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.
Issue Section:
Natural and Mixed Convection
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.html6.
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
.Copyright © 2016 by ASME
You do not currently have access to this content.
