Zero-flux Couette-Poiseuille flow
= Zero-flux Couette-Poiseuille flow
{title2=$u/U=3(y/h)^2-2y/h$}
Between a resting wall at $y=0$ and a wall moving at speed $U$ at $y=h$, an adverse <pressure gradient> $p_x=G$ gives
$$
u(y)=-\frac G{2\mu}y(h-y)+\frac{Uy}{h}.
$$
Zero <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 $h/3$.