The onset of thermal convection, due to heating from below in a system consisting of a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity, is investigated analytically. The porous medium is both anisotropic in permeability whose principal axes are oriented in a direction that is oblique to the gravity vector and in thermal conductivity with principal directions coincident with the coordinate axes. The Beavers-Joseph condition is applied at the interface between the two layers. Based on parallel flow approximation theory, a linear stability analysis is conducted to study the geothermal river beds system and documented the effects of the physical parameters describing the problem. The critical Rayleigh numbers for both the fluid and porous layers corresponding, to the onset of convection arising from sudden heating and cooling at the boundaries are also predicted. The results obtained are in agreement with those found in the past for particular isotropic and anisotropic cases and for limiting cases concerning pure porous media and for pure fluid layer. It has demonstrated that the effects of anisotropic parameters are highly significant .
Nield, D.A. and Bejan, A. (2006) Convection in Porous Media. 3rd Edition, Springer, New York.
Somerton, C.W. and Catton, I. (1982) On the Thermal Instability of Superposed Porous and Fluid Layers. Journal of Heat Transfer, 104, 160-165. https://doi.org/10.1115/1.3245044
Nishimura, T., Takumi, T., Shiraishi, M., Kawamura, Y. and Ozoe, H. (1986) Numerical Analysis of Natural Convection in a Rectangular Horizontally Cavity Divided into Fluid and Porous Region. International Journal of Heat and Mass Transfer, 26, 889-898. https://doi.org/10.1016/0017-9310(86)90184-5
Vasseur, P., Wang, C.H. and Sen, M. (1989) Thermal Instability and Natural Convection in a Fluid Layer over a Porous Substrate. Waerme-Stoffuebertragung, 24, 337-347. https://doi.org/10.1007/BF01592780
Chen, F., Chen, C.F. and Pearlsein, A.J. (1991) Convective Instability in Superposed Fluid and Anisotropic Porous Layers. Physics of Fluids, 4, 556-565. https://doi.org/10.1063/1.858117
Shivakumara, I.S., Suma, S.P. and Chavaraddi, K.B. (2006) Onset of Surface-Ten-sion-Driven Convection in Superposed Layers of Fluid and Saturated Porous Medium. Archives of Mechanics, 58, 71-92.
Jones, I.P. (1973) Low Reynolds Number Flow Past a Porous Spherical Shell. Proceedings of the Cambridge Philosophical Society, 73, 231-238. https://doi.org/10.1017/S0305004100047642
Valenca-Lopez, J.J. and Ochoa-Tapia, J.A. (2001) A Study of Buoyancy-Driven Flow in a Confined Fluid Overlying a Porous Layer. International Journal of Heat and Mass Transfer, 44, 4725-4736. https://doi.org/10.1016/S0017-9310(01)00105-3
Alazmi, B. and Vafai, K. (2001) Analysis of Fluid Flow and Heat Transfer Interfacial Conditions between a Porous Medium and a Fluid Layer. International Journal of Heat and Mass Transfer, 44, 1735-1749. https://doi.org/10.1016/S0017-9310(00)00217-9
Hirata, S.C., Goyeau, B. and Gobin, D. (2007) Stability of Natural Convection in Superposed Fluid and Porous Layers: Influence of the Interfacial Jump Boundary Condition. Physics of Fluids, 19, Article ID: 058102. https://doi.org/10.1063/1.2730877
Hirata, S.C., Goyeau, B., Gobin, D., Carr, M. and Cotta, R.M. (2007) Linear Stability of Natural Convection in Superposed Fluid and Porous Layers: Influence of the Interfacial Modeling. International Journal of Heat and Mass Transfer, 50, 1356-1367. https://doi.org/10.1016/j.ijheatmasstransfer.2006.09.038
Hirata, S.C., Goyeau, B. and Gobin, D. (2009) Stability of Thermosolutal Natural Convection in Superposed Fluid and Porous Layers. Transport in Porous Media, 78, 525-536. https://doi.org/10.1007/s11242-008-9322-9
Hirata, S.C., Goyeau, B., Gobin, D., Chandesris, M. and Jamet, D. (2009) Stability of Natural Convection in Superposed Fluid and Porous Layers: Equivalence of the One- and Two-Domain Approaches. International Journal of Heat and Mass Transfer, 52, 533-536. https://doi.org/10.1016/j.ijheatmasstransfer.2008.07.045
Alloui, Z. and Vasseur, P. (2010) Convection in Superposed Fluid and Porous Layers. Acta Mechanica, 214, 245-260. https://doi.org/10.1007/s00707-010-0284-y
Yovogan, J. and Degan, G. (2013) Effect of Anisotropic Permeability on Convective Heat Transfer through a Porous River Bed Underlying a Fluid Layer. Journal of Engineering Mathematics, 81, 127-140. https://doi.org/10.1007/s10665-012-9605-6
Yovogan, J., Degan, G. and Fagbemi, L. (2018) Effect of Constant Magnetic Field on Convective Heat Transfer through Anisotropic River Beds. Journal of Crystallization Process and Technology, 8, 57-71. https://doi.org/10.4236/jcpt.2018.82004
Beavers, G.S. and Joseph, D.D. (1967) Boundary Conditions at a Naturally Permeable Wall. Journal of Fluid Mechanics, 30, 197-207. https://doi.org/10.1017/S0022112067001375
Rudraiah, N. and Veerabhadraiah, R. (1977) Temperature Distribution in Couette Flow past a Permeable Bed. Proc. Indian Academy of Sciences, 86, 537. https://doi.org/10.1007/BF03046910
Rudraiah, N. and Veerabhadraiah, R. (1978) Effect of Buoyancy on the Free Surface Flow past a Permeable Bed. Waerme-Stoffuebertragung, 11, 265. https://doi.org/10.1007/BF02587790
Cormack, D.E., Leal, L.G. and Inberger, J. (1974) Natural Convection in a Shallow Cavity with Differentially Heated End Walls. Part 1. Asymptotic Theory. Journal of Fluid Mechanics, 65, 209-230. https://doi.org/10.1017/S0022112074001352
Sparrow, F.M., Goldstein, R.J. and Jonsson, V.K. (1964) Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Conditions and Nonlinear Temperature Profile. Journal of Fluid Mechanics, 18, 513-528. https://doi.org/10.1017/S0022112064000386
Vasseur, P., Wang, C.H. and Sen, M. (1989) The Brinkman Model for Natural Convection in Shallow Porous Cavity with Uniform Heat Flux. Numerical Heat Transfer, 15, 221-242. https://doi.org/10.1080/10407788908944686
Sen, M., Vasseur, P. and Robillard, L. (1987) Multiple Steady States for Unicellular Natural Convection in an Inclined Porous Layer. International Journal of Heat and Mass Transfer, 30, 2097-2113. https://doi.org/10.1016/0017-9310(87)90089-5
Sen, M., Vasseur, P. and Robillard, L. (1988) Parallel Flow Convection in a Tilted Two-Dimensional Porous Layer Heated from Below. Physics of Fluids, 31, 3480-3487. https://doi.org/10.1063/1.866916
Bahloul, A., Vasseur, P. and Robillard, L. (2007) Convection of a Binary Fluid Saturating a Shallow Porous Cavity Subjected to Cross Heat Fluxes. Journal of Fluid Mechanics, 574, 317-342. https://doi.org/10.1017/S0022112006004113
Nield, D.A. (1968) Onset of Thermohaline Convection in a Porous Medium. Water Resources Research, 4, 553-560. https://doi.org/10.1029/WR004i003p00553
Degan, G. and Vasseur, P. (2003) Influence of Anisotropy on Convection in Porous Media with Nonuniform Thermal Gradient. International Journal of Heat and Mass Transfer, 46, 781-789. https://doi.org/10.1016/S0017-9310(02)00352-6