Variable viscosity effects on the dissipation instability in a porous layer with horizontal throughflow

Abstract

The role of viscous heating in the onset of the instability in a liquid-saturated porous medium is studied. The change of the liquid viscosity with the temperature is taken into account by using a linear fluidity model. The system examined is a horizontal porous layer with an adiabatic lower boundary and an isothermal upper boundary. The combined effects of the viscous heating and of the variable viscosity yield a basic stationary and parallel throughflow in a horizontal direction. This basic solution may display singularities when the product between the Péclet number and the viscosity-temperature slope parameter exceeds the threshold value π/2. The linear stability of the basic solution is studied with respect to normal modes disturbances arbitrarily oriented to the basic flow direction. In all the physically realistic cases, the most unstable disturbances are proved to be the longitudinal rolls (the wave vector is perpendicular to the basic velocity). The instability to the longitudinal rolls occurs when the product between the Péclet number and the viscosity-temperature slope parameter exceeds its critical value. This critical value is smaller than π/2, for every nonzero value of the buoyancy parameter, viz., the Gebhart number. As a consequence, the parametric domain where the singularities of the basic solution arise is in fact included in the instability domain.