Zero-flux Couette-Poiseuille flow ($u/U=3(y/h{)}^2-2y/h$)

Between a resting wall at $y=0$ and a wall moving at speed $U$ at $y=h$, an adverse pressure gradient $p_x=G$ givesZero volume flux requires $G=6\mu U/h^2$, yielding $u/U=3(y/h{)}^2-2y/h$. Reverse flow below $2h/3$ balances the positive flux near the moving wall. The minimum is $-U/3$ at $h/3$.