Effect of Wave Number on Thermal
Convection in Couple-Stress Fluid in The Presence of Rotation and Magnetic Field
Kamal Singh1 and Ajaib S. Banyal2*
1Research Scholar, Department
of Maths, Singhania University, Pacheri Bari, Jhunjhunu-
333515 (Raj.)
2Department
of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP)-177033
*Corresponding
Author Email: singh_kamal1979@rediffmail.com; ajaibbanyal@rediffmail.com
ABSTRACT:
The
effect of wave number and rotation on the thermal convection in couple-stress
fluid heated from below in the presence of uniform magnetic field is
investigated. The paper established the regime for all non-oscillatory and
non-decaying slow motions starting from rest; the necessary condition for the
existence of ‘overstability’ and the sufficient
condition for the validity of the ‘exchange principle’ is also derived, when
both the bounding planes are rigid. Further, the stationary convection at
marginal state with free horizontal boundaries is analyzed numerically and
graphically, showing that the rotation has a stabilizing effect on the system.
In the presence of rotation; the magnetic field and couple-stress parameter has
stabilizing (or destabilizing) effect on the system depending upon conditions
satisfied. In the absence of rotation, both magnetic field and couple-stress
parameter has stabilizing effect. However, for the constant magnitude of any of
the couple-stress parameter, magnetic field and rotation, the wave number has a
destabilizing effect for a value less than the critical value, which varies
with the magnitude of the couple-stress parameter, magnetic field and rotation;
and for higher values than the critical value of the wave number; it has a
stabilizing effect on the system.
KEYWORDS: Thermal convection; Couple-Stress Fluid; Rotation; Magnetic Field; PES;
Chandrasekhar number; Taylor number. MSC 2000 No.:
76A05, 76E06, 76E15; 76E07.
1. INTRODUCTION:
The
thermal instability of a fluid layer with maintained adverse temperature
gradient by heating the underside plays an important role in Geophysics,
interiors of the Earth, Oceanography and Atmospheric Physics etc. A detailed
account of the theoretical and experimental study of the onset of Bénard Convection in Newtonian fluids, under varying
assumptions of hydrodynamics and hydromagnetics, has
been given by Chandrasekhar[1]. The use of Boussinesq
approximation has been made throughout, which states that the density changes are
disregarded in all other terms in the equation of motion except the external
force term. Sharma et al[2]has considered the effect of suspended
particles on the onset of Bénard convection in hydromagnetics. The fluid has been considered to be
Newtonian in all above studies. With the growing importance of non-Newtonian
fluids in modern technology and industries, the investigations on such fluids
are desirable. Stokes[3] proposed and postulated the theory of couple-stress fluid. One of
the applications of couple-stress fluid is its use to the study of the
mechanism of lubrication of synovial joints, which has become the object of
scientific research. According to the theory of Stokes[3], couple-stresses are found to
appear in noticeable magnitude in fluids having very large molecules. Since the
long chain hylauronic acid molecules are found as
additives in synovial fluid, Walicki and Walicka[4] modeled synovial fluid as couple-stress
fluid in human joints. An electrically conducting couple-stress fluid heated
from below in porous medium in the presence of uniform horizontal magnetic
field has been studied by Sharma and Sharma[5]. Sharma and Thakur[6] have studied the thermal convection in
couple-stress fluid in porous medium in hydromagnetics.
Sharma and Sharma[7] and Kumar and Kumar[8] have studied the effect of dust particles,
magnetic field and rotation on couple-stress fluid heated from below and for
the case of stationary convection, found that dust particles have destabilizing
effect on the system, where as the rotation is found to have stabilizing effect
on the system, however couple-stress and magnetic field are found to have both
stabilizing and destabilizing effects under certain conditions.
However,
in all above studies the case of two free boundaries which is a little bit
artificial except the stellar atmospheric case is considered. Banerjee et al[9] gave a new scheme for combining the
governing equations of thermohaline convection, which
is shown to lead to the bounds for the complex growth rate of the arbitrary
oscillatory perturbations, neutral or unstable for all combinations of
dynamically rigid or free boundaries and, Banerjee
and Banerjee[10] established a criterion on
characterization of non-oscillatory motions in hydrodynamics which was further
extended by Gupta et al. [11]. However no such result existed for
non-Newtonian fluid configurations, in general and for couple-stress fluid
configurations, in particular. Banyal[12] and Banyal
and Singh[13-14]
have characterized the non-oscillatory motions in couple-stress fluid.
Keeping
in mind the importance of non-Newtonian fluids, the present paper is an attempt
to characterize the onset of instability analytically, in a layer of
incompressible couple-stress fluid heated from below in the presence of uniform
vertical rotation opposite to force field of gravity, when the bounding
surfaces of infinite horizontal extension, at the top and bottom of the fluid
are rigid. It is shown that for the configuration under consideration that, if,
then an arbitrary neutral or unstable modes of the system are definitely non-oscillatory
and, in particular the PES is valid, where 
is the Taylor number,
Q is the Chandrasekhar number, 
is the magnetic Prandtl number and F is the couple-stress parameter.
2. FORMULATION OF THE PROBLEM AND
PERTURBATION EQUATIONS
5. CONCLUSIONS:
In this paper, the effect of wave number,
magnetic field, couple-stress parameter and rotation on a couple-stress fluid
heated from below is investigated and the immediate conclusions of the theorems
proved above; and numerical and graphical discussion, are as follows:
(a). The necessary condition for
the onset of oscillatory motions and ‘overstability’,
for configuration under consideration, is that the inequality (14) must be
satisfied. Thus, the sufficient condition for the non-existence of oscillatory
motions and hence the validity of ‘exchange principle’ is that, for the
configuration under consideration. The essential content of the theorem, from
the point of view of linear stability theory is that for the configuration of
couple-stress fluid of infinite horizontal extension heated form below, having
rigid top and bottom horizontal bounding surfaces; and the region outside is
perfectly conducting, in the presence of uniform vertical magnetic field and
rotation, parallel to the force field of gravity, an arbitrary neutral or
unstable modes of the system are definitely non-oscillatory in character if,
and in particular PES is valid.
(b). It is observed from figure 2 and Table 1,
that rotation has stabilizing effect on the onset of instability in the present
configuration. From figure 1 and figure 3; and table 1, magnetic field and the
couple-stress parameter has the stabilizing (or destabilizing) effect on the
onset of instability on the system depending upon the condition (20). However,
in the absence of rotation, magnetic field and couple-stress parameter has
stabilizing effect on the system.
(c). From table 1, it is clear that for the
stationary convection at marginal state for the constant magnitude of
couple-stress parameter, magnetic field and rotation, the wave number has a
destabilizing effect for a value less than the critical value, which varies
with the magnitude of couple-stress parameter, medium permeability and
rotation, and for higher value than the critical value of the wave number, it
has a stabilizing effect on the system when both the horizontal boundaries are
free.
Table 1
|
x |
R1 when F1 =
10, TA1=200 |
R1 when Q1=20,
TA1=200 |
R1 When Q1=20, F1 = 10 |
||||
|
Q1 = 20 |
Q1 = 40 |
Q1 = 60 |
F1=5 |
F1=15 |
TA1=400 |
TA1=600 |
|
|
0.1 |
449.8242 |
670.0442 |
890.0442 |
376.6192 |
532.0292 |
519.9285 |
590.0328 |
|
0.2 |
269.5100 |
389.5100 |
509.5100 |
217.6700 |
321.3500 |
306.7001 |
343.8902 |
|
0.3 |
214.9971 |
301.6631 |
388.3291 |
167.3958 |
262.5984 |
240.8025 |
266.6077 |
|
0.4 |
192.7380 |
262.7380 |
332.7380 |
144.7180 |
240.7580 |
212.5761 |
232.4141 |
|
0.5 |
184.0714 |
244.0714 |
304.0714 |
133.4464 |
234.6964 |
200.1428 |
216.2142 |
|
0.6 |
182.8162 |
236.1482 |
289.4802 |
128.2042 |
237.4282 |
195.8309 |
209.6840 |
|
0.7 |
186.3653 |
234.9353 |
283.5053 |
126.7092 |
246.0214 |
197.8302 |
209.2951 |
|
0.8 |
193.4413 |
238.4413 |
283.4413 |
127.8313 |
259.0513 |
203.3726 |
213.3040 |
|
0.9 |
203.3442 |
245.5662 |
287.7882 |
130.9440 |
275.7443 |
212.0451 |
220.7460 |
|
1.0 |
215.6923 |
255.6923 |
295.6923 |
135.6923 |
295.6923 |
223.3846 |
231.0769 |
|
1.5 |
308.7206 |
342.0526 |
375.3846 |
178.5174 |
438.9237 |
313.2867 |
317.8529 |
|
2.0 |
451.5100 |
481.5100 |
511.5100 |
249.0100 |
654.0100 |
454.5200 |
457.5301 |
|
2.5 |
647.5258 |
675.5258 |
703.5258 |
347.4008 |
947.6508 |
649.6516 |
651.7774 |
|
3 |
902.8886 |
929.5546 |
956.2206 |
476.2326 |
1329.5446 |
904.4665 |
906.0444 |
6. ACKNOWLEDGEMENT:
Authors are highly thankful to the
reviewers for their thoughtful and constructive comments which improves the
quality of the research paper significantly.
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V.K. Stokes, Couple-stress in fluids,
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|
Received on 17.12.2014 Accepted
on 23.12.2014 ©A&V
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