DOI : 10.17577/IJERTV4IS100189
- Open Access
- Total Downloads : 206
- Authors : S. Pranesh , R. V. Kiran
- Paper ID : IJERTV4IS100189
- Volume & Issue : Volume 04, Issue 10 (October 2015)
- DOI : http://dx.doi.org/10.17577/IJERTV4IS100189
- Published (First Online): 14-10-2015
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Rayleigh-Bénard Chandrasekhar Convection in An Electrically Conducting Fluid using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries
S. Pranesh Department of Mathematics, Christ University, Hosur Road,
Bangalore 560 029, India.
R.V. Kiran
Department of Mathematics, Christ Junior College,
Hosur Road, Bangalore 560 029, India.
Abstract – The effect of imposed time-periodic boundary
temperature (ITBT, also called temperature modulation) and magnetic field at the onset of convection is investigated by making a linear analysis. The classical Fourier heat law is replaced by the non-classical Maxwell-Cattaneo law. The
where, Q
dQ Q T dt
is the heat flux, is a relaxation time and is the
classical approach predicts an infinite speed for the propagation of heat. The adopted non-classical theory involves wave type of heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The Venezian approach is adopted in arriving at the critical Rayleigh number, correction Rayleigh number and wave number for small amplitude of ITBT. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures are modulated in phase, (b) asymmetric, corresponding to out-of phase modulation and (c) only the lower wall is modulated. The temperature modulation is shown to give rise to sub-critical motion. The shift in the critical Rayleigh number is calculated as a function of frequency and it is found that it is possible to advance or delay the onset of convection by time modulation of the wall temperatures. It is shown that the system is most stable when the boundary temperatures are modulated out-of-phase. It is also found that the results are noteworthy at short times and the critical eigenvalues are less than the classical ones.
Keywords: Time periodic boundary temperature, Rayleigh-Bénard Chandrasekhar convection, Magnetic field, Maxwell-Cattaneo law.
-
INTRODUCTION
The Classical Fourier law of heat conduction expresses that the heat flux within a medium is proportional to the local temperature gradient in the system. A well known consequence of this law is that heat perturbations propagate with an infinite velocity. This drawback of the classical law motivated Maxwell [1], Cattaneo [2], Lebon and Cloot [3], Dauby et al. [4], Straughan [5], Siddheshwar [6],
Pranesh [7], Pranesh and Kiran [8, 9, 10] and Pranesh and Smita[11] to adopt a non-classical Maxwell-Cattaneo heat flux law in studying Rayleigh-Bénard/Marangoni convection to get rid of this unphysical results. This Maxwell-Cattaneo heat flux law equation contains an extra inertial term with respect to the Fourier law.
heat conductivity. This heat conductivity equation together
with conservation of energy equation introduces the hyperbolic equation, which describes heat propagation with finite speed. Puri and Jordan [12, 13] and Puri and Kythe [14, 15] have studied other fluid mechanics problems by employing the Maxwell-Cattaneo heat flux law.
Thomson [16] and Chandrasekhar [17] studied
theoretically and Nakagawa [18, 19] and Jirlow [20] studied experimentally, the thermal stability in a horizontal layer of fluid with magnetic field and found that vertical magnetic field delays the onset of thermal convection. The application of magnetoconvection is primarily found in astrophysics and in particular by the observation of sunspots. Motivated by the above applications several authors investigated the suppression of convection by strong magnetic fields.
One of the effective mechanisms of controlling the convection is by maintaining a non-uniform temperature gradient across the boundaries. Such a temperature gradient may be generated by (i) an appropriate heating or cooling at the boundaries, (ii) injection/suction of fluid at the boundaries, (iii) an appropriate distribution of heat source, and (iv) radiative heat transfer. These methods are mainly concerned with only space-dependent temperature gradients. However in practice, the non-uniform temperature gradients find its origin in transient heating or cooling at the boundaries. Hence, basic temperature profile depends explicitly on position and time. This problem, called the thermal modulation problem, involves solution of the energy equation under suitable time-dependent boundary conditions. This temperature profile which is a function of both space and time can be used as an effective mechanism to control the convective flow by proper adjustment of its parameters, namely, amplitude and frequency of modulation. It can be used to control the convection in material processing applications to achieve higher efficiency and to advance
convection in achieving major enhancement of heat, mass and momentum transfer.
There are many studies available in the literature concerning how a time-periodic boundary temperature affects the onset of Rayleigh-Bénard convection. Since the problem
of Taylor stability and Bénard stability are very similar,
Maxwell Cattaneo heat flux law:
Q 1 Q Q T,
.
Magnetic induction equation:
(4)
Venezian [21] investigated the thermal analogue of Donnellys experiment [22] for small amplitude temperature modulations. Many researchers, under different conditions, have investigated thermal instability in a horizontal layer with temperature modulation. Some of them are Gershuni and
H
t
Equation of state:
(q.)H (H.)q m2H,
o[1 (T To )].
(5)
(6)
Zhukhovitskii [23], Rosenblat and Tanaka [24], Siddheshwar
where, q
is the velocity, H
is the magnetic field, T is the
and Pranesh [25, 26], Mahabaleswar [27], Bhatia and
Bhadauria [28, 29], Malashetty and Mahantesh Swamy [30], Bhadauria [31], Siddheshwar and Annamma [32], Bhadauria and Atul [33], Pranesh and Sangeetha [34, 35], Pranesh [36] and Pranesh and Riya[37].
The objective of the present problem is to investigate the combined effect of magnetic field and temperature modulation on the onset of Raleigh-Bénard magnetoconvection using Maxwell-Cattaneo law.
-
MATHEMATICAL FORMULATION
Consider an infinite horizontal layer of a
temperature, P is the hydromagnetic pressure, is the density, o is the density of the fluid at reference
temperature T To , m 1 , m is magnetic
m
permeability, g is the acceleration due to gravity, is the thermal conductivity, is the coefficient of thermal
Q
expansion, is the dynamic viscosity, 1 1 q, is
2
the heat flux vector and is the constant relation time.
Boussinesquian, electrically conducting fluid, of depthd. Cartesian co-ordinate system is taken with origin in the lower boundary and z-axis vertically upwards. Let T be the
The lower wall z 0 and upper wall subjected to the temperatures
1
z d are
temperature difference between the lower and upper boundaries. (See Fig. 1.)
and
T T0 2 T[1 cos(t)],
(7)
H0 Z
1
T T0
1 T[1 cos(t )].
2
(8)
Z =d
T T0 2 T[1 cos(t )]
respectively, where is the small amplitude, is the frequency and is the phase angle.
Z = 0
Y
s(
O T T 1 T[1 co t)]
-
BASIC STATE
The basic state of the fluid being quiescent is described by:
0 2 X
Q 0, 0, Qb (z) , T Tb (z, t), ,
(9)
Figure 1. Schematic diagram of the Rayleigh Bénard situation.
The basic governing equations are:
Hb H0k
The temperature T , pressure P , heat flux and density
Continuity equation:
.q 0,
(1)
b satisfy
b b Qb
T 2T
Conservation of linear momentum:
b b ,
(10)
q
t z2
,
o t (q.)q P gk
(2)
-
Pb
bg,
(11)
2q
z
m H. H
Q
Tb ,
(12)
Conservation of energy:
b z
T qT
(3)
[1 (T T )].
(13)
t .Q,
b 0 b 0
The solution of (10) that satisfies the thermal boundary conditions of (7) and (8) is
0T.
(25)
Tb Ts (z) T1(z, t),
(14)
Operating divergence on the (23) and substituting
in (22), on using (20), we get
The steady temperature field Ts and an oscillating field T1
T 1 T
,
are given by
1 1 b 2W
where,
T
t t 2 z
T
(26)
Ts T0
(d 2z),
(15)
2T 1
W b
2d
z
z
where, 1
t
z
T )e d a()e
d e it
(16)
1 Rea(
.
The perturbation (21), (24) and (26) are non- dimensionalised using the following definitions:
In the (16)
d
(1 i)
1
2 2
,
(17)
(x*,y*,z*) (x, y, z) , q*
d
q
,
,
d
2
W t
W*
, t*
,
d2
(27)
T e i e d
a()
e e .
(18)
* T
2
H
*
and Re stands for the real part.
We now superpose infinitesimal perturbation on this basic state and study the stability of the system.
-
-
LINEAR STABILITY ANALYSIS
T T , H H
o
Using (25) in (21), operating curl twice on the resulting equation and non-dimensionalising the resulting
equation and also (24) and (26), using (27), we get:
Let the basic state be disturbed by an infinitesimal
thermal perturbation. We now have
1 (2
W) R2T 4
q q b q, P Pb P,
Pr t
Pr 2 H
1
W
,
(28)
b ,
Q Qb Q, ,
(19)
Q
z
T T
H k
Pm
z
b T , Hb
0 H
Hz
W Pr 2
The primes indicate that the quantities are infinitesimal perturbations and subscript b indicates basic state value.
z z
T
Pm
T0
Hz ,
2
(29)
2
Substituting (19) in to (1) to (6) and using the basic state solutions (10) to (13). We get the linearised
1 2C t t
z C
W T
.
(30)
equations governing the infinitesimal perturbations in the form
T0
z 1
-
2C W
t
.q 0,
q 2
(20)
where, the asterisks have been dropped for simplicity and the non-dimensional parameters
0 t P q gk
R, Q, Pr, Pm and C are as defined as
m 0 H
-
H
z
, (21)
ogT d3
R
(Rayleigh Number),
T
t
-
W Tb
.Q ,
z
(22)
m 2H02d2
Q
(Chandrasekhar Number)
1
Tb q
Pr
(Prandtl Number),
1 t Q
2
z z
W
,
(23)
o
H
H
T
W k
2
(24)
Pm
m
(Magnetic Prandtl Number) and
t 0 z
m H ,
C
2d2
(Cattaneo Number).
R, W R0 , W0 R1, W1 2 R 2 , W2 ….
(37)
In (30), T0 is the non dimensional from of Tb ,
The expansion (37) is substituted into (35) and
where,
z
z
the coefficients of various powers of are equated on either side of the equation. The resulting system of equation is
z
T0 1 f (z),
(31)
LW0 0,
(38)
,
LW Pr 2 C2 1 2C
f (z) Re
A()ez A()ez eit , (32)
1 t Pm
t (39)
ei e
f R
2 W
R 2 W
and
A()
2
.
(33)
0 1 0
1 1 0
e e
LW Pr 2 C2 1 2C
t
Equations (28) to (30) are solved subject to the following conditions:
2 t Pm
2W H
f R 2 W
R 2 W .
W T z 0 at z 0,1
(34)
0 1 1
1 1 1
z2 z
f R 2 W
R 2 W
Eliminating T and Hz from (28) to (30), we get a differential equation of order 8 for W in the form
where,
1 1 0
2 1 0
(40)
2 L M12M2 M3 R 2 M4 M3
4
[K ]K 2 .K1
2 W R2 T0 K
W, (35)
0 1
Pr 2 2
, (41)
Q Pr 2K
1 z 3
Q M5 2
Pm 2 z2
Pm z
Pr
M 1 2
,
where,
K1 t
2 , Pm
1
Pr t
M 1 2C
2 ,
K 2 1 2C
t t
2 ,
2
M
t t
Pr 2
,
K3 K1 C2 1 2C ,
3 t Pm
t
1 2
M 4 C2 1 2C ,
K 4 Pr t .
t
In dimensionless form, the velocity boundary
M 1 2C 2
conditions for solving (35) are obtained from (28) to (30) and (34) in the form:
5
t t .
2W
W
z2
4W
z4
6W
z6
0 at z 0, 1
(36)
-
SOLUTION TO THE ZEROTH ORDER PROBLEM
The zeroth order problem is equivalent to the Rayleigh-Bénard problem with magnetic field using Maxwell-Cattaneo law in the absence of temperature modulation. The stability of the system in the absence of
V. METHOD OF SOLUTION
We seek the eigen-function W and eigenvalues R
thermal modulation is investigated by introducing vertical velocity perturbation W0 lowest mode of convection as
of the (35) for the basic temperature distribution (31) that
W0 Sin (z) expil x m y,
(42)
departs from the linear profile
T0 1by quantities of
z
where, l and m are wavenumbers in xy-plane with
a 2 l2 m2. Substituting (42) into (38) we obtain the
order . Thus, the eigenvalues of the present problem differ from those of the ordinary Bénard convection by quantities of
expression for Rayleigh number in the form
K6 Q2K 2
order . We seek the solution of (35) in the form
R 0 1 1,
1
(43)
a 2 CK 2 1
1
where, K2 2 a 2 and a 2 l2 m2.
Setting
Q 0 and C 0, the (43)reduces the classical
and K2 n22 a 2 (see Siddheshwar and Pranesh [26]).
Rayleigh-Bénard result.
The equation for W2 , then becomes
K6
R 0 1 .
a 2
n
(44)
where,
LW2
A1R
2a 2W0
-
-
A2
a 2f R
0W1,
(48)
-
SOLUTION TO THE FIRST ORDER PROBLEM
A2 CK 2 1 Pr K 2 2C 2
Equation (39) for W1 now takes the form
n
Pm n
(49)
LW1 A1fR0a 2 sin(z) R1a 2 sin(z), (45)
i CK 2 1 2 Pr CK 2
n Pm n
where,
A1 CK 2 1 Pr 2
We shall not solve (48), but will use this to determine. The
1 Pm K1 .
solvability condition requires that the time-independent part
If the above equation is to have a solution, the right hand side must be orthogonal to the null-space of the operator L . This implies that the time independent part of the RHS of
of the right hand side of (48) must be orthogonal to
sin(nz) , and this results in the following equation,
2
R a 2 B () 2 A 2
the (45) must be orthogonal to sin(z) . Since f varies
sinusoidal with time, the only steady term on the RHS of
R 0 Re n 2 Y1 ,
2 L (, n) 2 2
(50)
(45) is
A1R1a 2 sin( z), so that R 1
1
*
0 . It follows that
Y1 L1(, n) L (, n),
all the odd coefficients i.e. R1 R3 …………. 0 in (37).
To solve (45), we expand the right-hand side using Fourier series expansion and obtain W1 by inverting the operator L term by term as
1
1
L (, n) L(, n)A* ,
2
2
A* and L* (, n) are the conjugates of
1
L1(, n) respectively.
A2 and
W1 A1R 0
where,
a 2 Re Bn () eit sin( L(, n)
z), (46)
-
MINIMUM RAYLEIGH NUMBER FOR CONVECTION
The value of Rayleigh number R obtained by this
Bn () A()gn1() A()gn1(),
2n22 e e 1n e i e i Bn () e e 2 n 12 2 2 n 12 2 ,
(47)
L, n X1 R 0a 2 X 2 Q Pr n 2 2 X3
procedure is the eigenvalue corresponding to the eigen- function W which, though oscillating, remains bounded in time. Since, R is a function of the horizontal wave number a and the amplitude of modulation , we have
Ra, R0 a 2R 2 a …………… (51) It was shown by Venezian [21] that the critical value
Pm
of thermal Rayleigh number is computed up to
O(2
) , by
iX 4 R 0a 2 X5 Q Pr n 2 2 ,
evaluating R 0 and R 2 at a a0 . It is only when one wishes
Pr
Pm
Pr
to evaluate
R 4 that
a 2 must be taken into account where
2C 2 K 6 2 K 4 K 6
a a
minimizes R . Evaluate the critical value of R
Pm n
n Pm n 2 2 2
X1
2C 4
1 K 2
K 4
,
K 4
(denoted by R 2c ) one has to substitute a a0 in R 2 , where
2
2
Pm n Pm n
X CK 2 1 Pr K 2 2C 2 ,
Pr n
a 0 is the value at which R 0 given by (44) is minimum.
We now evaluate R 2c for three cases:
2 n
Pm n
-
When the oscillating field is symmetric, so that the
X K 6 2C 2 K 2 ,
wall temperatures are modulated in phase with 0 . In this
3 n
1 4
n
2 2
2 1 4
case, Bn () bn
odd.
or 0, accordingly as n is even or
X
Pm
Pm K n Pr K n 2C
4
K n
.
-
When the wall temperature field is antisymmetric, corresponding to out-of phase modulation with
2C 2 K 4
Pr K 6 K 6
. In this case,
Bn () 0 or bn , accordingly
n Pm n n
X CK 2 1 2 Pr CK 2
as n is even or odd.
-
When only the temperature of the bottom wall is
5 n
Pm n
modulated, the upper plate being held at constant
temperature, with i . In this case,
bn
classical Fourier law of heat conduction by non-classical Maxwell-Cattaneo law.
Bn ()
for integers values of n.
2
The analysis presented in this paper is based on the
The bn are given by
4n22
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
bn 2 n 12 2 2 n 12 2
The variable defined in (17), in terms of the
1
)
dimensionless frequency, reduces to (1 i 2 and thus
on the value of the modulating frequency . When 1, the period of modulation is large and hence the disturbance grows to such an extent as to make finite amplitude effects important. When , R 2c 0 , thus the effect of modulation becomes small. In view of this, we choose only
2
2 16n 24 2
bn 2 n 14 4 2 n 14 4 .
moderate values of in our present study.
The results have been presented in Fig. 2 to Fig. 13, from these figures we observe that the value of R 2c may be
Hence from (50) and using the above expression
positive or negative. The sign of the correction Rayleigh
of Bn () , we can obtain the following expression for
R 2c
number R 2c
characterizes the stabilizing or destabilizing
in the form
effect of modulation on R 2c . A positive
R 2c
means the
R 0a 2
Bn () 2 A2 2 Y1
modulation effect is stabilizing while a negative
R 2c means
R 2c
2 Re
L1(, 2
2 . (52)
the modulation effect is destabilizing compared to the system in which the modulation is absent.
1
where, Y1 L1(, n) L* (, n)
Fig. 2 is the plot of R 2c versus frequency of
In (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 (52) converges rapidly in all cases. The variation of R 2c with for different values of C, Pr, Pm and Q are depicted in figures.
-
-
SUBCRITICAL INSTABILITY
The critical value of Rayleigh number R c is determine to be of order 2 , by evaluating R 0c and R 2c ,
modulation for different values of Cattaneo number C , in the case of in-phase modulation. In the figure, we observe that as C increase R 2c become more and more negative, which represents Cattaneo number has a destabilizing influence. Increase in Cattaneo number leads to narrowing of
the convection cells and thus lowering of the critical Rayleigh number. It is also observed from the figures that influence of Cattaneo number is dominant for small values because the convection cells have fixed aspect ratio. It is interesting to
note that for a given value of C, R 2c decreases for small
and is of the form
Rc R0c 2R 2c
(53)
values of and increases for the moderate values of . Thus small values of destabilize the system and moderate values
where, R 0c and R 2c can be obtained fom (44), (52) respectively.
If R 2c is positive, sub critical instability exists and
R c has a minimum at 0 . When R 2c is negative, sub
critical instabilities are possible. In this case from (47) we have
of stabilize the system. 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 in phase, 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
2
R 0c
R 2c
(54)
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
Now, we can calculate the maximum range of , by assigning values to the physical parameters involved in the above condition. Thus, the range of the amplitude of modulation, which causes sub critical instabilities in different physical situations, can be explained.
-
RESULTS AND DISCUSSION
In this paper we make an analytical study of the effects of temperature modulation and magnetic field at the onset of convection in Newtonian fluid by replacing the
predicted by the linear theory.
Fig. 3 is the plot of R 2c versus frequency of modulation for different values of Chandrasekhar number
Q , in the case of in-phase modulation. In the figure, we
observe that as Q increases R 2c becomes more and more negative, for small values of and becomes more and more positive for moderate values of . In making the conclusions
from the figure we should recollect that Q influences R 0c .
We find that
R 2c
increases with increase in Q . When the
The effect of
1 on R with respect to out-of-phase
magnetic field strength permeating the medium is modulation and only lower wall modulation is illustrated in
considerably strong, it induces viscosity into the fluid, and the magnetic lines are distorted by convection. Then these magnetic lines hinder the growth of disturbances, leading to the delay in the onset of instability.
Fig. 4 is the plot of R 2c versus frequency of
modulation for different values of Prandtl number Pr,in the case of in-phase modulation. In the figure, we observe that as Pr increases R 2c becomes more and more negative.
Fig. 15 and Fig. 16. We observe that as 1 increases R increases thus increase in 1 stabilizes the system. It is also observed from these figures that the frequency of modulation
destabilizes the system.
The results of this study are useful in controlling the convection by thermal modulation with Maxwell-Cattaneo law.
We can infer from this is that the effect of increasing the
viscosity of the fluid is to destabilize the system.
In Phase Temperature
100
Fig. 5 is the plot of R 2c versus frequency of
Modulation
C = 0.001 80
modulation for different values of magnetic Prandtl
C = 0.01
C = 0.05
60
number Pm, in the case of in-phase modulation. In the figure, C = 0.03
we observe that as Pm increase R 2c becomes more and more negative.
We now discuss the results pertaining to out-of- phase modulation. Comparing Fig. 2 to Fig. 5 and Fig. 6 to
C = 0.07
Pr = 0.9, Q = 10, Pm = 5
40
20
Fig. 9 respectively we find that
R 2c
is positive for the out-
0
R
of-phase modulation where as it is negative for in-phase
-12 -10 -8 -6 -4 -2 0 2 4 6 8
modulation. Thus,
Q, Pr and Pm have opposing influence 2c
in in-phase and out-of-phase modulations. However, C has identical influence on R 2c in both in-phase and out-of-phase and these can be seen in Fig. 2 and Fig. 6. The above results are due to the fact that the temperature field has essentially a linear gradient varying in time, so that the instantaneous Rayleigh number is super critical for half cycle and subcritical during the other half cycle.
The above results on the effect of various parameters on R 2c for out-of-phase modulation do not qualitatively change in the case of temperature modulation of just the lower boundary. This is illustrated with the help of Fig. 10 to Fig. 13.
Figure 2: Plot of R2c versus frequency of modulation for different values of
Cattaneo number C.
100
In Phase Temperature Modulation
Q = 0 80
Q = 10
Q = 25
Pr = 0.9, C = 0.01, Pm = 5 60
40
20
From the above result, we can conclude that the system is more stable when boundary temperature is modulated in out-of-phase in compare to only lower wall modulation and in-phase modulation. In-phase temperature modulation leads to sub critical motions. Sub critical motions
0
R
-15 -10 -5 0 5
2C
Figure 3:
are ruled out in the case of out-of-phase modulation and lower wall temperature modulation.
Fig. 14 is the plot of Rayleigh number R versus
Plot of R2c versus frequency of modulation for different values of Chandrasekhar number Q.
amplitude of modulation
1 for different values of , in the
case of in-phase modulation. From the figure we observe that
has amplitude of modulation 1
increases the Rayleigh
number R also increases thus amplitude of modulation stabilizes the system. It can be clearly seen that as increases R increases for smaller values of and decreases for
moderate values of which reconforms our earlier discussion on this.
In Phase Temperature Modulation
Pr = 0.5
Pr = 0.7
Pr = 0.9
100
80
100 Out of Phase Temperature Modulation
80 Q = 0
Q = 10
Q = 25
Q = 10, C = 0.01, Pm = 5 60
40
60 Pr = 0.9, C = 0.01, Pm = 5
40
20
20
0
R
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
2c
Figure 4: Plot of R2c versus frequency of modulation for different values
0
R
0 50 100 150 200 250
2c
Figure 7:
of Prandtl number Pr.
Plot of R2c versus frequency of modulation for different values of Chandrasekhar number Q.
In Phase Temperature Modulation
Pm = 05
Pm = 15
Pm = 25
Pr = 0.9, Q = 10, C = 0.01
100
80
60
40
100
80
60
Out of Phase Temperature Modulation
Pr = 0.5
Pr = 0.7
Pr = 0.9
Q = 10, C = 0.01, Pm = 5
40
20
0
R
-6 -5 -4 -3 -2 -1 0 1 2 3
2c
Figure 5:
20
0
R
0 50 100 150 200 250
2c
Plot of R2c versus frequency of modulation for different values of Magnetic Prandtl number Pm.
Figure 8: Plot of R2c versus frequency of modulation different values of Prandtl number Pr.
Pr = 0.9, Q = 10, Pm = 5
40
Out of Phase Temperature Modulation
100
Out of Phase Temperature
Modulation
100
C = 0.001
80
C = 0.01
C = 0.03
80
C = 0.05
60
C = 0.07
Pm = 05
Pm = 15
60 Pm = 25
Pr = 0.9, Q = 10, C = 0.01
40
20 20
0
R
0 50 100 150 200 250 300
2c
0
R
0 50 100 150 200 250
2c
Figure 6: Plot of R2c versus frequency of modulation for different values of Cattaneo number C.
Figure 9: Plot of R2c versus frequency of modulation for different values of Magnetic Prandtl number Pm.
100
80
60
40
Lower Wall Temperature Modulation
= 0.001
C = 0.01
C = 0.03
C = 0.05
C = 0.07
Pr = 0.9, Q = 10, Pm = 5
100
80
60
40
Lower Wall Temperature Modulation
Pm = 05
Pm = 15
Pm = 25
Pr = 0.9, Q = 10, C = 0.01
20 20
0
R
0 10 20 30 40 50 60 70
2c
0
R
0 10 20 30 40 50 60
2c
Figure 10: Plot of R2c versus frequency of modulation for different values of Cattaneo number C.
Figure 13: Plot of R2c versus frequency of modulation for different values of
Magnetic Prandtl number Pm.
100
80
60
40
Lower Wall Temperature Modulation
Q = 0
Q = 10
Q = 25
Pr = 0.9, C = 0.01, Pm = 5
791.4
791.2
791.0
790.8
R
790.6
c
790.4
In Phase Temperature Modulation
Pr = 0.5, Pm = 5, Q = 10, C = 0.01
790.2
20
790.0
0
R
0 10 20 30 40 50 60 70
2c
Figure 11: Plot of R2c versus frequency of modulation for different values of
Chandrasekhar number Q.
100
789.8
0.0 0.2 0.4 0.6 0.8 1.0
1
Figure 14: Plot of critical Rayleigh number Rc R0c 2R 2c versus
amplitude of modulation 1 for in phase temperature modulation for different values of frequency of modulation .
Lower wall Temperature
80 Modulation
Pr = 0.5
Pr = 0.7
60 Pr = 0.9
Q = 10, C = 0.01, Pm = 5
40
815
810
805
R
c 800
Out of Phase Temperature Modulation
Pr = 0.5, Pm = 5, Q = 10, C = 0.01
20 795
0
R
0 10 20 30 40 50 60
2c
790
0.0 0.2 0.4 0.6 0.8 1.0
Figure 12: Plot of R2c versus frequency of modulation different values of Prandtl number Pr.
Figure 15: Plot of Rc versus 1 for out of phase temperature modulation for different values of .
797
Lower Wall Temperature Modulation
-
G. Lebon and A. Cloot, A nonlinear stability analysis of the BénardMarangoni problem, J. Fluid Mech., vol. 145, 1984, pp. 447-469.
796
-
P.C. Dauby, M. Nelis and G. Lebon, Generalized Fourier
equations and thermo-convective instabilities, Revista
795
794
R
793
c
792
791
790
789
Pr = 0.5, Pm = 5, Q = 10, C = 0.01
0.0 0.2 0.4 0.6 0.8 1.0
Mexicana de Fisica., vol. 48, 2001, pp. 57-62.
-
B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction, Ricerche mat., vol. 58, 2009, pp. 157-162.
-
P.G. Siddheshwar, Rayleigh-Bénard Convection in a second order ferromagnetic fluid with second sounds, Proceedings of 8th Asian Congress of Fluid Mechanics, Shenzen, December 6-10, 1999, pp. 631.
-
S. Pranesh, Effect of Second sound on the onset of Rayleigh-Bénard convection in a Coleman Noll Fluid, Mapana Journal of sciences, vol. 13, 2008, pp. 1-9.
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S. Pranesh and R.V. Kiran, Study of Rayleigh-Bénard magneto convection in a Micropolar fluid with Maxwell- Cattaneo law, Applied Mathematics, 1, 2010, pp. 470-
Figure 16: Plot of Rc versus 1 for lower wall temperature modulation for different values of .
-
-
CONCLUSIONS Following conclusions are drawn from the problem:
-
The system is more stable when boundary temperature is modulated in out-of-phase.
-
In-phase temperature modulation leads to sub critical motions.
-
The results of the study throw light on an external means of controlling Rayleigh-Bénard convection either advancing or delaying convection by thermal modulation.
-
It is observed that for large frequencies, the effect of modulation disappears.
-
The non classical Maxwell Cattaneo heat flux law involves a wave type heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The classical Fourier flux law overpredicts the critical Rayleigh number compared to that predicted by the non-classical law. Overstability is the preferred mode of convection.
ACKNOWLEDGMENT
Authors would like to acknowledge management of Christ University and Christ Junior College for their support in completing the work and also to Prof. Pradeep. G. Siddheshwar, Professor, Department of Mathematics, Bangalore University, Bangalore for his valuable suggesting during the completion of their work which increased the quality of the paper.
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