As a follow-up to Prob. 9–46, calculate the volume flow rate per unit width into the page of Fig. P9–46 from first principles (integration of the velocity field). Compare your result to that obtained directly from the stream function. Discuss.

Data from Problem 46

Consider fully developed Couette flow—flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary as illustrated in Fig. P9–46. The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity field is given by

Generate an expression for stream function ψ along the vertical dashed line in Fig. P9–46. For convenience, let ψ = 0 along the bottom wall of the channel. What is the value of ψ along the top wall?

FIGURE P9–46

Transcribed Image Text:

V = (u, v) (Vy/h)i + Oj.