 3 1 1 3

1

1 1 1 1 3 1 1 1 1 1 3 1

(1 ) (1 ) (1 )

1 (1 ) 1 (1 )

b x M x x MR

x h K x M M x h K x M M

...(66)

Which is the expression for Rayleigh number for

ferromagnetic dusty fluid in a porous medium. Further if we set 3 0M in equation (66), we get

2

11

1 1 1

(1 )xR

x h K

...(67)

which is the expression for the classical Rayleigh-

Bénard number for Newtonian dusty fluid in

porous medium.

Now to investigate the effects of medium

permeability parameter 1( )K non-buoyancy

magnetization parameter 3( )M , micropolar

coupling parameter 1( )N , spin-diffusion parameter

2( )N , micropolar heat conduction parameter 3( )N

, dust particles parameter 1( )h and rotation

parameter 1

,AT we examine the behaviour of

1 1 1 1 1 1

1 3 1 2 3 1

, , , , ,d R d R d R d R d R d R

d K d M d N d N d N d h and 1

1A

d R

d T

analytically as follows

From equation (64), we have

11 3 1 22

11

1 311 1 3 1 1 1 2

11 2

1 (1 ) 2

Nb x M N b N

Kd R

N b Nd Kx x M M h N b N

2 3 3 3 2 2 32 1 1 1 1 2 1 1 1 2

1 2 21 1 1 1

22

1 11 2

1

1 2 4 (1 ) 22

1( 2 )

ATN b N N N N b N N b NN b N b

K K K K

N N bN b N

K

If

2 22

1 1 1 2

12 1 1 1 1 1

1 21 1 1 1, m a x . 1 , 1

2A

N N K NT

N K N N N N

and 1 3

1 2

N Nh N

, then 1

1

0d R

d K

,

thus, under these conditions the medium

permeability has stabilizing effect.




In the absence of rotation (i.e., 1

0 )AT , 1

1

0d R

d K

when 2 1 2

21 1

2 N K N

N N

and

1 3

1 2

N Nh N

, thus,

under these conditions, the medium permeability

has destabilizing effect in the absence of rotation.

Again from equation (64), we have

22

22 1 1 1 1 3 1 11 2 1 22

11 3 1 11

223 1 31 1 1 3 1

1 1 2 1 1 2

1

1 1 (1 )( 2 ) 2

(1 ) (1 )

1 1 (1 )( 2 ) ( 2 )

ATN N b x M M xb N b N b N b N

Kx M x Md R

d M N b NN N b x M Mx N b N h N b N

K

Clearly 1

3

0d R

d M when

1

1

2

b

K

and

1 3

2

1

N NN

h

, thus the non-buoyancy magnetization

has destabilizing effect under the conditions

1

1

2

b

K

and 1 3

2

1

N NN

h

. In the absence of

rotation, 1

3

0d R

d M , under the same conditions,

which implies that the destabilizing behaviour of

non-buoyancy magnetization remains unaffected

by rotation.

In the absence of rotation and micropolar viscous

effects 1( 0 )N , 1

3

d R

d M is always negative,

implying the destabiizing effect of non-buyancy

magnetization, which is derived by Sunil et al.[19].

Again from (64), we have

1 31

1 1 1 3 1

(1 )

1 (1 )

b x Md R

d N x x M M

2 2

1 31 1 1 11 2 1 1 2 1 2

1 1

2

1 2 1 1 1 1 11 2 1 22

1 1 1

1 12 2 2 ( 2 )

2 2 (1 ) 2 14 ( 2 ) ( 2 )

A

N b NN N b N N bN b N h N b N b N b N

K K

TN b N N N b N N bN b N b N b N

K K K

2

211 22

2

1 3 31 2 1 1 1 11 1 2 1 2 1

1 1 1

22

11 11 2 1 1 2

1

2

2 2 (1 ) 2 1( 2 ) ( 2 ) 2

12 ( 2 )

AT

N b N

N b N b NN b N N N b N N bh N b N N b N h

K K K

N bN N bN b N h N b N

K

2

3N

If 1

2 b

K

or

1

1

2

b

K

,

3

12 0b N

h

or 3 3 1

1 1

2

,2

b N N Nh h

N

and

1

1

ATb

K

or

2

1 2

1

,A

bT

K

then

1

1

0d R

d N , thus the coupling parameter has a stabilizing effect under the conditions

3 1 3

1

1 2

1,

2 2

N N b Nbh

K N

and

2

1 2

1

A

bT

K

In the absence of rotation, 1

1

d R

d N will always be positive if

1

1

2

b

K

and

3

1

b Nh

, which is derived by

R. Mittal et. al[9].

Again from (64), we have




1 31

2 1 1 3 1

1

1 (1 )

b x Md R

d N x x M M

222

2 3 3 1 31 11 2 1 1 1 22 1

1 1 1

2 221 3 1 31 2 1

1 1 2 1 1 1 23 11 1 1

1

1

12 ( 2 )

2 22 2

1

A

A

N N b N NN N bb N b N N h T N b N

K K K

b N N N NN b N b NbN T N b N h N b h N

K K K

N

K

222

1 311 2 1 1 22 2

N b NN bN b N h N b N

If 3

1

1

0N

hK

or 3

1

1 1

2, 0

N bh

K K

or

1

1

2

b

K

and

1 3

1 2 0N N

h N

or 1 3

1

2

N Nh

N

, then 1

2

0d R

d N

,

which implies that the spin-diffusion parameter has a destabilizing effect when

1 3 3

1

2 1

N N Nh

N K

and

1

1

2

b

K

In the absence of rotation, 1

2

d R

d N is always negative, when 3

1

1

Nh

K

which is derived by R. Mittal et. al[9].

Thus the presence of rotation affects the dust particles i.e., it gives also the lower bound of 1h but in the absence

of rotation, only upper bound of 1h was obtained.

From (64), we have

1 1

3 1

d R N b

d N x

22

22 1 1 11 3 1 2 1 3 1 22

1

2

2 1 31 2 1 21 1 3 1 1 1 2

1 1 1

11 2 1 2

2 21 (1 ) 2


K

N b NN b N N b N bN x M M h N b N

K K K

If 1

20

b

K

or 1

1

2

b

K

, then 1

3

0d R

d N

, thus the micropolar heat conduction parameter has a stabilizing

effect when 1

1

2

b

K

, which is derived by R. Mittal et. al[9]. Hence the stabilizing behaviour of micropolar

heat conduction parameter is independent of presence of rotation.

From (64), we have

22

22 1 1 11 3 1 2 1 3 1 22

11 21

21 1 1 3 1 2 1 31 2 1 2

1 1 1 2

1 1 1

1(1 ) 2 1 2

2

1 (1 ) 2 22


KN b Nd R

d h x x M M N b NN b N N b N bN h N b N

K K K

If 1

20

b

K

or 1

1

2

b

K

, then 1

1

0d R

d h , thus the dust particles has a destabilizing effect when

1

1

2

b

K

.

Hence the destabilizing behaviour of dust particles is independent of rotation.

From (64) we have

2

1 3 1 21

2 2 1 31 2 1 211 1 1 3 1 1 1 1 2

1 1 1

(1 ) 2

2 21 (1 ) 2

A

b x M N b Nd R

d T N NN b N N b N bx N x M M h N b h N

K K K




If 1

20

b

K

or 1

1

2

b

K

and

1 3

1 2

N Nh N

, then 1

1

0

A

d R

d T , thus the rotation has a stabilizing effect when

1

1

2

b

K

and

1 3

1 2

N Nh N

. In the absence of micropolar viscous effect 1( 0 )N , 1

1A

d R

d T is always positive

which is derived by Sunil et. al[19].

For sufficiently large value of 1 ,M we obtain the result for magnetic mechanism as follows:

22

22 1 1 11 2 1 22

11 3

2 21 3 1 31 1

1 2 1 1 2

1

1( 2 ) 2

1

12 2

A

m

b TN N bb N b N N b N

Kx MR

x M N b NN N bN b N h N b N

K

1

1

1is la rg e s o th a t 0M

M

Where mR is the magnetic thermal Rayleigh number.

4 3 2

1 3 1 1 1 2 1 3 1 4

2 2

1 3 1 5 1 6 7

(1 ) (1 ) (1 ) (1 ) (1 )

(1 ) (1 )m

x M x S x S x S x S

R

x M x S x S S

...(68)

Where 2 22 2 2

22 2 1 1 1 1 2 2 1 1 11 2 2

1 1 1

2 2, 2 ,

AT NN N N N N N N N N NS S

K K K

2 22 21 2 1 1 31 1 1 1 2 2 1 1

3 4 5 1 22 21 1

4 42 2, , ,

A AN N T T N N NN N N N N NS S S h N

K K

2 2 2

1 31 1 2 2 1 1 1 16 1 2 1 1 7 1 1

1 1 1

2 2 2 22 , 2

N NN N N N N N N NS h N h N S h N

K K K

If 1 0N , then equation (69) reduces to

2

12 11 3 1 12

2

1 3 1 1

(1 ) (1 ) (1 )A

m

T K

x M x x

R

x M h K

Which is similar to expression derived by Sunil et. al[19]. For very large value of 3M , equation (68) reduces to

4 3 2

1 1 1 2 1 3 1 4

12

1 1 5 1 6 7

(1 ) (1 ) (1 ) (1 )

(1 ) (1 )m

x S x S x S x S

R R

x x S x S S

(in the absence of magnetic parameter)

Thus, in case of stationary convection, the

ferromagnetic micropolar fluid behaves like an

ordinary micropolar fluid for sufficiently large

values of 1M and 3 .M

Now differentiating w.r.t. to 1x on both sides of

(68), we have

6 5 4

1 3 1 3 1 1 5 1 2 5 1 6 1 3 5 2 6 1 7

3 2

1 4 5 3 6 2 7 1 4 6 3 7 1 4 7

2 5 4 3

1 3 1 3 1 1 5 1 2 5 1 6 1 2 6 1 7

2

1

1

( 2 ) (1 ) (1 ) (1 )

(1 ) (1 ) (1 )

(1 ) (1 ) ( 2 ) (1 ) 3 (1 ) 2 4

(1 )m

x M x M x S S x S S S S x S S S S S S

x S S S S S S x S S S S x S S

x M x M x S S x S S S S x S S S S

x Sd R

d x

3 6 2 7 4 5 1 3 7 7 4

24 2 2

1 3 1 5 1 6 7

3 (1 ) 2

(1 ) (1 )

S S S S S x S S S S

x M x S x S S




For minimum value of mR we must have 1

0md R

d x

7 2 6 2

1 3 5 3 1 5 1 3 5 5 6 1 5 1 64 2 4 2x M S M S S x M S S S S S S S

2 2

3 5 6 5 6 5 7 1 5 2 5 1 6 3 5 2 6 1 75

1

1 5 2 5 1 6

6 6 2 5 7 5 3

2

M S S S S S S S S S S S S S S S S S Sx

S S S S S S

2 24 5 6 5 6 5 7 6 7 2 5 1 6 3 5 2 6 1 7 4 51 3

2 7 4 5

4 2 6 4 2 4 8 4 2 8

2

S S S S S S S S S S S S S S S S S S S Sx M

S S S S

2 2 2

5 6 7 5 6 5 7 6 7

3

1 5 2 5 1 6 3 5 1 7 1 3 1 5 2 5 1 6 3 5 1 7

4 5 3 6 2 7 4 5 4 6 3 7

2 2 2

1 0 6 2 2 2 5 6 2 6 6

3 3 3

S S S S S S S S S

S S S S S S S S S S x M S S S S S S S S S S

S S S S S S S S S S S S

1 5 2 5 1 6 3 5 2 6 1 7 4 5 3 6 2 7 4 52 0 1 4 2 8 2 4 2S S S S S S S S S S S S S S S S S S S S

2

1 3 1 5 2 5 1 6 3 5 2 6 4 5 3 6 4 6 4 54 4 2 4 2 3 2 2x M S S S S S S S S S S S S S S S S S S

1 5 2 5 1 6 3 5 2 6 4 5 3 6 4 6 4 52 0 1 6 8 1 2 6 6 4 2 3S S S S S S S S S S S S S S S S S S

1 1 1 5 2 5 1 6 3 5 2 6 1 7 4 5 3 6 2 7 4 6 3 7 4 7x M S S S S S S S S S S S S S S S S S S S S S S S S

1 5 2 5 1 6 3 5 2 6 1 7 4 5 3 6 2 7 3 7 4 71 0 9 7 8 6 4 7 5 3 2S S S S S S S S S S S S S S S S S S S S S S

1 5 2 5 1 6 3 5 2 6 1 7 4 5 3 6

2 7 4 6 3 7 4 7

2 2 2 2 2 2 2 20

2 2 2 2

S S S S S S S S S S S S S S S S

S S S S S S S S

...(69)

Hence the magnetic thermal number mR has its minimum value when condition (69) holds.

IX. OSCILLATORY CONVECTION Comparing the imaginary parts of both sides of equation (63), we get

6 4 2

1 1 1 3 1 5 1 7 0A A A A

...(70)

Where 3 2 2 2 22 3

2 1 2 1 1 5 2 1 1 5 2 1 1 3 52 1 11 2 2 2

2 2L L j L L j L L j L LL jA b b

2 2 2 22 2 2 3

1 1 2 3 1 2 1 31 1 2 1 2 1 1 2 1 1

2 2

r rr r rj M P L j M P Lj M P L M P j M P L

3 2 2 23 2 2 2 3

3 2 1 2 1 5 1 1 2 3 1 1 1 2 3 1 12 1 1 1 1 2 1 1 1 2 13 2 2 2 2

L L L N L L L j L L L jL j N L L j L LA b b

2 2 2 2 2 2 2 22 2 2 2

1 2 3 1 1 1 1 5 1 1 3 5 2 3 1 1 2 5 1 1 12 4 1 1 2 4 1 1

2 2

L L L j N L N L L L L j L L j NL L j L L j

2 2 2 22 3 2 2 22 22 3 1 1 1 2 3 1 1 1 1 1 2 31 2 1 1 1 2 1 1 1 1 1 2

2 3 1 12 2 2

rr rL L j L L L j j N M P LL L j L M P N j N M P

L L j

2 2 2 2 3 22 2 3 2 2 2

2 5 1 1 1 2 5 3 1 1 1 1 2 5 1 11 2 1 1 2 1 1 1 1 1 2 1

2 2 2 2 2

2 2r rL L j N L L L j N L L L Nj M P N L N j N M P j

2 2 2 2 2 23 2

3 1 2 1 1 1 2 4 1 1 5 1 2 3 5 1 1 2 3 5 1 1 1 2 4 5 1 11 1 2 1

2

r rL j M P N L L L j L L L L L L L L L j L L L L jL M P N

b

2 2 2 2 2

21 2 3 5 1 1 1 2 5 1 1 2 3 5 1 2 4 5 1 1 1 2 3 4 1 1

1 2 3 5 1 12

L L L L j L L L L L L L L L L j L L L L jL L L L j

2 2

2 2 2 22 4 5 1 1 1 2 4 5 1 1 1 2 3 5 1 1

2 3 4 5 1 1 1 2 3 5 1 1 2 3 5 1

L L L j L L L L j L L L L jL L L L j L L L L j L L L j




2 2 22 2

2 1 2 4 5 1 1 1 3 2 1 1 1 3 2 11 4 1 1 2 1 4 1 2 12 3 4 5 1 1

r rr rL L L L j L L M P j L L M PL L j M P L L j M P

L L L L j

2 22 2 22 21 3 1 1 2 1 3 1 2 1 3 1 1 21 1 2 4 1 1 2

1 3 1 1 2 3 4 1 1 22

r r rr rr r

L L j M P L L M P L L j M PL M P L j M PL L j M P L L j M P

2

2 2 2 21 3 1 1 24 1 1 2 1 4 1 1 21 3 1 1 2 3 1 2 3 4 1 1 2

rr rr r r

L L j M PL j M P L L j M PL L j M P L j M P L L j M P

33 32

2 1 12 1 1 1 1 1 3 1 1 1 1 1 1 1 31 4 1 1 2 1

2 2 2

(1 ) ArL j TL N j x R N M N j x R M NL L j M P

2 2 2

3 21 2 1 2 1 2 1 1 1 1 11 1 1 1 2 1 1 1 1 2

2 (1 )(1 ) (1 ) (1 ) (1 )

L L j L j L j x R h Mx R M M x R M M

3 3

2 2 1 2 1 1 1 1 12 3 1 1 1 1 1 2 1 1 1 2 1 1 1 2(1 )(1 ) (1 )(1 ) (1 )

L L j L jL L j x R M M x R M M x R M M

2 2 2 2 3

2 21 1 1 1 1 1 11 1 1 2 1 1 1 3 1 1 1 1 1 2 1 1 1 2

2(1 ) (1 ) (1 )

j j L jx R M M x R M h L j x R M M x R M M

2 2

2 5 1 1 1 1 221 1 11 1 1 1 1 2 1 2 5 12 2 2

2 2A A r A

r

T L L j T j M P T

j j L M P L L

3 2 2 24 2 32 1 1 1 1 1 1 2 1 1 1 1 2 1 1

5 1 1 1 32 2

L N j N L L N L L NA b N b L

2 2

22 1 1 1 2 1 1 1 2 1 1 1 13 1 4 1 3 1 4 1 3 1 1( )

L j N L j N L N j NL L L j L L L N

4 2 4 2 2 2 4 4 4

2 1 1 5 2 1 5 1 2 1 1 1 1 1 1 2 2 1 11 32 2 2 2

2 2 2 r rL N L L N L L N L N M P M P N

j L

2 2 2

2 1 1 2 1 1 1 2 4 1 1 1 2 4 11 2 1 4 1 3 1 3 3 5 1 4 5

L L N L L L j L L Lb L L j L L L L L L L L

2

1 51 2 1 1 11 3 5 1 2 1 3 3 1 4 2 1 1 4 1 3 3 1 4

LL L N LL L L L j L L L L j j L L L L L

2 2

2 4 5 1 1 1 1 2 1 1 1 12 1 1 4 1 3 3 5 1 4 5 2 4 1 1 3

L L L j N L j N LL j L L L L L L L L L j j L

2 2 2 2

2 31 1 1 1 1 2 1 1 1 1 2 4 1 12 3 2 1 4 2 3 1

2 1 4

rr rr r r

r

M P LL N L N M P j N M P L j NM P L M P L M P L

M P L

2 3

21 1 2 5 2 1 1 1 11 4 1 3 1 1 1 3 1 1 1 3 12

2(1 ) (1 )

N L L L N jj L L L x R N M x R N M

2 2 2 3 3 2

1 2 4 5 1 1 2 1 5 1 1 1 1 1 1 21 3 1 1 1 3 1 4 1 32

2 rL L L L N L N L L N N M P

j L x R M N j L L L

2 2 2

1 4 2 1 1 1 2 1 11 3 1 2 1 4 1 3 3 5 1 4 5

r rL L M P N N M P Lj L b L L j L L L L L L

3 2

2 1 51 2 1 11 1 1 2 1 2 4 1 3 5(1 ) (1 )

LL L Nx R M M L L L L L

1 11 2 4 5 1 3 2 4 5 1 1 4 1 3

LL L L L j L L L L j j L L L




2

2 1 1 11 1 1 1 2 1 1 1 1 1 2 4 1 3 5 1 4 52 (1 ) (1 ) (1 )

L j Nx R M M x R h M L L L j L L L L

2 2

2 1 1 11 1 1 2 1 1 4 1 3 2 3 2 1 4(1 ) (1 ) ( ) r r

L j Nx R M M L j L L L M P L M P L

3 2

21 1 1 2 11 1 1 2 1 4 2 3 1(1 )

rr

L N M Px R M M L L M P L

1 11 1 3 2 4 1 4 2 1 4 1 3( )r r

LL j L M P L j L M P j L L L

2

1 1 11 1 1 1 2 1 1 1 1 1 1 4 2 3 2 1 42 (1 ) r r

j Nx R M M x R M h j L L M P L M P L

2

2 21 1 2 11 1 1 1 2 1 4 1 3 1 1 1 3 1(1 ) (1 )

j N L Nx R M M j L L L x R N M

1 1 2 1 1 1 11 1 3 1 1 1 3 1 1 3

2(1 ) (1 )

j L N Lx R N M x R N M j L

21 1 1 1

1 1 1 3 1 4 1 3 1 1 1 3 12

N j Nx R M N j L L L x R M N

2 21 1 1 1 1

1 3 1 1 1 3 1 1 1 1 12

22 2

AT LN Lj L x R M N j j L

2 1 2 211 1 1 1 1 12

4AL T

j L L j

2 1 2 1 1

1 2 1 4 1 3 1 1 1 1 2 1 1 1 2 1 1 1(1 ) (1 ) (1 ) (1 ) (1 )L L j

L L j L L L x R M M x R M M x R h M

1 11 2 1 3 1 1 1 1 2 1 1 1 1( 2 (1 ) (1 ) (1 )

LL L j L x R M M x R h M

2 1 1 4 1 3 1 1 1 1 2 1 1 1 1( ) 2 (1 ) (1 ) (1 )L j j L L L x R M M x R h M

2 1 1

2 1 1 4 1 1 1 1 2 2 1 1 3 1 1 1 2 1 1 1(1 ) (1 ) (1 ) (1 ) (1 )L

L j L L x R M M L j j L x R M M x R h M

2 1 1 1

1 1 4 1 3 1 1 1 1 2 1 1 1 1 1 1 2( ) (1 ) (1 )L j

L j L L L x R M M x R M h x R M M

1 1 1 1 21 1

1 1 3 1 1 4 1 3 1 1 1 1 2 1 1 1 1

1 1 1 1

2 (1 )( ) 2 (1 )

x R M MLL j L j j L L L x R M M x R M h

x R M h

21 1

1 1 3 1 1 1 1 1 1 2 1 1 4 1 1 1 1 2(1 ) (1 )L

j j L x R M h x R M M j L L x R M M

2 2 21 1

1 1 1 1 2 1 2 5 1 2 2 5 1 1 1 1 1 12 22 2 ( ) ( 4 )

A A

r r

T T

L j L M P L L M P L L j L L j

4 2 2 24 32 1 1 1 2 4 1 1 1 2 1 2 4 1 1

7 3 1 42

3( )

L N L L L N L L N L L j NA b b L L

42 4 2

22 5 12 1 1 2 1 2 4 1 11 4 1 3 1 2 4 1 4 1 32 2

2rL L NL N N M P L L L N

j L L L b L L L j L L L

2 2 3

2 21 2 4 5 1 1 4 1 2 2 1 11 2 4 3 1 4 1 2 4 1 1 1 3 12

(1 )r

L L L L N L L N M P L NL L L L L L L L j x R N M

23 3 3 2

1 2 4 5 12 1 1 1 1 1 1 1 4 1 21 1 3 1 1 1 1 3 1 1 1 32 2 2

2 2(1 )

rL L L L NL N N N L L N M P

x R N M x R M N x R M N




2 21 1 1 21 2 1 1 1 2 1

1 1 1 2 1 1 1

1 1 1 1

2 (1 ) (1 )(1 ) (1 ) (1 )

(1 )

x R M ML L N L L Nb x R M M x R h M

x R h M

+

2 22 2 2 1 1 1 1 11 2 4 5 1 1 1 2 1 1 1 1 1 1 1 1 1 2(1 ) (1 ) (1 ) (1 )

L j N L NL L L L x R M M x R h M x R M h x R M M

2 2 22 21 1 1 1 1 1

1 1 1 1 2 1 1 1 1 1 4 2 1 1 1 1 1 1 2

2(1 ) (1 )r

L N L N j Nx R M M x R M h L L M P x R M h x R M M

1 1 1 1 32 1 1 2 4 1 1

1 1 3 1 1 4 1 3 1 1 3 1 1 4 1 3

2(1 ) (1 ) ( )

N x R M NL N L L L Nx R N M j L L L x R N M j L L L

2

1 2 1 2 1 2 1 4 1 3 1 1 1 221 4 1 1 1 11 1 1 3 1 1 1 12 2

1 1 1

{ (1 )(1 )22 2

(1 )}

A AL L T L T L L j L L L x R M ML L Nx R M N j L j L

x R h M

2

1 2 4 1 1 1 1 2 1 1 1 1 1 2 4 1 1 1 1 2 1 1 12 (1 ) (1 ) (1 ) (1 ) (1 ) (1 )L L L x R M M x R h M L L L j x R M M x R h M

2

1 1 4 1 3 1 1 1 1 1 1 2 1 4 1 1 1 1 2 1 1 1 1( ) (1 ) 2 (1 )L j L L L x R M h x R M M L L x R M M x R M h

2

1 11 1 4 1 1 1 1 1 1 2 2 5 22

(1 )A

r

L T

j L L x R M h x R M M L L M P

It is clear from equation (70) that 1 may be either zero or non-zero, which implies that the modes may be

either non-oscillatory or oscillatory.

In the absence of dust particles 1( 0 , 0 , 0 )h f and rotation 1

( 0 )AT , the equation (70) becomes

2 2

2 2 4 2 3 1 1

1 3 1 2 2 5 1 1 2 2 5 1 3 4 1 2 2 5

24r r r

L j N bb L j M P L L M P L L L L L j b M P L L

2 2 2 2 2 2

4 1 2 2 5 1 3 2 2 5 1 2 3 12r rL j b M P L L L L b M P L L L L L j b

2 2 2

2 3 4 1 3 1 1 1 2 2 1 12 (1 ) (1 )L L L j b L j x R M L M M

2 3 2 3 4 3

2 2 21 2 3 1 2 4 1 1 12 2 5 1 2 4 1 1 2 3 42

22 2r

L L L N b L L j N b N bM P L L L L L j b L L L L b

22 2 2 1 1

1 4 1 2 2 5 1 4 2 2 5 1 1 2 2 1 12 (1 ) (1 )r r

j NL L N b M P L L L L b M P L L x R M b L M M

1 31 1

4 1 1 3 2 1 1 1 1 1 3 2 1 1(1 ) (1 )L Lj N

L x R N b L M M N x R N b L M M

2

1 4 1 1 1 2 2 1 1 1 2 3 1 1 1 22 (1 ) (1 ) (1 ) (1 ) 0L L j x R M L M M L L L x R M M

...(71)

In the absence of micro-viscous effect 1( 0 )N and microinertia 1( 0 )j , equation (71) reduces to

2 2 2 2 2 2 2 2

1 1 1 3 2 5 2 1 4 2 5 2 1 2 3 4 1 2 3 1 1 1 22 (1 ) (1 ) 0r rL L b L L M P L L b L L M P L L L L b L L L x R M M

...(72)

Now equation (72) yields that

1 0 when 2

2 22

3 1

1 12

r r

r

M P Pa

M P K

Thus, in the absence of dust particles, rotation, micro-viscous effect and microinertia, the sufficient condition for

non-oscillatory modes is given by

2

2 22

3 1

1 12

r r

r

M P Pa

M P K

...(73)

X. OBSERVATIONS In stationary convection, the variations of thermal Rayleigh number R, with respect to the variations of

Medium permeability ( 1K ), (see fig. 2 to 3); Non-Buoyancy magnetization (M3), (see fig. 4 to 5): Micropolar




heat conduction parameter ( ) (see fig. 6 to 7); Dust particles parameter (h1), (see fig. 8); and Taylor number

1( )AT , (see fig. 9); respectively have been predicted by the graphs given below:

Fig. 2: Marginal instability curve for the variation of R vs 1K for =0.5,

1AT =100, h1=1.2, N1=0.2,

N2=2, N3=0.5, M1=100, M3=5.

Fig. 3: Marginal instability curve for the variation of R vs 1K for =0.5,

1AT =0, h1=1.2, N1=0.2,

N2=2, N3=0.5, M1=100, M3=5.

Fig. 4: Marginal instability curve for the variation of R vs M3 for =0.5,

1AT =100, h1=1.2, N1=0.2,

N2=2, N3=0.3, M1=100, K1=0.5.




Fig. 5: Marginal instability curve for the variation of R vs M3 for =0.5, h1=1.2, N1=0.2, N2=2,

N3=0.3, M1=100, K1=0.5, 1AT =0.

Fig. 6: Marginal instability curve for the variation of R vs N3 for =0.5, h1=1.2,

1AT =10,

N1=0.2, N2=2, M1=100, K1=0.5, M3=5.

Fig. 7: Marginal instability curve for the variation of R vs N3 for =0.5, h1=1.2,

1AT =10, N1=0.2,

N2=2, M1=100, a=1, K1=0.5.




Fig. 8: Marginal instability curve for the variation of R vs h1 for =0.5,

1AT =100, N1=1.2, N2=2,

N3=0.3, K1=0.5, M1=100, a=1.

Fig. 9: Marginal instability curve for the variation of R vs

1AT for =0.5, N1=0.2, N2=2, N3=0.3,

M1=1000, K1=0.5, h1=1.2, a=1.

XI. CONCLUSIONS

For Stationary Convection:

1. The medium permeability has stabilizing effect when

22

1 1

12 1

1,A

N NT

N K

2

1 2

1 1 1 1

21 1 1 1m a x . 1 , 1

2

K N

N N N N

and 1 3

1 2

N Nh N

(see

fig. 2) 2. In the absence of rotation, the medium permeability has destabilizing effect when

2 1 2

21 1

2

2

N K N

N N

and

1 3

1 2

N Nh N

(see fig. 3)

3. The non-buoyancy magnetization has destabilizing effect when

1

2K

b

and 1 3

1

2

N Nh

N

(see fig. 4)

Also the destabilizing behaviour of non-buoyancy

magnetization remains unaffected by rotation under

the same conditions (see fig. 5)

4. The coupling parameter has a stabilizing effect

when




21 3 3

1 11 221

2, ,

2A

N N b NbT K h

b NK

5. The spin-diffusion parameter has a destabilizing effect when

1 3 3

1

2 1

N N Nh

N K

and 1

2K

b

6. The micropolar heat conduction parameter has

a stabilizing effect when 1

2K

b

(see fig. 6

& 7) 7. The dust particles has a destabilizing effect

when 1

2K

b

(see fig. 8)

8. The rotation has a stabilizing effect when

1

2K

b

and 1 3

1

2

N Nh

N

(see fig. 9)

9. In case of stationary convection, the

micropolar ferromagnetic fluid behaves like an

ordinary micropolar fluid for sufficiently large

values of 1M and 3M .

10. The magnetic thermal Rayleigh number mR

has its minimum value when equation (70) is

identically satisfied.

For Oscillatory Convection:

In the absence of dust particles, rotation, micro-

viscous effect and microinertia, the sufficient

condition for non-oscillatory modes is given by

2

2 22

3 1

1 12

r r

r

M P Pa

M P K

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