 Solution

Take $x$ vertically downward along the wall and $y$ normally outward, with the wall at $y=0$ and free surface at $y=h$. The expected steady velocity is $\mathbf{u}=u(y){\mathbf{e}}_x$: it vanishes at the wall and is opposed by an upward air shear $ku(h)$ at the surface. Gravity supplies the downward body force $\rho g{\mathbf{e}}_x$, while the surface normal stress is the atmospheric pressure $p_0$.