 Pressure-driven thinning of a uniform Stokes layer

Let a viscous layer occupy $0<y<h(t)$, with no slip at $y=0$, zero tangential traction at $y=h$, and exterior pressure $p_0-{\rho }_a{E}^2{x}^2/2$. The Cartesian streamfunction ansatz $\psi =xf(y)$ reduces the Stokes equations to $f^{(4)}=0$. The resulting surface velocity isso the positive thinning rate is $-\dot{h}={\rho }_a{E}^2{h}^3/(3\mu )$.