= Lubrication boundary condition for a crack in a Darcy medium
{title2=$\partial_x(h^3p_x/12)+k[p_y]^+_-=0$}
In a thin fluid-filled crack, <lubrication theory> gives flux per unit depth $q=-h^3p_x/(12\mu)$. The surrounding <Darcy flow> has normal velocity $-kp_y/\mu$. Conservation in the crack gives $q_x=-[u_y]^+_-$, yielding the displayed condition. Pressure is continuous across the crack. This reduced model neglects transverse pressure variation and assumes wall slip lengths are small compared with $h$.
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