 Moving thermal front in a porous current

A localized temperature adjustment separates regions with different fluid density, viscosity and Darcy speed. Mass conservation in the moving frame gives the displayed plateau-depth relation. A speed stated as $V_T=\Gamma u_c$ with $u_c$ a Darcy velocity has effective retardation parameter $\varphi \Gamma$. Setting both laboratory-frame mass fluxes equal neglects storage associated with passage of the temperature front.