 Viscous mountain spreading over an underthrust plate

For a Newtonian layer with density $\rho$, dynamic viscosity $\mu$, thickness $h$ and a plate moving at $U$, a stress-free roof and no-slip base give $q=Uh-\rho g{h}^3{h}_x/(3\mu )$. Zero net mountain-material volume flux produces $h^3=9\mu Ux/(\rho g)$ when the contact height vanishes. Very near its idealized contact line the slope violates the lubrication approximation.