Discussion
Consider steady, two-dimensional, incompressible flow over a non-porous flat plate as shown in the figure. For the control volume PQRS, the speed, \(u_{\infty}\), at section PQ is uniform and the speed at section RS is given by \(u(y) = A_{0}\left(\frac{y}{h}\right)^{n}\), where \(n\) is a positive integer. The value of \(A_{0}\) for which the flow through section PS will vanish is:
Answer: B
The flow through PS vanishes when \(A_{0} = u_{\infty}(n + 1)\).
The plate is non-porous, so no fluid crosses QR. If nothing crosses PS either, the mass entering through PQ must all leave through RS. Per unit depth:
\[ \begin{aligned} u_{\infty} h &= \int_{0}^{h} A_{0}\left(\frac{y}{h}\right)^{n} dy \\ &= \frac{A_{0} h}{n + 1} \\ A_{0} &= u_{\infty}(n + 1) \end{aligned} \]
\(u_{\infty}/(n + 1)\) is the average speed of a profile with \(A_{0} = u_{\infty}\), the result inverted. The options with n - 1 come from a slip in the power of y when integrating the profile.
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