Discussion
An incompressible fluid flows between a pair of infinite plates separated by a distance L. The top plate is moving with a constant velocity U, whereas the bottom plate is stationary, as shown in the figure. The difference of the stream functions \((\psi_{T}-\psi_{B})\) at the two plates for a laminar and fully-developed flow is equal to
Answer: A
The difference is \(\frac{UL}{2}\).
In plane Couette flow with no pressure gradient the profile is linear, \(u = Uy/L\). The difference in stream function between two streamlines equals the volume flow rate per unit width between them:
\[ \begin{aligned} \psi_{T}-\psi_{B} &= \int_{0}^{L}u\,dy \\ &= \int_{0}^{L}\frac{Uy}{L}\,dy = \frac{UL}{2} \end{aligned} \]
Equivalently, the average velocity is U/2 across a gap L. UL would need the whole gap to move at U, and 2UL and 4UL exceed even that.
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