Solution
= Solution
The leading streamwise momentum equation in <lubrication theory> is
$$
0=-p_x+\mu u_{yy},
$$
with $p=p(x)$. The <no-slip boundary condition> is
$$
u(x,0)=U,\qquad u(x,h(x))=0.
$$
Twice integrating in $y$ gives the local <Couette-Poiseuille flow in a thin gap>
$$
\boxed{
u(x,y)=U\left(1-\frac yh\right)
+\frac{p_x}{2\mu}y(y-h).}
$$
Its volume flux per unit span is
$$
\boxed{
Q(x)=\int_0^hu\,dy
=\frac{Uh}{2}-\frac{h^3p_x}{12\mu}.}
$$