Darcy porous convection with a Jeffreys heat model

We develop and analyse a theory for thermal convection in a Darcy porous material when the heat transfer is based on a Jeffreys theory. The memory terms in the Jeffreys heat constitutive law lead to the important result that the structure of the instability hinges on the relative size of two non-dimensional parameters, the Maxwell coefficient and the Jeffreys coefficient. This leads to a fundamentally different theory of thermal convection in a porous material. When the Jeffreys parameter is greater than or equal to the Maxwell one we show the linear operator attached to the stability problem is symmetric and instability will only arise via stationary convection. In this case the linear instability limit is the same as the nonlinear stability threshold and so linear theory completely captures the onset of the physics of convection. On the other hand, when the Maxwell parameter exceeds the Jeffreys one a skew symmetric term is present and this allows oscillatory convection to happen with suitable parameters. Physically, this is to be expected, since a large Maxwell parameter is leading to a hyperbolic theory of heat flow and hence oscillatory instabilities, whereas when the Jeffreys parameter is large this represents a very strong form of heat dissipation. With modern heterogeneous porous materials like ceramics or foams different time scales for heat transfer at the molecular level are likely and thus a Jeffreys theory with relaxation time for both heat flux and temperature gradient is useful. This work shows that the memory effect of the heat flux equation leads to a strong structural change in the instability behaviour.