 Solution

Integrate the axial ordinary differential equation twice: $u=-G{y}^2/(2\mu )+C_{1}y+C_2$. The no-slip boundary condition at zero gives $C_2=0$, and the one at $h$ gives $C_1=U/h+Gh/(2\mu )$. HenceThe first term is Couette flow; the second is Poiseuille flow driven in positive $x$ when $G>0$. Their addition is valid because the reduced Stokes equation is linear.