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ISSN 2229 – 5046209 International Journal of Mathematical Archive- 5(3), March – 2014
THERMAL INSTABILITY IN A ROTATING MICROPOLAR VISCOELASTIC FLUID LAYER
UNDER EFFECT OF ELECTRIC FLIED
Sayed A. Zaki
*, Mohamed I. Othman and Adel Y. Elmekawy
Department of Mathematics and statistics, Faculty of Science Taif University, Taif, Saudi Arabia.
(Received on: 15-02-14; Revised & Accepted on: 08-03-14)
ABSTRACT
T
he problem of onset of convective instability in a dielectric micropolar viscoelastic fluid (Walters’ liquidB
')
heated from below confined between two horizontal plates under the simultaneous action of the rotation of the system, vertical temperature gradient, one relaxation time and vertical electric field is considered. Linear stability theory is used to derive an eigenvalue of twelve order, and an exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wave number for the onset of instability are presented graphically as a function of rotation at a certain value of the Prandtl number, for various values of the relaxation time, the Rayleigh electric number, the elastic parameter and micropolar parameters.Keywords: Instability, Viscoelastic, Rotation, micropolar, The power series method.
NOMENCLATURE
L
distance between two rigid boundaries)
σ
,
σ
,
σ
(
1 2 3=
σ
microrotatingρ
densityμ
coefficient of viscosityP
pressureγ
,
β
,
α
,
k
material constants of the heat conducting micropolar fluid defined in Eq. (3)v
C
specific heat at constant volumev
k
thermal conductivityj
microinertia)
w
,
v
,
u
(
=
v
velocityε
dielectric constante
coefficient of relative variation of the dielectric constant with temperatureο
τ
relaxation timeg)
0,
(0,
−
=
g
the gravitational accelerationο
α
coefficient of relative variation of the density with temperature T temperature)
E
0,
(0,
z=
E
electric fieldο
K
the elastic constant of Walters’ liquidB
′
οη
the limiting viscosity at small rate of shearο
ρ
η
ν
=
ο the kinematic viscosityCorresponding author: Sayed A. Zaki
*210
© 2014, IJMA. All Rights Reserved ο
ρ
ν
k
K
=
the thermal diffusivity2 ο * ο
L
ρ
K
K
=
the elastic parameterδ
coefficient giving account of the coupling between the spin flux and the heat flux1. INTRODUCTION
In recent years, using the theory of micropolar fluids developed by Eringen [1, 2], several authors [3-5] have investigated problems related to stability and turbulence. As the theory of micropolar fluids encompass a wide variety of fluids (for example: liquid crystals, polymers, animal blood, etc.), in which randomly oriented bar like elements, dumbbell molecules or spherical particles are present, and as each volume element of the fluid undergoes translation as well as rotation, the analysis of the problems of stability revealed a number of interesting physical phenomena which are unseen in Newtonian fluids.
Initiating the study of thermal instability of a micropolar fluid layer heated from below, Ahmadi [6] has shown that there exists cellular convection at the onset of instability. Assuming that the boundaries are free from shear stress and microrotation, he has obtained an analytical solution in the case of free boundaries. His analysis shows that the micropolar fluids are more stable than Newtonian ones. Datta and Sastry [7] have extended the analysis of Ahmadi to the case of heat conducting micropolar fluids. They have found that the heat induced by microrotation causes instability of the layer, whether the fluid is heated from below or above. The instability for heating from above is quite a novel phenomenon as it does not have analogous in Newtonian fluid. While analysing the problem of convective instability of a micropolar fluid layer confined between rigid boundaries, Walzer [8] has mentioned that the analysis of the instability finds applications in the area of Geophysics, for example, in understanding the phenomenon of rising of valconic liquid with bubbles, and convective process inside the earth’s mantle. However, he has concluded his analysis without any calculation of eigenvalue. Rama Rao [9] has examined the onset of instability in a heat conducting micropolar fluid layer confined between rigid boundaries. On obtaining a numerical solution of the eigenvalue problem, he has shown that, in the case of adverse temperature gradient, the convective cells at the onset of instability are more elongated than those in the case of positive temperature gradient.
The effect of rotating on thermal convection in micropolar fluids is important in certain chemical engineering and biochemical situations. Sharma and Kumar [10] studied the effect of uniform rotation on thermal instability micropolar fluid. They found that the present of coupling between thermal and micropolar effect might introduce oscillatory motion in the system.
In technological field there exists an important class of fluids, called non-Newtonian fluids, which are also being studied extensively because of their practical applications. One such fluid is called viscoelastic fluid. Walters [11] and Beard and Walters [12] deduced the governing equations for the boundary flow for a prototype viscoelastic fluid which they have designated as liquid
B
′
when this liquid has a very short memory. Singh and Singh [13] have studied the magneto-hydrodynamic flow of a viscoelastic fluid past an accelerated plate. Othman [14] has studied the stability of a rotating layer of viscoelastic dielectric liquid (Walters’ liquidB
′
) heated from below. Othman [15] investigated the convective stability of a horizontal layer of viscoelastic conducting liquid (Walters’ liquidB
′
) heated from below and rotating about a vertical axis in the presence of a magnetic field and thermal relaxation. In these works, more general model of magneto-hydrodynamic free convection flow which also includes the relaxation time of heat convection and the electric permeability of the electromagnetic field are used. The inclusion of the relaxation time and electric permeability modify the governing thermal and electromagnetic equations, changing them from parabolic to hyperbolic type, and there by eliminating the unrealistic result that thermal disturbance is realized instantaneously everywhere within a fluid.An important stability problem is the thermal convection in a horizontal thin layer of fluid heated from below. A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below, under varying assumptions of hydrodynamics, has been given by Chandrasekhar [16]. Othman [17] analyzed the problem of the onset of stability in a horizontal layer of viscoelastic dielectric fluid (Walters’ liquid
B
′
) under the simultaneous action of a vertical ac electric field and a vertical temperature gradient without rotation.211
© 2014, IJMA. All Rights Reserved
2. FORMULATION OF THE PROBLEM
We consider an incompressible, dielectric and infinite micropolar viscoelastic fluid layer confined between two horizontal surfaces separated by a distance L. Choosing the origin on the lower boundary, let us introduce the Cartesian co-ordinate system x,y,z in which z is measurement at right angles to the boundaries. Let the system be rotating (round the z-axis) with a uniform angular velocity
Ω
=
(
0,0,
Ω
)
. The lower bounding surface atz
=
0
and the upper bounding surface atz
=
L
are maintained at constant temperaturesT
ο andT
1, respectively. The lower surface is grounded and the upper surface is kept at high alternating (60 HZ) potential whose root-mean-square value isφ
1.Under the foregoing assumptions the basic equations can be written as [17]
0
x
v
i i
=
∂
∂
(1)
ο ο
2 2
i i i i
k i
k k
k k
k i
e i
v
v
P
v
v
ρ
v
ρg
η
x
x
f
K
x
x
t
x
x
t
∂
∂
∂
∂
∂
∂
+
=
−
+
−
−
∂
∂
∂
∂ ∂
∂ ∂
∂
+
2
3 2
i m i
i m
m m k k
k m k k m k
2
x
x
x
x
x
x
x
x
x
v
v
v
v
v
v
∂
∂
∂
∂
∂
∂
∂ ∂
−
∂ ∂
−
∂
∂
∂
∂
2
ijk j k ijk ij
e
v
ke
x
σ
ρ
∂
+
Ω +
∂
, (2)( ) (
) (
)
2(
)
.
.
2
,
j
v
k
v
k
t
ρ
∂
+ ∇
σ α β
+
∇ ∇
σ
+ ∇ + ∇ ∧ −
γ σ
σ
∂
(3)C
ρ
v
(
) T
t
v.
∂
+
∇
∂
(
)
2
.
v
k
T
δ
T
σ
= ∇ + ∇
∇ ∧
ρ
Cτ
ο(
) T,
v
t
t
v.
∂ ∂
+
+
∇
∂
∂
(4)( )
.
ε
E
0,
∇
=
(5)and
0
E
∇ ∧ =
orE
= −∇
ϕ
.
(6) where,f
eiis the force of electric origin which may be expressed as Landau and Lifshitz [18]ei
f
=
e i 21
ρ E
2
E
x
iε
∂
−
∂
i1
2
x
∂
+
∂
2
ε
ρ
E
ρ
∂
∂
(7)taking into account the fact that the free charge density
ρ
e is zero. If we replace the pressure*
1
P
P
2
= −
ρ
ε
E
2
ρ
∂
∂
(8)The electrostriction term disappear from Eq. (2) which can be rewritten in the form:
*
2
ο ο
i
2 2
i i i i
k i
k k
k k
k i
v
v
P
ε
v
1
v
ρ
v
ρg
η
x
x
E
K
x
x
t
x
x
2
x
t
∂
∂
∂
+
∂
=
−
∂
+
−
∂
−
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+
2
3 2
i m i
i m
m m k k
k m k k m k
2
x
x
x
x
x
x
x
x
x
v
v
v
v
v
v
∂
∂
∂
∂
∂
∂
∂
∂
−
∂
∂
−
∂
∂
∂
∂
2
ijk j k ijk ij
e
v
ke
x
σ
ρ
∂
+
Ω +
212
© 2014, IJMA. All Rights Reserved The mass density and the dielectric constant are assumed to be functions of temperature as follows:
(
)
[
ο ο]
ο
1
α
T
T
ρ
ρ
=
−
−
,α
ο> 0 (10)(
)
[
ο]
ο
1
e
T
T
ε
ε
=
−
−
, e > 0 (11)where
α
οand e are usually positive.It is clear that there exist the following steady solutions (denoted by an over bar):
0,
u
= = =
v
w
, (12)0,
σ
=
(13)0 0
,
T
= −
T
β
z
(14)],
z
β
α
1
[
ρ
ρ
=
ο+
ο ο (15)]
z
β
e
1
[
ε
ε
=
ο+
ο , (16)x
E
=
0,
E
y=
0,
(
0 0)
,
1
z
E
E
e
β
z
=
+
(17)ο
ο
E
φ
Log (1 e β z )
e
= −
+
(18)here,
ο 1 ο
T
T
β
,
L
−
=
(19)(
0)
0
0
,
log 1
e
E
e
z
ϕ β
β
= −
+
(20) are the adverse temperature gradient and the root-mean-square value of the electric field at z = 0. If necessary, the modified pressureP
*can be determined from2
1
0
2
i z
i i
p
g
E
x
x
ε
ρ
∂
∂
=
−
∂
∂
. (21)
Let this initial steady state be slightly perturbed where the simple relation
ψ
=
ψ
+
ψ
′
can be expressed any physical quantitiesψ
after perturbation and prime refers to perturbed quantities. Following the usual steps of linear stability theory we can obtain the following main equations:u
v
w
0
x
y
z
′
′
′
∂
∂
∂
+
+
=
∂
∂
∂
, (22)2 2 '
2 2 ' 0 0 2 ' 0 0 2 0 4 '
1 1 ' '
0 0 0
2
e E
eE
K
w
g
T
w
t
z
t
z
ε
β
ε
β
ϕ
ς
υ
α
ρ
ρ
ρ
∂
∂
∂
∂
− ∇ ∇
−
+
∇
−
∇
+
∇
+ Ω
∂
∂
∂
∂
2 ' 3 0
0,
k
ρ
−
∇ Ω =
(23)
'
2 0 2
' '
0
2
w
K
,
t
υ
ς
z
ρ
t
ς
∂
∂
∂
− ∇
= Ω
−
∇
∂
∂
∂
(24)2 2
3
ο
Ω
3 3ρ j
γ
Ω
k[ w 2Ω ]
t
∂
′
= ∇
− ∇
+
′
213
© 2014, IJMA. All Rights Reserved
2
ο v ο ο v ο 3
T
ρ c 1 τ
β w
k
T δβ Ω
t
t
′
+
∂
∂
−
′
= ∇
′
−
∂
′
∂
′
, (26)0
z
T
E
e
φ
ο 2=
′
∂
′
∂
+
′
∇
. (27)where,
(
v
)
zy
u
x
v
ζ
=
∇
∧
′
∂
′
∂
−
′
∂
′
∂
=
is the z-component of vorticity.
2 1
3 z
σ
σ
Ω
(
)
x
y
′
′
∂
∂
=
−
= ∇ ∧
′
′
∂
∂
σ
, 22 2 2 2 1
y
x
∂
∂
+
∂
∂
=
∇
, 2 2 2 1 2z
∂
∂
+
∇
=
∇
.The boundary conditions appropriate for the problem are given by [10]
0
Ω
z
ζ
T
z
w
w
3 2 2=
=
′
∂
∂
=
ϕ′
=
′
=
′
∂
′
∂
=
′
atz
′
=
0,
L
(28)Now, introducing the nondimensional variables given by
2
2 2
,
v,
,
,
v, ,
v,
,
v vk
L
k
k
k
L
L
L
L
L k
β
L
L
β
L
2 0 0
,
3,
v
k
eE
L
L
β
2 vk
and
L
Ω =
as units of length, velocity, time, temperature, vorticity, electro potential, microrotation and rotation of the fluid respectively, we obtain the equations governing the disturbances as:
1 r 2 2
P
w
t
−
∂
− ∇
∇
∂
(
H E)
E2 2
1 1
φ
R
T R
z
R
∂
−
−
∂
+
∇
∇
* 1ο r 4
K P
w
t
−∂
+
∇
∂
ζ
2
Ω
z
∂
+
∂
K
Ω
30
2
=
∇
−
(29) 2 rP
t
ς
∂
− ∇
=
∂
2
rw
P
z
∂
Ω
−
∂
* 2 0ζ,
K
t
∂
∇
∂
(30)]
Ω
2
w
[
K
Ω
C
t
Ω
j
3=
ο∇
2 3−
1∇
2+
3∂
∂
, (31)
2
0 0 3
1
T
T
1
w
t
t
t
τ
∂ ∂
τ
∂
δ
+
− ∇
= +
− Ω
∂
∂
∂
(32)0
z
T
φ
2=
∂
∂
+
∇
. (33)where,
ν
k
βL
g
α
R
v 4H
=
is the Rayleigh heat number,ν
k
ρ
L
β
E
e
ε
R
v ο 4 2 2 ο 2 οE
=
is the Rayleigh electric number,v r
k
ν
P
=
is the Prandtl number,2 v ο v ο 1 v 2 ο ο 2
L
c
ρ
δ
δ
,
k
ρ
k
K
,
k
L
ρ
γ
C
,
L
j
j
=
=
=
=
.3. NORMAL MODE ANALYSIS
Following the normal mode analysis we assume that the solutions of Eqs. (29-33) are given by:
214
© 2014, IJMA. All Rights Reserved where,
λ
=
a
2+
b
2 is the wave number and c is the stability parameter which is, in general, a complex constant. For solutions having the dependence of the form (34), Equestion. (29-33) yield(
) (
)
(
)
(
)
(
)
1 2 2 2 2 2
2
* 1 2 2 2 2
0
2
0,
r H E E
r
P c
D
D
w
R
R
R D
K P c D
w
DZ
K
D
G
λ
λ
λ
λ
λ
−
−
−
−
−
+
+
Θ +
Φ
+
−
+ Ω
−
−
=
(35)(
2 2)
*(
2 2)
0
2
,
r r
c
P D
λ
z
P
DW
K c D
λ ζ
−
−
=
Ω
−
−
(36)(
)
(
2 2)
(
2 2)
2
,
c
A
D
λ
G
A D
λ
W
+
−
−
= −
−
(37)(
)
(
2 2)
0
1
c
τ
c
D
λ
+
−
−
Θ
= 1
(
+
τ
0c W
)
−
δ
G
,
(38)(
2 2)
0.
D
−
λ
Φ + Θ =
D
. (39) where 1K
A
j
=
, 1ο
K
A
,
C
=
D
d
.
d z
=
(40)In seeking solutions of these equations we must impose certain boundary conditions at the lower surface
z
=
0
and the upper surfacez
=
1
.
The most realistic boundary conditions may be written as0
G
Z
Φ
Θ
W
D
W
=
=
=
=
=
=
atz
=
0,1
(41) In this paper, however, we shall use somewhat different boundary conditions given by [21]0
G
Z
D
Φ
Θ
W
D
W
=
2=
=
=
=
=
atz
=
0,1
(42) This case, although admittedly an artificial one to consider, is of importance since its exact solution is readily obtained. Furthermore, from past experience with problems of this kind (see for example, Chandrasekhar [16] and Turnbull [20]), one may feel fairly confident that the general features of the physical situation will be disclosed by a discussion of this case equally as well as by a discussion of solutions satisfying less artificial boundary conditions.Eliminating
Z
,
Θ
,
Φ
and
G
from Equation (35-39), we obtain:(
)(
)
(
)
(
)
(
)
(
) (
)
(
)(
)
(
)(
)
(
) (
)
(
)
(
)(
) (
)
(
)(
)
(
)
(
)
* 2 2 2 2 1 2 2
ο r
2
2 2 2 2
2 * 2 2
ο r
2 2 2 2
2
2 * 2 2 2 2
ο r
* 1 * 2 2 2 2
ο ο r
K
P
2
1
1
K
P
2
K
P
K
K
P
1
r
H E
H E
r
c
c
D
c
A
D
P c
D
c
D
D
R
R
c
c
c
D
c
A
D
D
A R
R
c
c
D
D
P c c
c
D
c
c
D
ο
ο
ο
λ
λ
λ
τ
λ
λ
λ
τ
λ
λ
λ
δλ
λ
λ
λ
τ
λ
− −
+
−
−
+
−
−
−
−
×
+
−
−
−
+
+
+
+
−
−
×
+
−
−
−
+
+
+
−
−
−
+
+
−
−
+
−
−
×
(
) (
)
(
)
(
)(
)
(
)
(
)(
) (
)
(
)
(
)
(
) (
)
(
)(
)
(
)
(
) (
)
32 2 2 2 2 * 2 2
ο r
2 2 2 2 * 2 2 2 2 2
ο r
2 2 2 2 2 2 2 2
3
* 2 2 2 2 2 2
ο r
2
1
K
P
2
K
P
4
2
1
K
P
1
E
E
r
c
A
D
D
R
c
c
c
D
c
A
D
D
A R
c
c
D
D
D
P
c
A
D
c
c
D
D
D
KA c
c
D
c
c
D
D
ο
ο
ο
λ
λ
λ
τ
λ
λ
δλ
λ
λ
λ
τ
λ
λ
λ
τ
λ
λ
+
−
−
−
−
+
+
−
−
×
+
−
−
−
+
−
−
−
+ Ω
+
−
−
+
−
−
−
+
+
−
−
+
−
−
−
0
W
=
215
© 2014, IJMA. All Rights Reserved It can be shown from equation (43) that all even order derivatives of W vanish on the boundaries. The proper solution for W characterizing the lowest mode is:
z
π
sin
W
W
=
ο , (44) whereW
οis a constant. Substituting (44) in (43) and puttingπ
2+
λ
2=
b
, we obtain:2
R
Hλ
{
2(
*)
(
)
(
*)
[
]
}
0 r
1
0 0 r2
b A c b K c
P
b
c
c b K c
P
c
A b
δ
−
−
−
+
τ
−
−
+
+
[
]
(
)
(
)
[
]
(
)
(
)
(
)
(
)
* 2 *
0 0 0
2
2 * 2 2 *
0 0 0
1
2
R
2
1
r r
H
r r
b
c
c b K c
P
c
A b
b A c b K c
P
c
A b
c
c b K c
P
b A
c b K c
P
τ
δ
λ
π
τ
δ
π
+
−
−
+
+ −
−
−
+
+
+
−
+
−
−
+
−
−
(
)
* 3 1 *
0 r 0 r
K b P c c b K c
−
P
+
−
−
c
(
1
+
τ
0c
)
+
b
[
c
+
2
A b
+ +
]
KAb
3(
*)
0 r
c b K c
P
−
−
×
c
(
1
+
τ
0c
)
+
b
−
(
)
2 *
0 r
b
c b K c
−
−
P
[
c
+
2
A b cP
+
]
r−1+
b
c
(
1
+
τ
0c
)
+
b
[
]
2 2
4
A
π
P b c
r2
A b
−
Ω
+
+
c
(
1
+
τ
0c
)
+
b
. (45)4. OVERSTABILITY MOTIONS
Since c is, in general, a complex constant we put
c
=
c
r+
i
ω
, wherec
rand
ω
are real. The marginal state is reached whenc
r=
0
; ifω
=
0
, one says that principle of exchange of stabilities is valid otherwise we have overstability and thenc
=
i
ω
at marginal stability.Putting
c
=
i
ω
in equation (44), the real and imaginary parts of equation (45) yield:ωY
i
X
R
=
+
(46)There, X and Y are real-valued functions of
P
r,
τ λ
0, , ,
Ω
R
E, ,
A K
, ,
δ
K
0*,
andω
, and explicit expansions for thesefunctions are follows:
)
A
A
(
λ
A
A
ω
A
A
X
22 2 1 2
4 2 2 3 1
+
+
=
, (47))
A
A
(
λ
A
A
A
A
Y
22 2 1 2
3 2 4 1
+
−
=
(48)where
(
)
{
2 *}
3{
2 2 *(
)
(
2)
}
21 r
1
0 0 0 02
0 r 02
A
=
P
δ
A
− −
ω τ
K
b
+
ω τ
−
ω
K
+
A
τ
+
P
ω τ
−
A b
2{
}
02
A
b
,
ω
τ
+
+
(49)
(
)
{
*}
3{
(
)
*(
2)
(
)
}
22 0
1
r 01
02
0 r2
0A
= −
K
δ
A
− −
P
τ
b
+
δ
A
− +
K
A
−
ω τ
−
P
+
A
τ
b
+
{
ω τ
2 0+
2
A b
}
,
(50){
2 *2 1}
6{
(
)
2 *(
1 2 1 1)
}
53 0
2
02
2
0 02
02
r r
r r r r r r
A
=
ω
K P P b
−+
AP K
− −
ω
K
P
−+ +
ω
K
τ
P
−−
AK P
−+
b
(
)
(
)
(
)
(
)
4 * * 1 * 1 1 2 * 1
0 0 0 0 0 0 0 4
2 2 *
0 0
2
2
2
4
1
1 2
r r r r
r
K
K P
K AP
P
K A
P
b
AK K
P
ω
τ
τ
τ
ω
ω
ω
τ
− − − −
−
−
+
−
+
+
+
+
−
(
)
(
)
(
)
(
)
2 2 *2 4 * 1
0 0 0 0 3
4 1 2 2 1 1 2 2
0 0
2
2
1
4
2
4
E r r r
r r r
R
P
AP
K
K P
A
b
P
A
P
P A kA
P
λ
δ
ω
τ
ω
τ
ω τ
ω
ω
τ
π
−
− − −
−
+
+
+
+
−
+
+
−
+
−
−
Ω
216
© 2014, IJMA. All Rights Reserved
(
)
(
)
(
)
(
)
2 2 * 2 2 2 *
0 0 0 0 0 2
2 2
2 0
2
1
2
4
2
E r r r
r
R
K
A
P
AP
P
K
AP
b
P
A
λ
ω
τ
ω τ
π
ω
τ δ
π
ω τ
+
−
+
+
−
+
−
+
+
Ω
−
(
)
(
)
(
)
2 2 2 * * 2 2
0 0 0 0 0 0
2 2 2
0
2
2
2
4
2
E r r
r
R
K
K A
P
A
P A
b
P
A
λ
ω π τ
τ
τ
ω
τ
π
π ω
τ
−
−
+
−
+
−
+
+
Ω
+
+
π λ ω
2 2 2R
E(
+
2
A
τ
0)
,
(51)(
)
(
)
(
)
{
}
*2 6 2 *2 1 *2 2 5
4
2
01
0 r r 04
2
02
A
=
K b
+
+
ω
K P
−−
P
+
K
A
−
ω τ
−
KA
−
b
(
)
(
)
(
)
(
)
(
)
2 *2 1 2 2 * 1 2 1
0 0 0 0 0 0 4
1
2
2
2
2
4
1
r r r r
r r
K P
A
K
P
KA
P
P
b
A K
KP
P
ω
−ω τ
ω
−τ
ω
τ
−τ
−
−
−
+
−
+
−
+
+
+
−
− − +
+
{
λ
2R
E(
P
rτ
0−
K
0*+
δ
K A
0*)
+
2
ω
2K P
0* r−1(
ω τ
2
0−
2
A
)
+
ω
2(
2
P
r−1
+ +
4
A
τ
0+
P
r−1−
KA
τ
0)
}
b
3
(
)
(
)
(
)
(
)
(
)
2 2 * 2 * * 2
0 0 0 0 0 0
2
2 1 2 2 2
0
1
2
2
2
4
1
E
r r
R
P
A
A
K
P
K
A K
A
b
P
A
P
λ
τ
δ
π
τ
π
δ
ω τ
ω
−ω τ
π
+
+
−
+
−
−
+
−
+
+
−
−
Ω
+
(
)
(
)
2 2 2 * 2 *
0 0 0 0 0
2 2 2
0
2
1 2
2
4
2
E r r
r
R
A
A
K A
K
P
P A
b
P
A
λ
ω τ
π δ
ω
τ
τ
π
ω τ
−
+
− +
−
−
−
+
+
Ω
−
+
π λ
2 2R
E(
ω τ
2
0−
2
A
)
. (52) It is apparent from Eq. (45) that for arbitrary assigned values ofP
r,
R
E,
τ
ο,
λ
,
Ω
,
K
*ο,
A
,
K
,
δ
,
andω,
R
Hwill be complex but the physical meaning of R required it to be real. Consequently, from the condition that R must be real, so we have either
H
R
=
X
and
ω
=
0
(53) or0
H
R
=
X and Y
=
. (54)From Eq. (53) we obtain the eigenvalue equation for a natural stationary instability,
1 2
3 H
A
λ
A
R
=
. (55) In this case(
)
3 21 r
1
2
rA
=
P A
δ
−
b
−
AP b
(56)(
)
(
)
6 5 2 2 2 3
3 r
2
E r1
4
A
= −
P b
+
AP K
−
b
+
λ
R P
−
δ
A
−
π
Ω
P b
+
λ
2R P
E r(
π δ
2A
−
2
A
−
π
2−
8
π
2Ω
2P A
r)
b
2
−
2
π
2P
rλ
2R Ab
E . (57) For Newtonian viscous fluidR
E= =
A
K
= = Ω = =
δ
ω
0,
Eq. (55) reduces to2 3
H
λ
b
R
=
. (58) hich agrees with the classical result (Chandrasekhar [16]). Equation (55) will give the critical Rayleigh heat numberHC
217
© 2014, IJMA. All Rights Reserved On the other hand Eq. (54) leads,
)
A
A
(
λ
A
A
ω
A
A
R
2 2 2 1 2
4 2 2 3 1
H
+
+
=
, (59)and
0
A
A
A
A
1 4−
2 3=
. (60) For assigned values ofP K
r,
0*,
τ
0, , ,
Ω
A K
, ,
δ
and R
E Eqs. (59) and (60) defineR
H as a function ofλ
, the minimum of this function determines the critical Rayleigh numberR
HC for the onset of oscillatory convection (i.e. overstability) should be compared with that the onset of stationary convection (i.e. ordinary instability). The type of instability, which takes place in practice, will be that corresponding to the lower value of the critical Rayleigh heat number.5. NUMERICAL RESULTS
In order to determine the conditions under which instability sets in overstability
P K
r,
0*,
τ
0, , ,
Ω
A K
, ,
δ
and R
Ewere assigned fixed values, and the value of
ω
was evaluated numerically from Eq. (60). Using this value ofω
, the value ofR
H was evaluated numerically from Eq. (59). The procedure was then repeated for various values ofλ
in order to locate the minimum ofR
H. The critical Rayleigh heat numberR
HC obtained for both stationary instability and overstability is shown in Figs.1-4.We have plotted the variation of the critical Rayleigh heat number
R
HCwith the rotationΩ
using Eq. (59) satisfying (60) for the onset of over stable case for values of the dimensionless parametersP
r=
100,
δ
=
1, 0.5, 0.1,
*
0
0.1, 0.8,
1,
K
=
K
=
=
1
,τ
0=
0.02, 0.05
andA
=
0.2, 0.5,
Figure 1 represents the dependence ofR
HC onΩ
for three values ofR
E=
0,1000, 2000
,τ
0=
0.02, 0.05,
δ
=
1
,K
0*=
0.1
andA
=
0.2,
Figure 2 represents the dependence ofR
HConΩ
in the case ofR
E=
1000.
Figure 3 represents the dependence ofR
HC onΩ
in the case ofR
E=
1000.
,τ
0=
0.02, 0.05
,A
=
0.2,
K
0*=
0.1, 0.8,
andδ
=
1
. Figure 4 represents the dependence ofHC
R
onΩ
in the case ofR
E=
1000
,A
=
0.2,
τ
ο=
0
.
02
,
0
.
05
,K
0*=
0.1
andδ
=
0.1, 0.5
. The flow is stable ifR
H<
R
HC and otherwise unstable.Figures 1-4 reveal that the critical Rayleigh heat number
R
HC decreases with an increase the Rayleigh electric numberH
R
and the elastic parameterK
0*, whileR
HC increases with an increase the relaxation timeτ
ο, the rotationΩ
and the parameters A,δ
(i.e. the onset of stability is delayed asR
E andK
*ο increase, while the onset of instability is delayed asτ
ο,Ω
,δ
and A increase). The value ofR
HC for an oscillatory instability is greater than that of a stationary instability.In figure 5 we have exhibited the dependence of critical wave number
λ
C onΩ
for three values ofδ
=
1, 0.5, 0.1
,,
05
.
0
,
02
.
0
τ
ο=
*0
0.1
K
=
,R
E=
1000
and A
=
0.2,
Figure 5 reveals that the critical wave numberλ
C decreases or increases asδ
orτ
ο increases. This implies that the width of the cell at the onset of instability increases with the heat imparted by microrotation, while it reduces as the relaxation timeτ
ο increases.218
© 2014, IJMA. All Rights Reserved the case of high rotation the motion of the fluid prevails essentially in the horizontal plates. This motion is reduced by the presence of micropolar additives. Thus the component of the velocity perpendicular to the horizontal plates enhances, thereby the system is prone to instability.
6. CONCLUSION
Natural convection of a rotating micropolar viscoelastic fluid heated from below in the presence of electric field has been analyzed numerically. The study focused on the effect of a rotating micropolar fluid, elastic parameter, electric field and relaxation time on the convection phenomenon. From the above analysis, we conclude that the micropolar additives, the rotation of the system and the relaxation time have stabilizing effect while the elastic parameter and the presence of electric field have destabilizing effect. It is also noted from Figs. 1-4 that the critical Rayleigh heat number for overstability is always greater than the critical Rayleigh heat number for stationary convection.
7. REFERENCES
1. Eringen, A. C., Şuhubi, E. S.: Nonlinear theory of simple micro-elastic solids, Int. J. Engng. Sci., 2, (1964) 189-203.
2. Eringen , A. C.: Theory of micropolar fluids, J. Math. Mech. 16, (1966) 1-18.
3. Jean, S. K., Bhattacharyya, S. P.: The effect of microstructure on the thermal convection in a rectangular box of fluid heated from below, Int. J. Engng. Sci. 24, (1986) 69.
4. Hsu, T. H., Chen, C. K.: Natural convection of micropolar fluids in a rectangular enclosure, Int. J. Eng. Sci. 34, (1996) 407-415.
5. Peddiesou,J.: Boundary-layer theory for a micropolar fluid, Int. J. Engng. Sci.10, (1972) 23. 6. Ahmadi, G.: Stability of micropolar fluid layer heated from below, Int. J. Eng. Sci. 14, (1976) 81-89.
7. Datta, A. B., Sastary, V. U. K.: Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Engng. Sci. 14, (1976) 631-637.
8. Walzer, U.: Ger. Bcit. Geo-physik Leipzig, 85, (1976) 137.
9. Rama Rao, K. V.: Thermal instability in a micropolar fluid layer between rigid boundaries, Acta Mechanica, 32, (1979) 79.
10. Sharma, R. C., Kumar, P.: Effect of rotation on thermal convection in micropolar fluids, Int. J. Engng. Sci. 32, (1994) 545-551.
11. Walters, K.: Second-order effects in elasticity, Plasticity and fluid dynamics. p. 507, Oxford: Pergamon Press, 1964.
12. Beard, D. W., Walters, K.: Elastico-viscous boundary-layer flows, I. Two- dimensional flow near a stagnation point, Proc. Camb. Phil. Soc., 60, (1964), 667-671.
13. Singh, A., Singh, J.: Magnetohydrodynamic flow of a viscoelastic fluid past an accelerated plate, Nat. Acad. Sci. Lett. 6, (1983) 233-241.
14. Othman, M. I. A.: Elactrohydrodynamic instability of a rotating layer of a visco- elastic fluid heated from below, ZAMP, 55, (2004) 1-15.
15. Othman, M. I. A., Zaki, S. A.: The effect of thermal relaxation time on a elactro-hydrodynamic viscoelastic fluid layer heated from below, Can. J. Phys., 81, (5) (2003) 779-787.
16. Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability, Oxford Univ. Press, London, 1961.
17. Othman, M. I. A.: Electrohydrodynamic stability in a horizontal viscoelastic fluid layer in the presence of a vertical temperature gradient, Int. J. Engng. Sci. 39, (2001) 1217-1232.
18. Landau, L., Lifshitz, E.: Elactrohydrodynamics of continuous media, Pergamon Press, New York, 1960. 19. Bhattacharyya, S. P., Abbas, M.: On the stability of a hot rotating layer of micro-polar fluid, Lett. Appl.
Engng. Sci. 23, (1985) 371.
219
© 2014, IJMA. All Rights Reserved
8. FIGURES CAPTIONS
. 1 . 0 K and 5
ω
1,
δ
1, K 1, 0.2, A 100, P at R and τ of values various
for of function a as R number heat Rayleigh critical
the Represents
Fig.1
* ο
r E ο
HC
= =
= = = = =
Ω
. convection stationary
of onset the represents 0
ω
0. 100 R and 5
ω
1,
δ
1, K 1, , 1 . 0 K 100, P at A and τ of values various
for of function a as R number heat Rayleigh critical
the Represents
Fig.2
E
* ο r
ο
HC
= =
=
= = = = =
Ω
220
© 2014, IJMA. All Rights Reserved
. convection stationary
of onset the represents 0
ω
0. 100 R and 5
ω
1,
δ
1, K 1, , 2 . 0 A 100, P at K and τ of values various
for of function a as R number heat Rayleigh critical
the Represents
Fig.3
E
r * ο ο
HC
= =
=
= = = = =
Ω
. convection stationary
of onset the represents 0
ω
0. 100 R and 5
ω
, 2 . 0 A 1, K 1, , 1 . 0 K 100, P at
δ
and τ of values various
for of function a as R number heat Rayleigh critical
the Represents
Fig.4
E
* ο r
ο
HC
= =
=
= = = = =
Ω
221
© 2014, IJMA. All Rights Reserved
Source of support: Nil, Conflict of interest: None Declared 0.
100 R and 5
ω
, 2 . 0 A 1, K 1, , 1 . 0 K 100, P at
δ
and τ of values
various for of function a as
λ
number wave critical the Represents
Fig.5
E
* ο r
ο
C
= =
= = = = =
Ω
