Brian Straughan

We develop and analyse a theory for thermal convection in a Darcy porous material taking into account the concept of local thermal non-equilibrium whereby the temperature in the solid skeleton may be different from that in the saturating fluid. The thermal microstructure of the solid skeleton is such that we adopt a Jeffreys theory for heat conduction in this part of the continuous medium. The Jeffreys theory in the solid has a mathematical form analogous to that proposed by Sir Harold Jeffreys, F.R.S., for a fluid, although now rather than being a relation between stress and the symmetric part of the velocity gradient as in the original case, the relation is between the heat flux and the temperature gradient. A novel effect is that oscillatory convection requires the Maxwell coefficient to exceed the Jeffreys coefficient, while the precise stationary–oscillatory transition depends separately on both coefficients and on the strength of thermal interaction between the solid and fluid phases.

We develop and analyze a theory for thermal convection in a linearly viscous incompressible fluid employing the heat flux law attributed to Sir Harold Jeffreys, F.R.S. This theory employs two relaxation coefficients known as the Maxwell and Jeffreys coefficients and we derive conditions on these parameters, which allow stationary or oscillatory convection to occur. We present a sharp global nonlinear energy stability theory when the linear Jeffreys constitutive heat model is employed. The nonlinear analysis shows precisely how the Maxwell and Jeffreys parameters affect the conditions for the onset of instability. It is the ratio of the two parameters which is crucial. It is shown that oscillatory convection may only occur when the Maxwell parameter exceeds the Jeffreys one.