= Gravity film on an upward-moving wall
{title2=$h_t+\partial_x(-Uh+\rho gh^3/(3\mu))=0$}
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 $q=-Uh+\rho gh^3/(3\mu)$. 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 $q'(h)=-U+\rho gh^2/\mu$.
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