$9-49$ Consider fully developed, two$-$dimensional channel

flow$-$flow between two infinite parallel plates separated by

distance $h,$ with both the top plate and bottom plate station$-$

ary, and a forced pressure gradient $d\frac{P}{d}x$ driving the flow as

illustrated in Fig. P$9-49.(d\frac{P}{d}x$ is constant and negative.$)$ The

flow is steady, incompressible, and two$-$dimensional in the $xy-$

plane. The velocity components are given by $u=(\frac{1}{2})(d\frac{P}{d}x)$

$(y^2-hy)$ and $v=0,$ where $$ is the fluid's viscosity. Generate

an expression for stream function $$ along the vertical dashed

line in Fig. P$9-49.$ For convenience, let $=0$ along the bottom

wall of the channel. What is the value of $$ along the top wall?

FIGURE P$9-49$