Pressure-driven thinning of a uniform Stokes layer
= 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 $p_0-\rho_aE^2x^2/2$. The <Cartesian streamfunction> ansatz $\psi=xf(y)$ reduces the Stokes equations to $f^{(4)}=0$. The resulting surface velocity is
$$
\dot h=-\frac{\rho_aE^2}{3\mu}h^3,
$$
so the positive thinning rate is $-\dot h=\rho_aE^2h^3/(3\mu)$.