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An analytical investigation is conducted to study the effect of magnetic field on convection heat transfer through packed porous beds which consists of a horizontal fluid layer (river bed) and a porous zone with anisotropic permeability and underlined by a surface heated by a constant temperature T 1 . The free surface of the fluid layer overlying the horizontal porous layer receives solar rays to length of day and is then considered heated isothermally at temperature T 2 such as T 1 < T 2 . Flow in porous medium is assumed to be governed by the generalized Brinkman-extended Darcy law and in the fluid layer by the Navier-Stokes model. The Beavers-Joseph condition is applied at the interface between the two layers. The influence of Hartmann number and hydrodynamic anisotropy on the convective phenomenon is investigated analytically. It is found that the magnetic field, the anisotropic permeability and the thickness of the porous lining, ε, have a strong influence of the geothermal convective flow and the heat transfer rate. Journal of Crystallization Process and Technology presence of a uniform magnetic field, was investigated analytically by [2]. On the basis of the linear stability theory, the critical Rayleigh numbers for the onset of motion were obtained for various types of thermal and hydrodynamic boundary conditions. The case of a shallow cavity heated isothermally from the sides was considered by [3]. The effect of a magnetic field on the convective heat transfer was investigated analytically using matched asymptotic expansions. The results indicate that the retarding effect of the electromagnetic body Lorentz force decreases the strength of convection in the enclosure.Several studies have been made on the effect of the magnetic field on the thermal convection through porous cavity. Our present research concerns the study of the thermal convection in a horizontal fluid-superposed porous layer and few studies have been made in this sense (see our literature magazine [4]). * 1 2 K K K = and the orientation angle φ, defined as the angle between the horizontal direction and the principal axis with the permeability K 2 .Thus, the flow regime is divided into two zones:Zone 1 (fluid layer) from the free surface of the fluid to the surface of the J. Yovogan et al.

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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.

In this paper, an analytical study is reported on double-diffusive natural convection in a shallow porous cavity saturated with a non-Newtonian fluid by using the Darcy model with the Boussinesq approximations. A Cartesian coordinate system is chosen with the x-and y-axes at the geometrical center of the cavity and the y'-axis vertically upward. The top and bottom horizontal boundaries are subject to constant heat (q) and mass (j) fluxes. The porous medium is anisotropic in permeability whose principal axes are oriented in a direction that is oblique to the gravity vector. The permeabilities along the two principal axes of the porous matrix are denoted by K1 and K2. The anisotropy of the porous layer is characterized by the permeability ratio K*=K 1 /K 2 and the orientation angle φ, defined as the angle between the horizontal direction and the principal axis with the permeability K2. The viscous dissipations are negligible. Based on parallel flow approximation theory, the problem is solved analytically, in the limit of a thin layer and documented the effects of the physical parameters describing this investigation. Solutions for the flow fields, Nusselt and Sherwood numbers are obtained explicitly in terms of the governing parameters of the problem.