The instability of steady natural convection of a non-Fourier fluid of the single-phase lagging (SPL) type between two vertical surfaces maintained at different temperatures is studied. SPL fluids possess a relaxation time, which reflects the delay in the response of the heat flux and the temperature gradient. The SPL model is particularly relevant to low-temperature liquids, ultrafast processes, and nanofluids (with a retardation time added in this case). Linear stability analysis is employed to obtain the critical state parameters, such as critical Grashof numbers. For intermediate Prandtl numbers (Pr = 7.5), when non-Fourier level exceeds a certain value, the neutral stability curve comprises a Fourier branch and an oscillatory branch. In this case, oscillatory convection increasingly becomes the mode of preference, compared to both conduction and stationary convection. Critical Grashof number decreases for fluids with higher non-Fourier levels.

References

References
1.
Joseph
,
D.
, and
Preziosi
,
L.
,
1989
, “
Heat Waves
,”
Rev. Mod. Phys.
,
61
(
1
), pp.
41
73
.
2.
Cattaneo
,
C.
,
1958
, “
A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation
,”
C. R.
,
247
, pp.
431
433
.
3.
Vernotte
,
P.
,
1961
, “
Some Possible Complication in the Phenomena of Thermal Conduction
,”
C. R.
,
252
, pp.
2190
2191
.
4.
Chandrasekharaiah
,
D. S.
,
1998
, “
Hyperbolic Thermoelasticity: A Review of Recent Literature
,”
ASME Appl. Mech. Rev.
,
51
(
12
), pp.
705
729
.
5.
Meyer
,
R. J.
,
2002
, “
Ultrasonic Drying of Saturated Porous Solids Via Second Sound
,” http://www.freepatentsonline.com/6376145.html
6.
Herrera
,
L.
, and
Falcón
,
N.
,
1995
, “
Heat Waves and Thermohaline Instability in a Fluid
,”
Phys. Lett. A
,
201
(
1
), pp.
33
37
.
7.
Dai
,
W.
,
Wang
,
H.
,
Jordan
,
P. M.
,
Mickens
,
R. E.
, and
Bejan
,
A.
,
2008
, “
A Mathematical Model for Skin Burn Injury Induced by Radiation Heating
,”
Int. J. Heat Mass Transfer
,
51
(
23–24
), pp.
5497
5510
.
8.
Espinosa-Paredes
,
G.
, and
Espinosa-Martínez
,
E.-G.
,
2009
, “
Fuel Rod Model Based on Non-Fourier Heat Conduction Equation
,”
Ann. Nucl. Energy
,
36
(
5
), pp.
680
693
.
9.
Miranville
,
A.
, and
Quintanilla
,
R.
,
2009
, “
A Generalization of the Caginalp Phase-Field System Based on the Cattaneo Law
,”
Nonlinear Anal. Theory Methods Appl.
,
71
(5–6), pp.
2278
2290
.
10.
Mitra
,
K.
,
Kumar
,
S.
,
Vedevarz
,
A.
, and
Moallemi
,
M. K.
,
1995
, “
Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
117
(
3
), pp.
568
573
.
11.
Cai
,
R.
,
Gou
,
C.
, and
Li
,
H.
,
2006
, “
Algebraically Explicit Analytical Solutions of Unsteady 3-D Nonlinear Non-Fourier (Hyperbolic) Heat Conduction
,”
Int. J. Therm. Sci.
,
45
(
9
), pp.
893
896
.
12.
Cossali
,
G. E.
,
2004
, “
Periodic Conduction in Materials With Non-Fourier Behaviour
,”
Int. J. Therm. Sci.
,
43
(
4
), pp.
347
357
.
13.
Roetzel
,
W.
,
Putra
,
N.
, and
Das
,
S. K.
,
2003
, “
Experiment and Analysis for Non-Fourier Conduction in Materials With Non-Homogeneous Inner Structure
,”
Int. J. Therm. Sci.
,
42
(
6
), pp.
541
552
.
14.
Sahoo
,
N.
,
Ghosh
,
S.
,
Narasimhan
,
A.
, and
Das
,
S. K.
,
2014
, “
Investigation of Non-Fourier Effects in Bio-Tissues During Laser Assisted Photothermal Therapy
,”
Int. J. Therm. Sci.
,
76
, pp.
208
220
.
15.
Liu
,
L. H.
,
Tan
,
H. P.
, and
Tong
,
T. W.
,
2001
, “
Non-Fourier Effects on Transient Temperature Response in Semitransparent Medium Caused by Laser Pulse
,”
Int. J. Heat Mass Transfer
,
44
(
17
), pp.
3335
3344
.
16.
Ghazanfarian
,
J.
, and
Shomali
,
Z.
,
2012
, “
Investigation of Dual-Phase-Lag Heat Conduction Model in a Nanoscale Metal-Oxide-Semiconductor Field-Effect Transistor
,”
Int. J. Heat Mass Transfer
,
55
(
21–22
), pp.
6231
6237
.
17.
Chou
,
Y.
, and
Yang
,
R.-J.
,
2009
, “
Two-Dimensional Dual-Phase-Lag Thermal Behavior in Single-/Multi-Layer Structures Using CESE Method
,”
Int. J. Heat Mass Transfer
,
52
(
1–2
), pp.
239
249
.
18.
Antaki
,
P. J.
,
2005
, “
New Interpretation of Non-Fourier Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
127
(
2
), pp.
189
193
.
19.
Letfullin
,
R. R.
,
George
,
T. F.
,
Duree
,
G. C.
, and
Bollinger
,
B. M.
,
2008
, “
Ultrashort Laser Pulse Heating of Nanoparticles: Comparison of Theoretical Approaches
,”
Adv. Opt. Technol.
,
2008
, pp.
1
8
.
20.
Nie
,
X. B.
,
Chen
,
S. Y.
,
E
,
W. N.
, and
Robbins
,
M. O.
,
2004
, “
A Continuum and Molecular Dynamics Hybrid Method for Micro- and Nano-Fluid Flow
,”
J. Fluid Mech.
,
500
, pp.
55
64
.
21.
Bhattacharyya
,
S.
,
Zheng
,
Z.
, and
Conlisk
,
A. T.
,
2005
, “
Electro-Osmotic Flow in Two-Dimensional Charged Micro- and Nanochannels
,”
J. Fluid Mech.
,
540
, pp.
247
267
.
22.
Oliveira
,
M. S. N.
,
Rodd
,
L. E.
,
McKinley
,
G. H.
, and
Alves
,
M. A.
,
2008
, “
Simulations of Extensional Flow in Microrheometric Devices
,”
Microfluid. Nanofluid.
,
5
(
6
), pp.
809
826
.
23.
Peshkov
,
V.
,
1944
, “
Second Sound in Helium II
,”
J. Phys. USSR III
, Vol. 8, pp.
381
–389.
24.
Shimazaki
,
T.
,
Murakami
,
M.
, and
Iida
,
T.
,
1995
, “
Second Sound Wave Heat Transfer, Thermal Boundary Layer Formation and Boiling: Highly Transient Heat Transport Phenomena in He II
,”
Cryogenics (Guildf)
,
35
(
10
), pp.
645
651
.
25.
Zhang
,
P.
,
Murakami
,
M.
, and
Wang
,
R. Z.
,
2006
, “
Study of the Transient Thermal Wave Heat Transfer in a Channel Immersed in a Bath of Superfluid Helium
,”
Int. J. Heat Mass Transfer
,
49
(
7–8
), pp.
1384
1394
.
26.
Donnelly
,
R. J.
,
2009
, “
The Two-Fluid Theory and Second Sound in Liquid Helium
,”
Phys. Today
,
62
(
10
), pp.
34
39
.
27.
Vadasz
,
J. J.
,
Govender
,
S.
, and
Vadasz
,
P.
,
2005
, “
Heat Transfer Enhancement in Nano-Fluids Suspensions: Possible Mechanisms and Explanations
,”
Int. J. Heat Mass Transfer
,
48
(
13
), pp.
2673
2683
.
28.
Wang
,
L.
, and
Wei
,
X.
,
2009
, “
Heat Conduction in Nanofluids
,”
Chaos, Solitons Fractals
,
39
(
5
), pp.
2211
2215
.
29.
Stranges
,
D. F.
,
Khayat
,
R. E.
, and
Albaalbaki
,
B.
,
2013
, “
Thermal Convection of Non-Fourier Fluids. Linear Stability
,”
Int. J. Therm. Sci.
,
74
, pp.
14
23
.
30.
Niknami
,
M.
, and
Khayat
,
R. E.
,
2013
, “
Energy Growth of Disturbances in a Non-Fourier Fluid
,”
Int. J. Heat Mass Transfer
,
67
, pp.
613
626
.
31.
Chester
,
M.
,
1963
, “
Second Sound in Solids
,”
Phys. Rev.
,
131
(
5
), pp.
2013
2015
.
32.
Struchtrup
,
H.
, and
Taheri
,
P.
,
2011
, “
Macroscopic Transport Models for Rarefied Gas Flows: A Brief Review
,”
IMA J. Appl. Math.
,
76
(
5
), pp.
672
697
.
33.
Nettleton
,
R. E.
,
1960
, “
Relaxation Theory of Thermal Conduction in Liquids
,”
Phys. Fluids
,
3
(
2
), pp.
216
225
.
34.
Ali
,
M.
, and
Weidman
,
P. D.
,
1990
, “
On the Stability of Circular Couette Flow With Radial Heating
,”
J. Fluid Mech.
,
220
, pp.
53
84
.
35.
Bergholz
,
R. F.
,
1978
, “
Instability of Steady Natural Convection in a Vertical Fluid Layer
,”
J. Fluid Mech.
,
84
(
4
), pp.
743
768
.
36.
Choi
,
I. G.
, and
Korpela
,
S. A.
,
1980
, “
Stability of the Conduction Regime of Natural Convection in a Tall Vertical Annulus
,”
J. Fluid Mech.
,
99
(
4
), pp.
725
738
.
37.
Daniels
,
P. G.
,
1987
, “
Convection in a Vertical Slot
,”
J. Fluid Mech.
,
176
, pp.
419
441
.
38.
Elder
,
J. W.
,
1965
, “
Laminar Free Convection in a Vertical Slot
,”
J. Fluid Mech.
,
23
(
1
), pp.
77
98
.
39.
Lee
,
Y.
, and
Korpela
,
S. A.
,
1983
, “
Multicellular Natural Convection in a Vertical Slot
,”
J. Fluid Mech.
,
126
, pp.
91
121
.
40.
Takashima
,
M.
,
1993
, “
The Stability of Natural Convection in a Vertical Layer of Viscoelastic Liquid
,”
Fluid Dyn. Res.
,
11
(
4
), pp.
139
152
.
41.
Vest
,
C. M.
, and
Arpaci
,
V. S.
,
1969
, “
Stability of Natural Convection in a Vertical Slot
,”
J. Fluid Mech.
,
36
(
1
), pp.
1
15
.
42.
Hart
,
J. E.
,
1971
, “
Stability of the Flow in a Differentially Heated Inclined Box
,”
J. Fluid Mech.
,
47
(
3
), pp.
547
576
.
43.
McFadden
,
G. B.
,
Coriell
,
S. R.
,
Boisvert
,
R. F.
, and
Glicksman
,
M. E.
,
1984
, “
Asymmetric Instabilities in Buoyancy-Driven Flow in a Tall Vertical Annulus
,”
Phys. Fluids
,
27
(
6
), pp.
1359
1361
.
44.
Choi
,
I. G.
, and
Korpela
,
S. A.
,
1980
, “
Stability of the Conduction Regime of Natural Convection in a Tall Vertical Annulus
,”
J. Fluid Mech.
,
99
(
4
), pp.
725
738
.
45.
Bahloul
,
A.
,
Mutabazi
,
I.
, and
Ambari
,
A.
,
2000
, “
Codimension 2 Points in the Flow Inside a Cylindrical Annulus With a Radial Temperature Gradient
,”
Eur. Phys. J. Appl. Phys.
,
9
(
3
), pp.
253
264
.
46.
Christov
,
C. I.
,
2009
, “
On Frame Indifferent Formulation of the Maxwell–Cattaneo Model of Finite-Speed Heat Conduction
,”
Mech. Res. Commun.
,
36
(
4
), pp.
481
486
.
47.
Khayat
,
R. E.
, and
Ostoja-Starzewski
,
M.
,
2011
, “
On the Objective Rate of Heat and Stress Fluxes. Connection With Micro/Nano-Scale Heat Convection
,”
Discrete Contin. Dyn. Syst. B
,
15
(
4
), pp.
991
998
.
48.
Baranyai
,
A.
,
Evans
,
D.
, and
Daivis
,
P.
,
1992
, “
Isothermal Shear-Induced Heat Flow
,”
Phys. Rev. A
,
46
(
12
), pp.
7593
7600
.
49.
Todd
,
B. D.
, and
Evans
,
D. J.
,
1995
, “
The Heat Flux Vector for Highly Inhomogeneous Nonequilibrium Fluids in Very Narrow Pores
,”
J. Chem. Phys.
,
103
(
22
), pp.
9804
9809
.
50.
Todd
,
B. D.
, and
Evans
,
D. J.
,
1997
, “
Temperature Profile for Poiseuille Flow
,”
Phys. Rev. E
,
55
(
3
), pp.
2800
2807
.
51.
Uribe
,
F. J.
, and
Garcia
,
A. L.
,
1999
, “
Burnett Description for Plane Poiseuille Flow
,”
Phys. Rev. E
,
60
(
4
), pp.
4063
4078
.
52.
Aoki
,
K.
,
Takata
,
S.
, and
Nakanishi
,
T.
,
2002
, “
Poiseuille-Type Flow of a Rarefied Gas Between Two Parallel Plates Driven by a Uniform External Force
,”
Phys. Rev. E
,
65
(
2
), p.
026315
.
53.
Drazin
,
P. G.
, and
Reid
,
W. H.
,
2004
,
Hydrodynamic Stability
,
Cambridge University
,
Cambridge, UK
, p.
605
.
You do not currently have access to this content.