Linear instability of mixed convection of cold water in a porous layer induced by viscous dissipation

An analysis of linear stability of the stationary laminar Darcy flow in a horizontal porous layer is performed. The porous layer is saturated with cold water. The upper plane boundary is assumed to be subject to heat transfer with finite conductance to an environment at the temperature of maximum density of cold water. The lower plane boundary is adiabatic. Convective instabilities are caused by flow viscous dissipation, inducing a basic temperature distribution that decreases in the upward direction. For prescribed values of the Biot number Bi and the Gebhart number Ge, the critical values of the product R=Ge*Pe^4, where Pe is the Péclet number associated to the basic flow solution, are determined. Disturbances in the form of oblique rolls are analyzed. It is shown that: transverse rolls are preferred at the onset of convection; critical values of R are almost independent of Ge for realistic values of this parameter; critical values of R depend on Bi and lie in an interval 3