Thermal convection with a Jeffreys model

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.