RESEARCH ARTICLE OPEN ACCESS Effect of Rotation on … - 5/B0603050428.pdf · enough that the...
Transcript of RESEARCH ARTICLE OPEN ACCESS Effect of Rotation on … - 5/B0603050428.pdf · enough that the...
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
www.ijera.com 4 | P a g e
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
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
www.ijera.com 5 | P a g e
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
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
www.ijera.com 6 | P a g e
+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
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
www.ijera.com 7 | P a g e
(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)
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
www.ijera.com 8 | P a g e
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
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
www.ijera.com 9 | P a g e
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
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
www.ijera.com 10 | P a g e
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
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
www.ijera.com 11 | P a g e
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
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
www.ijera.com 12 | P a g e
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
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
www.ijera.com 13 | P a g e
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
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
www.ijera.com 14 | P a g e
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
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
www.ijera.com 15 | P a g e
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
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
www.ijera.com 16 | P a g e
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.
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
www.ijera.com 17 | P a g e
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
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
www.ijera.com 18 | P a g e
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
ATN N bb x M N b N b x M N b N
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
ATN N bb x M N b N b x M N b N
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
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
www.ijera.com 19 | P a g e
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
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
www.ijera.com 20 | P a g e
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
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
www.ijera.com 21 | P a g e
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
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
www.ijera.com 22 | P a g e
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
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
www.ijera.com 23 | P a g e
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
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
www.ijera.com 24 | P a g e
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.
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
www.ijera.com 25 | P a g e
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.
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
www.ijera.com 26 | P a g e
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
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
www.ijera.com 27 | P a g e
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
REFERENCES
[1] Abraham, “Rayleigh-Bénard convection in
a micropolar ferromagnetic fluid”, Int. J.
Engg. Sci., vol. 40(4), pp. 449-460,
(2002).
[2] B.A. Finlayson, “Convective instability of
ferromagnetic fluids”, J. Fluid Mech.
40(4), 753-767 (1970).
[3] D.A. Nield, A. Bejan, “Convection in
porous media”, third ed. Springer, New
York, 2006.
[4] J.W. Scanlon, L.A. Segel, “Some effects
of suspended particles on the onset of
Bénard convection”, Phys. Fluids 16,
1573-1578 (1973).
[5] M. Zahn, D.R. Greer,
“Ferrohydrodynamic pumping in spatially
uniform sinusoidally time-varying
magnetic fields”, J. Magn. Magn. Maths
149, 165-173 (1995).
[6] P.G. Siddheshwar, “Convective instability
of ferromagnetic fluids bounded by fluid-
permeable, magnetic boundaries”, J.
Magn. Magn. Maths. 149(1-2), 148-150
(1995).
[7] P.J. Stiles and M. Kagan,
“Thermoconvective instability of a
horizontal layer of ferrofluid in a strong
vertical magnetic field”, J. Magn. Magn.
Mater, vol. 85(1-3), pp. 196-198, (1990).
[8] R.E. Rosensweig, “Ferrohydrodynamics”
Dover Publication INC, Mineola, New
York (1985).
[9] Reena Mittal and U.S. Rana, “Effect of
dust particles on a layer of micropolar
ferromagnetic fluid heated from below
saturating a porous medium”, Applied
Mathematics and Computation, Vol. 215,
pp. 2591-2607, (2009).
[10] Reena, U.S. Rana, “Effect of dust particles
on rotating micropolar fluid heated from
below saturating a porous medium, “Int. J.
Appl. Math. (AAM) 4(1), 187-217 (2009).
[11] S. Venkatasubramaniam and P.N. Kaloni,
“Effect of rotation on the
thermoconvective instability of a
horizontal layer of ferrofluids”, Int. J. Eng.
Sci., vol. 32(2), pp. 237-256 (1994).
[12] Sunil, A Sharma, D. Sharma, R.C.
Sharma, “Effect of dust particles on
thermal convection in a ferromagnetic
fluid, “Z. Naturforsch. 60a, 494-502
(2005).
[13] Sunil, A Sharma, R.C. Sharma, “Effect of
dust particles on ferrofluid heated and
soluted from below”, Int. J. Thermal Sci.
45(4), 347-358 (2006).
[14] Sunil, A. Sharma, P. Kumar, “Effect of
magnetic field dependent viscosity and
rotation on ferroconvection in the
presence of dust particles”. Appl. Math.
Comput. 182, 82-88 (2006).
[15] Sunil, A. Sharma, P.K. Bharti, R.G.
Sandil, “Effect of rotation on a layer of
micropolar ferromagnetic fluid heated
from below saturating a porous medium”,
Int. J. Eng. Sci. 44, 683-698 (2006).
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
www.ijera.com 28 | P a g e
[16] Sunil, A. Sharma, P.K. Bharti, R.G.
Shandil, “Marginal stability of micropolar
ferromagnetic fluid saturating a porous
medium”, J. Geophys. Eng. 3, 338-347
(2006).
[17] Sunil, A. Sharma, R.G. Shandil, U. Gupta,
“Effect of magnetic field dependent
visocisty and rotation of ferroconvection
saturating a porous medium in the
presence of dust particles”. Int. Commun.
Heat Mass Transfer 32, 1387-1399 (2005).
[18] Sunil, C. Prakash, P.K. Bharti, “Double-
diffusive convection in a micropolar
ferromagnetic fluid”, Appl. Math.
Comput. 189 (2007).
[19] Sunil, Divya Sharma, R.C. Sharma,
“Effect of rotation of ferromagnetic fluid
heated and soluted from below saturating
a porous medium”, J. Geophys. Eng. 1(2),
116-127 (2004)
[20] Sunil, Divya Sharma, R.C. Sharma,
“Effects of dust particles on thermal
convection in ferromagnetic fluid
saturating a porous medium”, J. Magn.
Magn. Maths 288(1), 183-195 (2005).
[21] Sunil, Divya, R.C. Sharma, “Effect of dust
particles on a ferromagnetic fluid heated
and soluted from below saturating a
porous medium”, Appl. Math and
Comput. 169(2), 833-853 (2005).
[22] Sunil, P. Chand, P.K. Bharti, Amit
Mahajan, “Thermal convection in
micropolar ferrofluids in the presence of
rotation”, J. Magn. Magn. Mater. 320,
316-324 (2008).
[23] Sunil. A Sharma, P.K. Bharti, R.G. Sandil,
“Linear stability of double-diffusive
convection in a micropolar ferromagnetic
fluid saturating a porous medium”. Int. J.
Mech. Sci. 49, 1047-1059, (2007).