= Solution
The <lubrication theory>[lubrication approximation] requires $H/L\ll1$, small wall slope, and negligible inertia, for example $\rho UH/\mu\ll L/H$. To leading order,
$$
p_z=0,\qquad
\mu\mathbf u_{zz}=\nabla p,\qquad
\nabla\cdot\mathbf u+w_z=0.
$$
With no slip at $z=0,h$,
$$
\mathbf u=\mathbf U\left(1-\frac zh\right)
+\frac{z(z-h)}{2\mu}\nabla p.
$$
Integration gives
$$
\boxed{\mathbf q=\frac{\mathbf Uh}{2}-\frac{h^3}{12\mu}\nabla p}.
$$
Integrated incompressibility gives $\nabla\cdot\mathbf q=0$, so locally $\mathbf q=\nabla\wedge(0,0,\psi)$. Solving the flux law for $\nabla p$ and taking its curl yields
$$
\boxed{\nabla\cdot\left(h^{-3}\nabla\psi\right)
=-Uh^{-3}h_y}.
$$
Back to article page
