%0 Journal Article
%T Hall Effect on B¨¦nard Convection of Compressible Viscoelastic Fluid through Porous Medium
%A Mahinder Singh
%A Chander Bhan Mehta
%J Journal of Fluids
%D 2013
%R 10.1155/2013/910531
%X An investigation made on the effect of Hall currents on thermal instability of a compressible Walter¡¯s B¡ä elasticoviscous fluid through porous medium is considered. The analysis is carried out within the framework of linear stability theory and normal mode technique. For the case of stationary convection, Hall currents and compressibility have postponed the onset of convection through porous medium. Moreover, medium permeability hasten postpone the onset of convection, and magnetic field has duel character on the onset of convection. The critical Rayleigh numbers and the wave numbers of the associated disturbances for the onset of instability as stationary convection have been obtained and the behavior of various parameters on critical thermal Rayleigh numbers has been depicted graphically. The magnetic field, Hall currents found to introduce oscillatory modes, in the absence of these effects the principle of exchange of stabilities is valid. 1. Introduction The theoretical and experimental results of the onset of thermal instability (B¨¦nard convection) under varying assumptions of hydrodynamic and hydromantic stability have been discussed in a treatise by Chandrasekhar [1] in his celebrated monograph. If an electric field is applied at right angles to the magnetic field, the whole current will not flow along the electric field. This tendency of the electric current of flow across an electric field in the presence of a magnetic field is called Hall current effect. The Hall effect is likely to be important in many geophysical and astrophysical situations as well as in flows of laboratory plasma. The use of the Boussinesq approximation has been made throughout, which states that the variations of density in the equations of motion can safely be ignored everywhere except in its association with the external force. It has been shown by Sato [2] and Tani [3] that inclusion of Hall currents gave rise to a cross flow, that is, a flow at right angle to the primary flow through a channel in the presence of a transverse magnetic field. In particular, Tani [3] has found that Hall effect produces a cross-flow of double-swirl pattern in incompressible flow through a straight channel with arbitrary cross-section. This breakdown of the primary flow and formation of secondary flow may be presumably attributed to the inherent instability of the primary flow in the presence of Hall current. Sato [2] has pointed out that even if the distribution of the primary flow velocity is stable to external disturbances, the whole layer may become turbulent if the distribution of the
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