As a follow-up to Prob. 9–49, calculate the volume flow rate per unit width into the page of Fig. P9–49 from first principles (integration of the velocity field). Compare your result to that obtained directly from the stream function. Discuss.

Data from Problem 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 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

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