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 stationary, and a forced pressure gradient dP/dx driving the flow as illustrated in Fig. P9–49. (dP/dx is constant and negative.) The flow is steady, incompressible, and two-dimensional in the xy plane. The velocity components are given by u = (1/2μ)(dP/dx) (y² – hy) and v = 0, where m is the fluid’s viscosity. Generate an expression for stream function ψ along the vertical dashed line in Fig. P9–49. For convenience, let ψ = 0 along the bottom wall of the channel. What is the value of ψ along the top wall?

FIGURE P9–49

X (A)n