The linear stability of a parallel flow in a heterogeneous porous channel is analyzed by means of the Darcy law and the Oberbeck–Boussinesq approximation. The basic velocity and temperature distributions are influenced by the effect of the viscous dissipation, as well as, by the boundary conditions. A horizontal porous layer bounded by impermeable and infinitely wide walls is considered. The lower boundary is assumed to be thermally insulated, while the upper boundary is assumed to be isothermal. A transverse heterogeneity for the permeability and for the thermal conductivity is taken into account. The main task of this work is to investigate the role of this heterogeneity in changing the threshold for the onset of instability. A linear stability analysis by means of the normal modes method is performed. The onset of instability against oblique rolls is studied. The eigenvalue problem is solved numerically.
References
1.
Drazin
, P. G.
, and Reid
, W. H.
, 2004, Hydrodynamic Stability
, 2nd ed., Cambridge University Press
, Cambridge, UK
.2.
Nield
, D. A.
, and Bejan
, A.
, 2006, Convection in Porous Media
, 3rd ed., Springer-Verlag
, NY
.3.
Horton
, C. W.
, and Rogers
, F. T.
, 1945, “Convection Currents in a Porous Medium
,” J. Appl. Phys.
, 16
, pp. 367
–370
.4.
Lapwood
, E. R.
, 1948, “Convection of a Fluid in a Porous Medium
,” Proc. Cambridge Philos. Soc.
, 44
, pp. 508
–521
.5.
Gasser
, R. D.
, and Kazimi
, M. S.
, 1976, “Onset of Convection in a Porous Medium With Internal Heat Generation
,” ASME J. Heat Transfer
, 98
, pp. 49
–54
.6.
Barletta
, A.
, Celli
, M.
, and Rees
, D. A. S.
, 2009, “The Onset of Convection in a Porous Layer Induced by Viscous Dissipation: A Linear Stability Analysis
,” Int. J. Heat Mass Transfer
, 52
, pp. 337
–344
.7.
Barletta
, A.
, Celli
, M.
, and Rees
, D. A. S.
, 2009, “Darcy–Forchheimer Flow With Viscous Dissipation in a Horizontal Porous Layer: Onset of Convective Instabilities
,” ASME J. Heat Transfer
, 131
, p. 072602
.8.
Barletta
, A.
, and Nield
, D. A.
, 2009, “Effect of Pressure Work and Viscous Dissipation in the Analysis of the Rayleigh–Bénard Problem
,” Int. J. Heat Mass Transfer
, 52
, pp. 3279
–3289
.9.
Storesletten
, L.
, and Barletta
, A.
, 2009, “Linear Instability of Mixed Convection of Cold Water in a Porous Layer Induced by Viscous Dissipation
,” Int. J. Therm. Sci.
, 48
, pp. 655
–664
.10.
Barletta
, A.
, and Celli
, M.
, 2011, “Local Thermal Non-equilibrium Flow With Viscous Dissipation in a Plane Horizontal Porous Layer
,” Int. J. Therm. Sci.
, 50
, pp. 53
–60
.11.
Barletta
, A.
, and Nield
, D. A.
, 2011, “Thermosolutal Convective Instability and Viscous Dissipation Effect in a Fluid-Saturated Porous Medium
,” Int. J. Heat Mass Transfer
, 54
, pp. 1641
–1648
.12.
Nield
, D. A.
, and Kuznetsov
, A. V.
, 2000, “Effects of Heterogeneity in Forced Convection in a Porous Medium: Parallel Plate Channel or Circular Duct
,” Int. J. Heat Mass Transfer
, 43
, pp. 4119
–4134
.13.
Nield
, D. A.
, and Kuznetsov
, A. V.
, 2001, “Effects of Heterogeneity in Forced Convection in a Porous Medium: Parallel-Plate Channel, Asymmetric Property Variation, and Asymmetric Heating
,” J. Porous Media
, 4
, pp. 137
–148
.14.
Nield
, D. A.
, and Kuznetsov
, A. V.
, 2001, “The Interaction of Thermal Nonequilibrium and Heterogeneous Conductivity Effects in Forced Convection in Layered Porous Channels
,” Int. J. Heat Mass Transfer
, 44
, pp. 4369
–4373
.15.
Nield
, D. A.
, and Simmons
, C. T.
, 2007, “A Discussion on the Effect of Heterogeneity on the Onset of Convection in a Porous Medium
,” Transp. Porous Media
, 68
, pp. 413
–421
.16.
Nield
, D. A.
, Kuznetsov
, A. V.
, and Simmons
, C. T.
, 2009, “The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium
,” Transp. Porous Media
, 77
, pp. 169
–186
.17.
Nield
, D. A.
, and Kuznetsov
, A. V.
, 2010, “The Onset of Convection in a Heterogeneous Porous Medium With Transient Temperature Profile
,” Transp. Porous Media
, 85
, pp. 691
–702
.18.
Barletta
, A.
, 2009, “Local Energy Balance, Specific Heats and the Oberbeck–Boussinesq Approximation
,” Int. J. Heat Mass Transfer
, 52
, pp. 5266
–5270
.19.
Nield
, D. A.
, 2007, “The Modeling of Viscous Dissipation in a Saturated Porous Medium
,” ASME J. Heat Transfer
, 129
, pp. 1459
–1463
.20.
Joseph
, D. D.
, 1965, “Stability of Frictionally-Heated Flow
,” Phys. Fluids
, 8
, pp. 2195
–2200
.21.
Nield
, D. A.
, Barletta
, A.
, and Celli
, M.
, 2011, “The Effect of Viscous Dissipation on the Onset of Convection in an Inclined Porous Layer
,” J. Fluid Mech.
, FirstView, 1
–15
.22.
Barletta
, A.
, and Rees
, D. A. S.
, 2009, “Stability Analysis of Dual Adiabatic Flows in a Horizontal Porous Layer
,” Int. J. Heat Mass Transfer
, 52
, pp. 2300
–2310
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
