 Open Access
 Total Downloads : 451
 Authors : S. Pranesh
 Paper ID : IJERTV2IS70350
 Volume & Issue : Volume 02, Issue 07 (July 2013)
 Published (First Online): 15072013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
The Effect of Imposed TimePeriodic Boundary Temperature and Electric Field on the Onset of RayleighBenard Convection in a Micropolar Fluid
S. Pranesh
Department of Mathematics, Christ University, Bangalore, India.
ABSTRACT
The effect of imposed time periodic temperature of small amplitude and AC electric field on the onset of RayleighBÃ©nard convection in a micropolar fluid is investigated using linear stability analysis. A regular perturbation method is used to arrive at an expression for the correction Rayleigh number that throws light on the possibility of subcritical motions. The Venezian approach is adopted for obtaining eigen value of the problem. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures are modulated inphase, (b) asymmetric, corresponding to outofphase modulation and (c) only the lower wall is modulated. It is observed that the system is most stable when the boundary temperatures are modulated outofphase. This problem is an example of external control of the internal convection.

INTRODUCTION
One of the effective mechanisms of controlling convection is through the maintenance of a nonuniform temperature gradient which is only spacedependent. However, in many practical situations nonuniform temperature gradients find their origin in transient heating or cooling at the boundaries, hence the basic temperature profile depends explicitly on position and time. This problem, called the thermal modulation problem, involves the solution of the energy equation under suitable timedependent boundary conditions. These profiles can be used as an effective mechanism to control the convective flow by proper tuning of its parameters, namely, amplitude and frequency of modulation. There can be an appreciable enhancement of heat, mass, or momentum if an imposed modulation can destabilize an otherwise stable system. Similarly, if it can stabilize an otherwise unstable system, higher efficiency can be achieved in many processing techniques, particularly in solidification processes. For example, in crystal growth during solidification of metallic alloys, one can consider timedependent temperature gradient to influence the transport process, thus to control the quality and structure of the resulting solid. The thermal modulation can be used as a mechanism to delay convection in the case of material processing applications to attain
higher efficiencies and to advance it in achieving major enhancement of mass, heat and momentum transfer.
The pioneer work in this field is due to Venezian [1], who investigated the effect of timedependent heating on the onset of thermal convection in a horizontal fluid layer heated from below. Actually, the investigation due to Venezian was motivated by the experiment of Donnelly [2], in which, he investigated the effect of rotational speed modulation on the onset of instability in fluid flow between two concentric cylinders. Many authors (Bhaduria [3, 4, 5], Pranesh and Sangeetha [6], Siddheshwar et. al. [7, 8]) have considered the effect of temperature modulation under different conditions to study its effect on the onset of convection.
In most part of the last century the engineering applications of fluid mechanics were restricted to systems in which electric and magnetic fields played no role. In recent years, the study of the interaction of electromagnetic fields with fluids started gaining attention with the promise of applications in areas like nuclear fusion, chemical engineering, medicine and high speed noiseless printing. The investigation of convective heat transfer together with the electrical and magnetic forces in nonNewtonian fluids is of practical importance. A systematic study through a proper theory is essential to understand the physics of the complex flow behavior of these fluids and also to obtain invaluable scaled up information for industrial applications.
In dielectric fluids with low values of conductivity, the electric effects will essentially govern the motion. The forces that are exerted by an electric field on free charges present in a liquid are transmitted by collision to the neutral molecules. The fluid will be set in motion, thus changing the distribution of charges that in turn modifies the electric field. There is an analogy between RayleighBÃ©nard instability and pure electroconvection. In the latter case, the destabilizing force is proportional to the mean charge gradient. If alternating electric fields of sufficiently high frequency are employed, then Kelvin or polarization body force becomes the driving force for convection.
Onset of natural convection in the presence of an external electric field has been studied by Turnbull [911], Turnbull and Melcher[12], Takashima and Aldridge [13], Takashima and Gosh[14], Takashima and Hambata [15], Stiles et. al. [16], Ezzat and Othman [17], Siddheshwar [18], Siddheshwar and Abraham [19, 20], Siddheshwar and Chan [21], Shivakumara et. al. [22, 23], Rudraiah et. al. [24] and Siddheshwar and Radhakrishna [25].
The theory of micropolar fluid introduced by Eringen[26] have become an important field of research especially in many industrially important fluids like paints, polymeric
suspensions, colloidal fluids, and also in physiological fluids such as normal human blood and synovial fluids. The mathematical theory of equations of micropolar fluids and applications of these fluids in the theory of lubrication and porous media is presented by Lukaszewicz [27].
The RayleighBÃ©nard instability in a horizontal thin layer of fluid heated from below is an important particular stability problem. The theory of thermomicropolar convection heated from below was studied by many authors Datta and Sastry [28], Ahmadi [29], Bhattacharya and Jena [30], Siddheshwar and Pranesh [31 35] and Pranesh and Kiran [36]. The literature pertaining to temperature modulation in micropolar fluid is mainly concerned with magnetic field and a corresponding study for micropolar fluid with effect of electric field is missing despite its importance in understanding control of convection encountered in many scientific and technological problems.
Therefore, main object of this paper is to study the effect of imposed temperature modulation and electric field on the stability of convective flow in a micropolar fluid by considering freefree boundaries.

MATHEMATICAL FORMULATION
Consider a layer of Boussinesquian, micropolar fluid confined between two infinite horizontal surfaces separated by a distance d apart. The uniform AC electric field is directed along the zaxis. A Cartesian system is taken with origin in the lower boundary and zaxis
vertically upward (see figure 1).
Z
1
z = d
E
T0 2 T 1
cos t
g
z = 0
Y
Micropolar Fluid
X
T0 1 T1 cost 2
Figure 1: Physical configuration The basic governing equations are:
Continuity equation:
.q
0,
(1)
Conservation of linear momentum:
q (q.)q p gk (2 )2q ( .) ,
(2)

t P E
Conservation of angular momentum:
I

t
(q.) ( )(. ) (2 ) ( q 2 ),
(3)
Conservation of energy:
T q
t oCv
T
.
.
2T,
(4)
Equation of state:
o [1 (T To )],
(5)
Equation of state for dielectric constant:
r (1 e ) e(T T0 ),
(6)
Faradays law:
0
E (7)
E ,
Equation of polarization field:
.(0 ) 0
E P
,
(8)
P 0 (r 1)E
where, q is the velocity , 0 is density of the fluid at temperature T = T0, p is the pressure,
is the density, g is acceleration due to gravity, is coupling viscosity coefficient or vortex
P
P
E
E
viscosity, is dielectric polarization, is the electric field, and are the bulk and shear
spinviscosity coefficients, is the angular velocity, I is moment of inertia,
' and
' are
bulk and shear spinviscosity coefficients, T is the temperature, is the thermal conductivity,
is micropolar heat conduction coefficient, is coefficient of thermal expansion, is
electrical conductivity,
r dielectric constant, e =

r

r
T
, e electric susceptibility,
o
o
TT
o electric permittivity of free space and electric scalar potential.
The wall temperatures are time dependent, externally imposed and are taken as
T(0, t) T0 1 T1 cost
2
T(d, t) T0 1 T1 cos t
2
(9)
(10)
where, is the amplitude of modulation, is the frequency of modulation and is the phase angle.
We consider three types of thermal modulation namely: Case (a): Symmetric (inphase, 0 )
Case (b): Asymmetric (outofphase, ) and
Case (c): Only lower wall temperature is modulated while the upper wall is held at constant temperature ( i)

Basic State
The basic state of the fluid is quiescent and is described by:
E
E
E
E
P
P
P
P
q b (0,0,0),p pb (z, t), b (z, t), b (z), b (z),T Tb (z, t),
(11)
Substituting equation (11) into basic governing equations (1)(8), we obtain the quiescent state solutions as:
pb
z
Tb
bg Pb Eb ,
z
2Tb
(12)
,
z z 2
(13)
b o 1 (Tb To ),
r (1 e ) e(Tb T0 ),
(1 e )E 0
Eb
(1 )
e
eT h
k,
z
z
,
(14)
1
Pb 0 E 0 (1 e )1
eT
k,
(1 e ) z
h
The solution of equation (13) that satisfies the thermal boundary conditions (9) and (10) is
T 2z
z
z
T T 1 Re()e d
()e d
eit
(15)
b 0 2 d
,
where
1
1 i
1 i
,
,
d 2 2
(16)
2
T ei e
( )
2
e e ,
and Re stands for the real part.
We now superpose infinitesimal perturbations on this basic state and study the stability of the system.

Linear Stability Analysis
The stability of the basic state is analyzed by introducing the following perturbation
E
E
E
E
E
E
P
P
P
P
P
P
q q b q', p pb p', b ', T Tb T', b ', b ',
(17)
where the prime indicates that the quantities are infinitesimal perturbations. Let the
components of perturbed polarization and electric field be P' , P' , P
(z) P'
and
1
1
2
2
b
b
E' , E' , E
(z) E' .
1 2 b 3
3
3
The second equation of (8), on linearization yields
i
i
P'
0e '
for i = 1,2
E
E
P
P
i
i
3
3
'
0e '

e0E
0T'
(18)
E
E
3
3
where it has been assumed that eT (1 e ).
E
E
Equation (7) implies one can write
' '.
Substituting equation (17) into equations (1)(8) and using the basic state equations, we get
linearized equations governing the infinitesimal perturbations in the form:
.q ' 0,
(19)
q p gk (2 )2q ( ) (
.) ' ( '.) ,
(20)
o t
Pb E
P Eb
I ( )( ) (2 ) ( q 2 ),
(21)
o t
T T
,
(22)
t d q C
o v
' 0T' ,
' 0eT ' ,
E
E
P
P
.0 '' 0.
(23)
(24)
(25)
Introducing the electric potential
' , eliminating the pressure p in equation (20) and
incorporating the quiescent state solution, we obtain the perturbed state vorticity transport
equation in the form:
I( ) ( )2 ( ) (2 ) (2w 2 ),
(26)
o t
Using equation (18) on equation (25), we obtain:
1 e 2'eE0DT' 0.
(27)
The perturbation equations (22), (25), (26) and (27) are nondimensionalized using the following definitions:
(x*
, y*
, z*
' '
x
x
y
y
) ,
' '
z
z
t
t
, , t*
, w* w
, T*
T'
, *
'
.
(28)
d
d
d
d
'
'
d
d
d 2
d 2
d
d
T
T
eE 0 Td
1 e
to obtain the equations governing the infinitesimal perturbation (after dropping the asterisk).
1 (2W) R2T (1 N )4W N 2
L2T L
(2),
(29)
Pr t
1 1 1 z 1
z 1
N 2 z
Pr t
N32z N12 W 2N1z ,
(30)
T 2T (W N5z ) T0 ,
(31)
t z
2 T 0,
z
(32)
The nondimensional parameters R, L, N1, N2, N3, N5 and Pr are given as
1
1
N
(Coupling Parameter),
N 2 I
d 2
(Inertia Parameter),
N3
( )d 2
N5
o Cvd 2
ogTd3
(Couple Stress Parameter),
(Micropolar Heat Conduction Parameter),
R ( )
(Rayleigh number),
0
0
0e2 E 2 T 2d 2
L
(1 e )( )
(Electric number),
Pr
0
(Prandtl number).
In equation (31), T0 is the nondimensional form of Tb , where
z z
T0
z
1 f (z),
(33)
where
f (z) ReA()ez A()ez eit
ei e
(34)
and
A()
2
e e ,
(35)
Equations (29) to (32) are solved subject to the conditions
2 w
w
z2
0
z
at z = 0, 1 (36)
Eliminating T , z and from equations (28)(30), we get
2
4 1 (1 N
)2 N26
N
t
Pr t
1 1
2 N
2 2N
W
Pr t 3
1 T
(37)
L4 0
1 z
LN N 24 T0 W R22 T0 N2
N
2 2N N N 2 W
1 5 1 z
1 z Pr t 3
1 1 5
In dimensionless form, the velocity boundary conditions for solving equation (37) are obtained from equations (29) to (32) and (36) in the form:
2 w
w
z2
4 w
z4
6 w
z6
8w
0
0
z8
at z = 0, 1. (38)



METHOD OF SOLUTION
We now seek the eigenfunction w and eigenvalues R of the equation (37) for the
basic temperature distribution (33) that departs from the linear profile
T0
z
1 by quantities
of order . Thus, the eigenvalues of the present problem differ from those of the ordinary
BÃ©nard convection by quantities of order . We seek the solution of equation (37) in the
form:
(R, w) (R0 , w0 ) (R1, w1) 2 (R 2 , w 2 ) ………….
(39)
The expansion (38) is substituted into equation (36) and the coefficients of various powers of
are equated on either side of the equation. The resulting system of equation is
L1w0 0,
(40)
L w L4 N2 N
2 2N N N 2 fw
1 1 1
1 1 1
3
Pr t
1 1 5 0
(41)
22 N2 N
2 2N N N
2 fR w ,
1 Pr t 3
1 1 5
0 0
L w L4 N2 N
2 2N N N 2 fw
1 1 1
1 1 1
3
Pr t
1 1 5 1
(42)
22 N2 N
2 2N N N
2 fR w R w ,
1 Pr t 3
1 1 5
0 1 2 0
where
L N2
N
2 2N
2
4 1
(1 N
)2 N26 L4
1 Pr t 3
Pr t
1 1
1
(43)
1 t
1 t

LN N
24 R
22 N2
N
2 2N

N N
2
1 5 1 0
1 Pr t 3
1 1 5 .

Solution To The Zeroth Order Problem
The zeroth order problem is equivalent to the RayleighBenard problem of Micropolar fluid with electric field in the absence of temperature modulation. The linear analysis of RayleighBenard convection in micropolar fluid without electric field has been thoroughly investigated by Siddheshwar and Pranesh [31].
The stability of the system in the absence of thermal modulation is investigated by
introducing vertical velocity perturbation
w0 corresponding to lowest mode of convection as:
w0 sin(z) expi(lx my), (44)
where l and m are horizontal wave number in x and y direction.
Substituting equation (44) into equation (40) we obtain the expression for Rayleigh number
in the form
N3K 2 2N1 K8 1 N1 N2 La 4 LN1N5K 2a 4
R 0
1
K 2a 2 N1N5K 2 N3K 2 2N1
, (45)
where K2 2 a 2 and a 2 l2 m2 .
In the absence of the electric field (i.e. L = 0), equation (45) reduces to
N3K 2 2N1 K8 1 N1 N 2
R 0


K 2a
2 N1N5 K 2

N3K 2
1 ,
2N1
which is the expression for Rayleigh number discussed by [30, 32]. Setting N1 = 0 and keeping N3 and N5 arbitrary in the above expression we get
K 6
R 0 ,
a 2
which is the classical Rayleigh BÃ©nard result.

Solution To The First Order Problem
Equation (41) for
w1 now takes the form
where
L1w1 La 4f R 0a 2K 2f R1a 2K 2 A1w 0 , (46)
A1 N3K2 2N1 N1N5K2
If the above equation is to have a solution, the right hand side must be orthogonal to the null
space of the operator L1. This implies that the time independent part of the RHS of the
equation (46) must be orthogonal to
sin(z) . Since f varies sinusoidal with time, the only
steady term on the RHS of equation (45) is R1a 2K2 sin(z) , so that R1 0 . It follows that
all the odd coefficients i.e.
R1 R3 ……… 0
in equation (39).
To solve equation (46), we expand the righthand side using Fourier series expansion and
obtain
where
w1 by inverting the operator L1 term by term as
W1 A1a 2 La 2 R 0K2 Re Bn ei t sinnz
L , n
Bn () A()gn1() A()gn1()
2n22 e e 1n e i ei
(47)
e e 2 n 12 2 2 n 12 2
(48)
L1, n
X X
1 2
1 2
2 N
2
2
X3
Pr


La
4X2

LN1N5K 2 a 4
R 0K 2 a
n
n
4
4
2X
n
n
4 N2

LN2a 4

2 2 N2
i Kn
Pr X1
X2X3
Pr R 0Kna
, Pr
4
4 2
2 4
X1 Kn 1
N1 Kn
N1 Kn ,
Pr
n
n
X2 N3K 2 2N1 ,
4
2 Kn 2
2 2
X3 Kn 1
N1 Kn
N1 Kn ,
Pr
X4 N3K 2 2N1 N1N5K2 ,
and
n n
n
n
K 2 n 22 a 2 .
The equation for w2 , then becomes
L1w 2 La 4 R 0a 2K2 fA1w1 R 2a 2K2 w 0
(49)
We shall not solve equation (49), but will use this to determine R2. The solvability condition requires that the timeindependent part of the right hand side of equation (49) must be
orthogonal to sin(nz) , and this results in the following equation,
2La 2 R 0k 2 1
k
k
R 2 2
1
n A1fw1 sin
0
z dz
where overbar denotes the time average. From equation (46), we have,
A1fw1
A1L1w1
1
1


Roa 2k 2 La 4
simplifying we get,
2
2
Roa 2k 2 La 4
Bn *
A1fw1 sinz 1
Re
sinnz Bn sinnz ,
And finally,
n1 L1, n
n1
R
k2 La 2
B 2
R 0 1 La 4 R a 2 k2 Re n
2
0 n
2
0 n
2
2k1
n1
L1 , n
R k2 La 2
B 2
L , n L* , n
1
1
0 1 La 4 R
2
2
a 2 k2 n
1 1
2k1 n1
0 n L , n2
2
La 2 R K2 A2 2 Bn 2 L2 , n L , n
n
n
R 2c
0
2K2
La 4 R 0a 2K2
2L2
2
, n 2
(50)
where
A2 N3Kn 2 2N1 N1N5Kn 2 iN2 ,
Pr
L2 (, n) A2 L(, n) and
L2*(, n) is the conjugate of
L2 (, n) respectively.


MINIMUM RAYLEIGH NUMBER FOR CONVECTION
The value of Rayleigh number R obtained by this procedure is the eigenvalue corresponding to the eigen function w, which, though oscillating, remains bounded in time. Sinc R is a function of the horizontal wave number a and the amplitude of modulation , we
have
R(a, ) R0 (a) 2R 2 (a)
(51)
It was shown by [1] that the critical value of thermal Rayleigh number is computed up to
O(2 ) , by evaluating R0 and R2 at a a 0 . It is only when one wishes to evaluate R4 that a2
must be taken into account where
a a 2
minimizes R2. To evaluate the critical value of R2
(denoted by R2c) one has to substitute a a 0 in R2, where a0 is the value at which R0 given by equation (44) is minimum.
We now evaluate R2c for three cases:
Case (a): When the oscillating field is symmetric so that the wall temperatures are
modulated inphase with 0 . In this case,
Bn () bn
or 0, accordingly as n is
even or odd.
Case (b): When the wall temperature field is antisymmetric corresponding to outofphase
modulation with . In this case,
Bn () 0 or bn, accordingly as n is even or
odd.
Case (c): When only the temperature of the bottom wall is modulated, the upper plate being
held at constant temperature, with i.In this case,
Bn () bn , for integer
2
values of n.
where
4n22
bn .
[2 (1 n)2 2 ][2 (1 n)2 2 ]The variable defined in equation (15), in terms of the dimensionless frequency, reduces to
1
(1 i) 2 ,
2
and thus
bn 2
16n 24 2
.
[ 2 (1 n)4 4 ][ 2 (1 n)4 4 ]Hence from equation (50) and using the above expression of Bn () , we can obtain the
following expression for R2c in the form
La 2 R K 2 A2 2 Bn 2 L2 , n L , n
n
n
R 2c
0
2K 2
La 4 R 0a 2K 2
2L2
2
, n 2
(52)
In equation (52) the summation extends over even values of n for case (a), odd values of n for case (b) and for all values of n for case (c).The infinite series (51) converges rapidly in all cases. The variation of R2c with for different values of N1 , N3, N5 L and Pr are depicted in figures (2)(7).
Results and Discussions:
In this paper, an analytical study is made on the effects of temperature modulation and electric field on the onset of convection in a horizontal layer of a micropolar fluid. The
expression for the critical correction Rayleigh number R2c is computed as function of the frequency of the modulation and for different parameters. The value of R2c has been
calculated for the following three cases; (a) when the walls temperature is modulated in
phase i.e.,
0 , (b) when the walls temperature is modulated outofphase, i.e., and
(c) when only the lower wall temperature is modulated, the upper wall is held at constant temperature, i.e., i.
The analysis presented is based on the assumption that the amplitude of the modulating temperature is small compared with the imposed steady temperature difference. The validity of the results obtained here depends on the value of the modulating frequency . When 1, the period of modulation is large and hence the disturbance grows to such an
extent that it makes finite amplitude effects important. When
,
R2c 0 , thus the
effect of modulation becomes small. In view of this, we choose only moderate values of in our present study.
The results have been presented in figures (2)(15). From the figures it is observed that the value of R2c may be positive or negative. The sign of R2c characterizes the stabilizing or destabilizing effect of modulation. A positive R2c means the modulation effect is
stabilizing while a negative R2c means the modulation effect is destabilizing compared to the system in which the modulation is absent.
The effect of inphase modulation of wall temperature on the onset of convection in a horizontal layer of micropolar fluid with electric field is shown in figures (2) (7) for different values of coupling parameter N1, inertial parameter N2, coupling parameter N3, micropolar heat conduction parameter N5, electric Rayleigh number L and Prandtl number Pr. From these figures, we find that for low frequency , R2c becomes more and more negative indicating that inphase modulation for low values of is destabilizing and for moderate values of , R2c becomes less and less negative indicating that inphase modulation for moderate values of is stabilizing. Let c be the frequency at which the R2c changes from destabilizing to stabilizing, then the modulated system may be classified as destabilizing or stabilizing according as < c or > c when compared with the unmodulated system. For some particular value of , R2c becomes zero. This is due to the fact that when the frequency of modulation is low, the effect of modulation on the temperature field is felt throughout the fluid layer. If the plates are modulated inphase, the temperature profile consists of the steady straight line section plus a parabolic profile which oscillates in time. As the amplitude of modulation increases, the parabolic part of the profile becomes more and more significant. It is known that a parabolic profile is subject to finite amplitude instabilities so that convection occurs at lower Rayleigh number than those predicted by the linear theory. From the figures (2) (7) for inphase modulation the following points are noted:
Figure (2) is the plot of correction Rayleigh number R2c versus frequency of modulation for different values of electric Rayleigh number L in respect of inphase modulation. The electric Rayleigh number L is the ratio of electric force to gravitational force. We see from the figure when L is greater than 1803 super critical motions occur and R2c increases with an
increase in L at a given frequency . Hence L has a stabilizing effect on the flow. When L is
less than 1803 subcritical motions occurs. It is also interesting to see from the figure that for a given L (L<1803), R2c first decreases with increase in , reaches a minimum and then increases with increase in and for a given L (L>1803) R2c increases with increase in reaches the maximum and then decreases with increase in . This shows that for a weak dielectric fluid, the flow is destabilized for small values of and stabilized for large values of . This is due to the fact that when the frequency of modulation is low, the effect of
modulation is felt throughout the fluid.
Figure (3) is the plot of correction Rayleigh number R2c versus frequency of modulation for different values of coupling parameter N1 in respect of inphase modulation. We observe that as N1 increases, R2c increases in negative direction. The increase in N1 implies increase in the concentration of suspended particles. These suspended particles consume the greater part of the energy in forming the gyrational velocity and as a result R2c becomes more and more negative.
Figure (4) is the plot of correction Rayleigh number R2c versus frequency of modulation for different values of inertia parameter N2 in respect of inphase modulation. Increase in N2 is representative of the increase in inertia of the fluid due to the suspended particles. Thus, as is to be expected, we find that as N2 increases R2c becomes less and less negative thereby stabilizing the system. Since N2 essentially arises with the acceleration term, it does not have any influence on Roc. It influences only R2c.
Figure (5) is the plot of correction Rayleigh number R2c versus frequency of modulation for different values of couple stress parameter N3 in respect of inphase modulation. The role played by the shear stress in the conservation of linear momentum is played by couple stress in the conservation of angular momentum equation. Increase in N3 signifies decrease in gyrational velocities. Hene, as N3 increases, we observe that R2c becomes less and less negative.
Figure (6) is the plot of correction Rayleigh number R2c versus frequency of modulation for different values of micropolar heat conduction parameter N5 in respect of inphase modulation. An increase in N5 implies that the heat induced into the system also increases resulting in reduced heat transfer from bottom to top. As a result, we find from the figure that as N5 increases R2c increases in negative direction.
Figure (7) is a plot of R2c versus , for different values of Pr for inphase modulation. It is observed that as Pr increases R2c increases in negative direction. It can be inferred from this that the effect of increasing the concentration of the suspended particle is to destabilize the system. This means that the fluids with suspended particles are more vulnerable than clean fluids to destabilization by modulation. It is appropriate to note that Pr does not affect the R0 part of R.
The effect of outofphase modulation on the wall temperature on the onset of convection is shown in figures (8) and (9). It is found that R2c is positive for outofphase whereas it is
negative for inphase. Thus N1 and L have opposing influences for inphase and outofphase modulations. The above results are due to the fact that in the case of outofphase modulation the temperature field has essentially a linear gradient varying in time, so that the instantaneous Rayleigh number is supercritical for half a cycle and subcritical during the other half cycle.
The above results on the effect of various parameters on R2c for outofphase modulation do not qualitatively change in the case of temperature modulation of just the lower boundary. This is illustrated in the figures (9) and (10).
From the study following conclusion can be made:

The system is more stable when boundary temperature are modulated in outofphase.

Inphase temperature modulation leads to subcritical motions.

The results of the study throw light on an external means of controlling the internal convection with electric field, either advancing or delaying convection by temperature modulation in a micropolar fluid.

The suspended particles scale down the effect of temperature modulation.
Acknowledgement:
This work is funded by Research Department (Major Project), Christ University, Bangalore. Author would like to acknowledge management of Christ University for their support in completing the work.
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