$(20$ points$)$ As shown below, the plane Poiseuille flow is a pressure$-$driven flow in a horizontal parallel$-$plate channel with wall spacing $H$ and width $W.$ Assuming that the flow is steady with a mean velocity $u,$ the local velocity is found to be

$v_x(y)=6u[(\frac{y}{H})-(\frac{y}{H}{)}^2],v_y=v_z=0$

$($a$)$ Express the friction force$/$unit area on the lower surface of flow by the lower plate in terms of a component of stress. Use $T=(gradv+grad{v}^T).($b$)$ Evaluate the difference between the average pressures at surface $S_3$ and $S_4.$