A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal rotating fluid layer. The temperature at the boundaries has been taken to be time-periodic, governed by the sinusoidal function. A weakly nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex form of Ginzburg–Landau equation (CGLE), is calculated. The influence of external controlling parameters such as amplitude and frequency of modulation on heat transfer has been investigated. The dual effect of rotation on the system for the oscillatory mode of convection is found either to stabilize or destabilize the system. The study establishes that heat transport can be controlled effectively by a mechanism that is external to the system. Further, the bifurcation analysis also presented and established that CGLE possesses the supercritical bifurcation.

References

References
1.
Chandrasekhar
,
S.
,
1961
,
Hydrodynamic and Hydromagnetic Stability
,
Oxford University Press
,
Oxford, UK
.
2.
Drazin
,
P. G.
, and
Reid
,
D. H.
,
2004
,
Hydrodynamic Stability
,
Cambridge University Press
,
Cambridge, UK
.
3.
Donnelly
,
R. J.
,
1964
, “
Experiments on the Stability of Viscous Flow Between Rotating Cylinders III: Enhancement of Hydrodynamic Stability by Modulation
,”
Proc. R. Soc. London Ser. A
,
281
(
1384
), pp.
130
139
.
4.
Venezian
,
G.
,
1969
, “
Effect of Modulation on the Onset of Thermal Convection
,”
J. Fluid Mech.
,
35
(02), pp.
243
254
.
5.
Gershuni
,
G. Z.
, and
Zhukhovitskii
,
E. M.
,
1963
, “
On Parametric Excitation of Convective Instability
,”
J. Appl. Math. Mech.
,
27
(
5
), pp.
1197
1204
.
6.
Rosenblat
,
S.
, and
Herbert
,
D. M.
,
1970
, “
Low Frequency Modulation of Thermal Instability
,”
J. Fluid Mech.
,
43
(02), pp.
385
398
.
7.
Rosenblat
,
S.
, and
Tanaka
,
G. A.
,
1971
, “
Modulation of Thermal Convection Instability
,”
Phys. Fluids
,
14
(
7
), pp.
1319
1322
.
8.
Finucane
,
R. G.
, and
Kelly
,
R. E.
,
1976
, “
Onset of Instability in a Fluid Layer Heated Sinusoidally From Below
,”
Int. J. Heat Mass Transfer
,
19
(
1
), pp.
71
85
.
9.
Roppo
,
M. N.
,
Davis
,
S. H.
,
Rosenblat
,
S.
,
1984
, “
Bénard Convection With Time-Periodic Heating
,”
Phys. Fluids
,
27
(
4
) pp.
796
803
.
10.
Ahlers
,
G.
,
Hohenberg
,
P. C.
, and
Lucke
,
M.
,
1985
, “
Thermal Convection Under External Modulation of the Driving Force—I: The Lorenz Model
,”
Phys Rev. A
,
32
(
6
) pp.
3493
3518
.
11.
Niemela
,
J. J.
, and
Donnelly
,
R. J.
,
1987
, “
Externally Modulation of Rayleigh–Bénard Convection
,”
Phys. Rev. Lett.
,
59
(
21
) pp.
2431
2434
.
12.
Schmitt
,
S.
, and
Lucke
,
M.
,
1991
, “
Amplitude Equation for Modulated Rayleigh–Bénard Convection
,”
Phys. Rev. A
,
44
(
8
), pp.
4986
5002
.
13.
Or
,
A. C.
, and
Kelly
,
R. E.
,
1999
, “
Time Modulated Convection With Zero Mean Temperature Gradient
,”
Phys. Rev. E
,
60
(
2
), pp.
1741
1747
.
14.
Or
,
A. C.
,
2001
, “
Onset Condition of Modulated Rayleigh–Bénard at Low Frequency
,”
Phys. Rev. E
,
64
, p.
050201
.
15.
Bhatia
,
P. K.
, and
Bhadauria
,
B. S.
,
2000
, “
Effect of Modulation on Thermal Convection Instability
,”
Z. Naturforsch.
,
55
(11–12), pp.
957
966
.
16.
Bhatia
,
P. K.
, and
Bhadauria
,
B. S.
,
2001
, “
Effect of Low-Frequency Modulation on Thermal Convection Instability
,”
Z. Naturforsch.
,
56
(6–7), pp.
507
522
.
17.
Bhadauria
,
B. S.
, and
Bhatia
,
P. K.
,
2002
, “
Time Periodic Heating of Rayleigh–Bénard Convection
,”
Phys. Scr.
,
66
(
1
), pp.
59
65
.
18.
Bhadauria
,
B. S.
,
2006
, “
Time-Periodic Heating of Rayleigh–Bénard Convection in a Vertical Magnetic Field
,”
Phys. Scr.
,
73
(
3
), pp.
296
302
.
19.
Bhadauria
,
B. S.
,
Bhatia
,
P. K.
, and
Debnath
,
L.
,
2009
, “
Weakly Non-Linear Analysis of ‘Rayleigh–Bénard Convection With Time-Periodic Heating
,”
Int. J. Non-Linear Mech.
,
44
(
1
), pp.
58
65
.
20.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Weak Nonlinear Double Diffusive Magneto-Convection in a Newtonian Liquid Under Temperature Modulation
,”
Int. J. Eng. Math.
,
2014
, pp.
1
14
.
21.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2013
, “
Heat Transport in an Anisotropic Porous Medium Saturated With Variable Viscosity Liquid Under Temperature Modulation
,”
Transp. Porous Media
,
100
(
2
), pp.
279
295
.
22.
Kiran
,
P.
, and
Bhadauria
,
B. S.
,
2015
, “
Nonlinear Throughflow Effects on Thermally Modulated Porous Medium
,”
Ain Shams Engg. J.
, (in press).
23.
Kiran
,
P.
, and
Bhadauria
,
B. S.
,
2015
, “
Chaotic Convection in a Porous Medium Under Temperature Modulation
,”
Transp. Porous Media
,
107
(
3
), pp.
745
763
.
24.
Ecke
,
R. E.
,
Zhong
,
F.
, and
Knobloch
,
E.
,
1992
, “
Hopf Bifurcation With Broken Reflection Symmetry in Rotating Rayleigh–Bénard Convection
,”
EPL (Europhys. Lett.)
,
19
(
3
), pp.
177
182
.
25.
Hu
,
Y.
,
Ecke
,
R. E.
, and
Ahlers
,
G.
,
1993
, “
Convection Near Threshold for Prandtl Numbers Near 1
,”
Phys. Rev. E
,
48
(
6
), pp.
4399
4413
.
26.
Liu
,
Y.
, and
Ecke
,
R. E.
,
1997
, “
Heat Transport Scaling in Turbulent Rayleigh–Bénard Convection: Effects of Rotation and Prandtl Number
,”
Phys. Rev. Lett.
,
79
(
12
), pp.
2257
2260
.
27.
Hu
,
Y.
,
Pesch
,
W.
,
Ahlers
,
G.
, and
Ecke
,
R. E.
,
1998
, “
Convection Under Rotation for Prandtl Numbers Near 1: Küppers–Lortz Instability
,”
Phys. Rev. E
,
58
(
5
), pp.
5821
5833
.
28.
Vorobieff
,
P.
, and
Ecke
,
R. E.
,
2002
, “
Turbulent Rotating Convection: An Experimental Study
,”
J. Fluid Mech.
,
458
, pp.
191
218
.
29.
Rauscher
,
J. W.
, and
Kelly
,
R. E.
,
1975
, “
Effect of Modulation on the Onset of Thermal Convection in a Rotating Fluid
,”
Int. J. Heat Mass Transfer
,
18
(
10
), pp.
1216
1217
.
30.
Malashetty
,
M. S.
, and
Swamy
,
M.
,
2008
, “
Effect of Thermal Modulation on the Onset of Convection in a Rotating Fluid Layer
,”
Int. J. Heat Mass Transfer
,
51
(
11
), pp.
2814
2823
.
31.
Bhadauria
,
B. S.
,
Siddheshwar
,
P. G.
, and
Suthar
,
O. P.
,
2012
, “
Nonlinear Thermal Instability in a Rotating Viscous Fluid Layer Under Temperature/Gravity Modulation
,”
ASME J. Heat Transfer
,
134
(
10
), p.
102502
.
32.
Siddheshwar
,
P. G.
,
Bhadauria
,
B. S.
,
Mishra
,
P.
, and
Srivastava
,
A. K.
,
2012
, “
Study of Heat Transport by Stationary Magneto-Convection in a Newtonian Liquid Under Temperature or Gravity Modulation Using Ginzburg–Landau Model
,”
Int. J. Non-Linear Mech.
,
47
(
5
), pp.
418
425
.
33.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Weakly Nonlinear Oscillatory Convection in a Viscoelastic Fluid Saturating Porous Medium Under Temperature Modulation
,”
Int. J Heat Mass Transfer
,
77
, pp.
843
851
.
34.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Heat and Mass Transfer for Oscillatory Convection in a Binary Viscoelastic Fluid Layer Subjected to Temperature Modulation at the Boundaries
,”
Int. Commun. Heat Mass Transfer
,
58
, pp.
166
175
.
35.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid Layer Under Gravity Modulation
,”
Int. J. Non-Linear Mech.
,
65
, pp.
133
140
.
36.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid Saturated Porous Medium Under Gravity Modulation
,”
Transp. Porous Media
,
104
(
3
), pp.
451
467
.
37.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Chaotic and Oscillatory Magneto-Convection in a Binary Viscoelastic Fluid Under G-Jitter
,”
Int. J. Heat Mass Transfer
,
84
, pp.
610
624
.
38.
Malkus
,
W. V. R.
, and
Veronis
,
G.
,
1958
, “
Finite Amplitude Cellular Convection
,”
J. Fluid Mech.
,
4
(03), pp.
225
260
.
39.
Kim
,
M. C.
,
Lee
,
S. B.
,
Kim
,
S.
, and
Chung
,
B. J.
,
2003
, “
Thermal Instability of Viscoelastic Fluids in Porous Media
,”
Int. J. Heat Mass Transfer
,
46
(
26
), pp.
5065
5072
.
40.
Kapil
,
M. S.
,
Bajaj
,
A.
, and
Werner
,
G.
,
2002
, “
Rayleigh–Bénard Convection With Rotation at Small Prandtl Numbers
,”
Phys. Rev. E
,
65
, p.
056309
.
41.
Vadasz
,
P.
,
1998
, “
Coriolis Effect on Gravity-Driven Convection in a Rotating Porous Layer Heated From Below
,”
J. Fluid Mech.
,
376
, pp.
351
375
.
42.
Bhadauria
,
B. S.
, and
Kiran
,
P.
,
2014
, “
Effect of Rotational Speed Modulation on Heat Transport in a Fluid Layer With Temperature Dependent Viscosity and Internal Heat Source
,”
Ain Shams Engg. J.
,
5
(
4
), pp.
1287
1297
.
43.
Kuppers
,
G.
, and
Lortz
,
D.
,
1969
, “
Transition From Laminar Convection to Thermal Turbulence in a Rotating Fluid Layer
,”
J. Fluid Mech.
,
35
(03), pp.
609
620
.
44.
Yang
,
W. M.
,
1995
, “
Effect of Modulation on Radiation Induced Instability
,”
Int. J. Heat Mass Transfer
,
38
(
1
), pp.
47
53
.
45.
Siddheshwar
,
P. G.
,
Bhadauria
,
B. S.
, and
Suthar
,
O. P.
,
2013
, “
Synchronous and Asynchronous Boundary Temperature Modulations of Bénard–Darcy Convection
,”
Int. J Non-Linear Mech.
,
49
, pp.
84
89
.
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