 Solution

Suppress hats for this calculation. Transverse pressure uniformity and zero surface pressure give $p=0$ throughout the film. Thus $u_{zz}=-S$. Integrating twice, using $u_z(h)=0$ and $u(0)=-1$, givesThe flux is $q={\int }_0^{h}udz=-h+S{h}^3/3$. Integrate continuity across the layer and apply the wall and kinematic conditions to obtain $h_t+q_x=0$. Restoring hats yieldsThe cubic flux contains downward gravity-driven transport and upward wall entrainment.