Effects of viscous dissipation on the convective instability of viscoelastic mixed convection flows in porous media

The thermal instability induced by small-amplitude perturbations superposed to the basic horizontal through flow in a plane porous layer (Prats problem) is here revisited. The fluid saturating the porous medium is assumed to be viscoelastic, and described through the Oldroyd-B model. The effect of viscous dissipation is taken into account. The main features of the linear instability are first described for the special case of negligible viscous dissipation, namely in the limit of a vanishing Gebhart number. Transverse rolls emerge as the selected normal modes at onset of convection. This same feature also arises when viscous dissipation is taken into account. In the general case, neutral stability curves as well as critical values of the Darcy-Rayleigh number, wave number and frequency are obtained by the numerical solution of an eigenvalue problem. It is shown that an adequate description of the combined effects of viscoelasticity and viscous dissipation can be obtained with the large Péclet number approximation. Such an approximation allows a simplified numerical solution and an optimised scaling of the parameters governing the transition to convective instability.