The effect of local thermal nonequilibrium (LTNE) on the stability of natural convection in a vertical porous slab saturated by an Oldroyd-B fluid is investigated. The vertical walls of the slab are impermeable and maintained at constant but different temperatures. A two-field model that represents the fluid and solid phase temperature fields separately is used for heat transport equation. The resulting stability eigenvalue problem is solved numerically using Chebyshev collocation method as the energy stability analysis becomes ineffective in deciding the stability of the system. Despite the basic state remains the same for Newtonian and viscoelastic fluids, it is observed that the base flow is unstable for viscoelastic fluids and this result is qualitatively different from Newtonian fluids. The results for Maxwell fluid are delineated as a particular case from the present study. It is found that the viscoelasticity has both stabilizing and destabilizing influence on the flow. Increase in the value of interphase heat transfer coefficient Ht, strain retardation parameter Λ2 and diffusivity ratio α portray stabilizing influence on the system while increasing stress relaxation parameter Λ1 and porosity-modified conductivity ratio γ exhibit an opposite trend.

References

References
1.
Gill
,
A. E.
,
1969
, “
A Proof That Convection in a Porous Vertical Slab is Stable
,”
J. Fluid Mech.
,
35
(
03
), pp.
545
547
.
2.
Rees
,
D. A. S.
,
1988
, “
The Stability of Prandtl–Darcy Convection in a Vertical Porous Slot
,”
Int. J. Heat Mass Transfer
,
31
(
7
), pp.
1529
1534
.
3.
Lewis
,
S.
,
Bassom
,
A. P.
, and
Rees
,
D. A. S.
,
1995
, “
The Stability of Vertical Thermal Boundary Layer Flow in a Porous Medium
,”
Eur. J. Mech. B,
14
(
4
), pp.
395
408
.
4.
Straughan
,
B.
,
1988
, “
A Nonlinear Analysis of Convection in a Porous Vertical Slab
,”
Geophys. Astrophys. Fluid Dyn.
,
42
(
3–4
), pp.
269
275
.
5.
Qin
,
Y.
, and
Kaloni
,
P. N.
,
1993
, “
A Nonlinear Stability Problem of Convection in a Porous Vertical Slab
,”
Phys. Fluids A
,
5
(
8
), pp.
2067
2069
.
6.
Barletta
,
A.
, and
de B. Alves
,
L. S.
,
2014
, “
On Gill's Stability Problem for Non-Newtonian Darcy's Flow
,”
Int. J. Heat Mass Transfer
,
79
, pp.
759
768
.
7.
Vafai
,
K.
, and
Sozen
,
M.
,
1990
, “
Analysis of Energy and Momentum Transport for Fluid Flow Through a Porous Bed
,”
ASME J. Heat Transfer
,
112
(
3
), pp.
690
699
.
8.
Minkowycz
,
W. J.
,
Haji-Sheikh
,
A.
, and
Vafai
,
K.
,
1999
, “
On Departure From Local Thermal Equilibrium in Porous Media Due to a Rapidly Changing Heat Source: the Sparrow Number
,”
Int. J. Heat Mass Transfer
,
42
(
18
), pp.
3373
3385
.
9.
Duval
,
F.
,
Fichot
,
F.
, and
Quintard
,
M.
,
2004
, “
A Local Thermal Non-Equilibrium Model for Two-Phase Flows With Phase-Change in Porous Media
,”
Int. J. Heat Mass Transfer
,
47
(
3
), pp.
613
639
.
10.
Rees
,
D. A. S.
, and
Pop
,
I.
, “
Local Thermal Non-Equilibrium in Porous Medium Convection
,”
Transport Phenomena in Porous Media III
,
D. B.
Ingham
and
I.
Pop
, eds.,
Elsevier
,
Oxford, UK
, pp.
147
173
.
11.
Rees
,
D. A. S.
,
2010
, “
Microscopic Modelling of the Two-Temperature Model for Conduction in Heterogeneous Media
,”
J. Porous Media
,
13
(
2
), pp.
125
143
.
12.
Nield
,
D. A.
, and
Bejan
,
A.
,
2013
,
Convection in Porous Media
,
4th ed.
,
Springer
,
New York
.
13.
Banu
,
N.
, and
Rees
, D. A. S.
,
2002
, “
Onset of Darcy–Benard Convection Using a Thermal Non-Equilibrium Model
,”
Int. J. Heat Mass Transfer
,
45
(
11
), pp.
2221
2228
.
14.
Malashetty
,
M. S.
,
Shivakumara
,
I. S.
, and
Kulkarni
,
S.
,
2005
, “
The Onset of Lapwood–Brinkman Convection Using a Non-Equilibrium Model
,”
Int. J. Heat Mass Transfer
,
48
(
6
), pp.
1155
1163
.
15.
Shivakumara
,
I. S.
,
Malashetty
,
M. S.
, and
Chavaraddi
,
K. B.
,
2006
, “
Onset of Convection in a Viscoelastic Fluid Saturated Sparsely Packed Porous Layer Using a Thermal Non-Equilibrium Model
,”
Can. J. Phys.
,
84
(
11
), pp.
973
990
.
16.
Straughan
,
B.
,
2010
, “
Green-Naghdi Fluid With Non-Thermal Equilibrium Effects
,”
Proc. R. Soc. A
,
466
(
2119
), pp.
2021
2032
.
17.
Celli
,
M.
,
Barletta
,
A.
, and
Storesletten
,
L.
,
2013
, “
Thermoconvective Instability and Local Thermal Non-Equilibrium in a Porous Layer With Isoflux-Isothermal Boundary Conditions
,”
31st UIT Heat Transfer Conference
,
Como
,
Italy
, June 25–27, pp.
45
53
.
18.
Barletta
,
A.
, and
Rees
,
D. A. S.
,
2015
, “
Local Thermal Non-Equilibrium Analysis of the Thermoconvective Instability in an Inclined Porous Layer
,”
Int. J. Heat Mass Transfer
,
83
, pp.
327
336
.
19.
Barletta
,
A.
,
Celli
,
M.
, and
Lagziri
,
H.
,
2015
, “
Instability of a Horizontal Porous Layer With Local Thermal Non-Equilibrium: Effects of Free Surface and Convective Boundary Conditions
,”
Int. J. Heat Mass Transfer
,
89
, pp.
75
89
.
20.
Rees
,
D. A. S.
,
2011
, “
The Effect of Local Thermal Nonequilibrium on the Stability of Convection in a Vertical Porous Channel
,”
Transp. Porous Media
,
87
(
2
), pp.
459
464
.
21.
Scott
,
N. L.
, and
Straughan
,
B.
,
2013
, “
A Nonlinear Stability Analysis of Convection in a Porous Vertical Channel Including Local Thermal Nonequilibrium
,”
J. Math. Fluid Mech.
,
15
(
1
), pp.
171
178
.
22.
Straughan
,
B.
,
2015
,
Convection With Local Thermal Nonequilibrium and Microfluidic Effects
,
Springer
,
Heidelberg
.
23.
Malashetty
,
M. S.
,
Tan
,
W.
, and
Swamy
,
M.
,
2009
, “
The Onset of Double Diffusive Convection in a Binary Viscoelastic Fluid Saturated Anisotropic Porous Layer
,”
Phys. Fluids
,
21
(
8
), p.
084101
.
24.
Li
,
Z.
, and
Khayat
,
R. E.
,
2005
, “
Finite-Amplitude Rayleigh–Benard Convection and Pattern Selection for Viscoelastic Fluids
,”
J. Fluid Mech.
,
529
, pp.
221
251
.
25.
Sheu
,
L. J.
,
Tam
,
L. M.
,
Chen
,
J. H.
,
Chen
,
H. K.
,
Lin
,
K. T.
, and
Kang
,
Y.
,
2008
, “
Chaotic Convection of Viscoelastic Fluids in Porous Media
,”
Choas Solitons Fractals
,
37
(
1
), pp.
113
124
.
26.
Malashetty
,
M. S.
, and
Kulkarni
,
S.
,
2009
, “
The Convective Instability of Maxwell Fluid-Saturated Porous Layer Using a Thermal Non-Equilibrium Model
,”
J. Non-Newtonian Fluid Mech.
,
162
(
1–3
), pp.
29
37
.
27.
Gozum
,
D.
, and
Arpaci
,
V. S.
,
1974
, “
Natural Convection of Viscoelastic Fluids in a Vertical Slot
,”
J. Fluid Mech.
,
64
(
03
), pp.
439
448
.
28.
Takashima
,
M.
,
1993
, “
The Stability of Natural Convection in a Vertical Layer of Viscoelastic Liquid
,”
Fluid Dyn. Res.
,
11
(
4
), pp.
139
152
.
29.
Alishaev
,
M. G.
, and
Mirzadjanzade
,
A. Kh.
,
1975
, “
For the Calculation of Delay Phenomenon in Filtration Theory
,”
Izvestya Vuzov Nefti Gaz.
6
, pp.
71
78
.
30.
Khuzhayorov
,
B.
,
Auriault
,
J. L.
, and
Royer
,
P.
,
2000
, “
Derivation of Macroscopic Filtration Law for Transient Linear Viscoelastic Fluid Flow in Porous Media
,”
Int. J. Eng. Sci.
,
38
(
5
), pp.
487
504
.
31.
Akhatov
,
I. Sh.
, and
Chembarisova
,
R. G.
,
1993
, “
The Thermoconvective Instability in Hydrodynamics of Relaxational Liquids
,”
Instabilities in Multiphase Flows
,
G.
Gouesbet
and
A.
Berlemont
, eds.,
Plenum Press
,
New York
.
32.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
,
2007
,
Transport Phenomena
,
Wiley
,
New York
.
33.
Hirata
,
S. C.
,
de B. Alves
,
L. S.
,
Delenda
,
N.
, and
Ouarzazi
,
M. N.
,
2015
, “
Convective and Absolute Instabilities in Rayleigh–Bénard–Poiseuille Mixed Convection for Viscoelastic Fluids
,”
J. Fluid Mech.
,
765
, pp.
167
210
.
You do not currently have access to this content.