A two-dimensional layer of very viscous fluid of uniform thickness $h(t)$ sits on a stationary, rigid surface $y=0$. It is impacted by a stream of air (which can be assumed inviscid) such that the air pressure at $y=h$ is $p_{0}-\frac{1}{2} \rho_{a} E^{2} x^{2}$, where $p_{0}$ and $E$ are constants, $\rho_{a}$ is the density of the air, and $x$ is the coordinate parallel to the surface.

What boundary conditions apply to the velocity $\mathbf{u}=(u, v)$ and stress tensor $\sigma$ of the viscous fluid at $y=0$ and $y=h$ ?

By assuming the form $\psi=x f(y)$ for the stream function of the flow, or otherwise, solve the Stokes equations for the velocity and pressure fields. Show that the layer thins at a rate

$$V=-\frac{\mathrm{d} h}{\mathrm{~d} t}=\frac{1}{3} \frac{\rho_{a}}{\mu} E^{2} h^{3}$$