Gravity film on an upward-moving wall ($h_t+{\partial }_x(-Uh+\rho g{h}^3/(3\mu ))=0$)

Take the downward coordinate along a vertical wall that moves upward with speed $U$. For a stress-free thin Newtonian film with negligible inertia and surface-pressure gradients, the lubrication approximation gives $\mu u_{zz}=-\rho g$, $u(0)=-U$, and $u_z(h)=0$. Thus $u=-U+(\rho g/\mu )(hz-z^2/2)$ and the volume flux per unit width is $q=-Uh+\rho g{h}^3/(3\mu )$. Integrating incompressibility across the material film gives the displayed conservation law. Gravity drives fluid downward, while the moving wall entrains it upward; the local characteristic speed is $q^{\prime }(h)=-U+\rho g{h}^2/\mu$.