Solution
= Solution
The steady parallel-flow <Navier-Stokes equations> reduce to
$$
p_y=0,\qquad \mu u''+\rho g=0.
$$
The boundary conditions are
$$
p(h)=p_0,\qquad u(0)=0,\qquad \mu u'(h)=-ku(h).
$$
The last condition balances the liquid's tangential traction with the opposing air stress. Hence $p=p_0$ and
$$
\boxed{
u(y)=-\frac{\rho g}{2\mu}y^2
+\frac{\rho gh}{\mu+kh}\left(1+\frac{kh}{2\mu}\right)y.}
$$