Thermal instability of viscoelastic fluids in horizontal porous layers as initial value problem

The instability of a viscoelastic fluid saturating a horizontal porous layer heated from below is studied theoretically with a dynamical system approach. The viscoelastic character of the flow is taken into account by a modified Darcy's law. The conservation equations of mass, momentum and energy are approximated by a reduced-order system of nonlinear, ordinary differential equations, which is similar to the well-known system derived by Lorenz to describe atmospheric convection. Equilibrium points and their stability are expressed as functions of a dimensionless heat capacity and of two relaxation parameters. Qualitative expressions of the Nusselt number when the system is out of equilibrium are also derived.