Couette-Poiseuille flow ($u(y)=Uy/a+(G/2)y(a-y)$)

Couette-Poiseuille flow superposes wall-driven shear and pressure-driven parabolic flow between no-slip parallel plates. With one plate moving at $U$, separation $a$, and $dp/dx=-\rho \nu G$, the displayed profile has volume flux $Ua/2+G{a}^3/12$ per unit width. Requiring zero flux gives $G=-6U/a^2$ and $u=U[3(y/a{)}^2-2y/a]$, with a pressure gradient induced by return-flow constraints.