= Darcy-Bénard convection
{c}
= Rayleigh-Darcy convection
{c}
{synonym}
Darcy-Bénard convection is <thermal convection> in a porous layer heated from below, with velocity governed by <Darcy's law>. With fixed temperatures and impermeable boundaries, a normalized two-dimensional model uses <streamfunction> $\psi$ and temperature $\theta$:
$$
\nabla^2\psi=-R\theta_x,\qquad \theta_t+\psi_z\theta_x-\psi_x\theta_z=\nabla^2\theta.
$$
The control parameter is a porous-medium <Rayleigh number>. On $0<z<1$ with temperature values $0,-1$, the conductive state is $\psi=0$, $\theta=-z$.
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