Paper 1, Section I, D

Show that the flow with velocity potential

$\phi=\frac{q}{2 \pi} \ln r$

in two-dimensional, plane-polar coordinates $(r, \theta)$ is incompressible in $r>0$. Determine the flux of fluid across a closed contour $C$ that encloses the origin. What does this flow represent?

Show that the flow with velocity potential

$\phi=\frac{q}{4 \pi} \ln \left(x^{2}+(y-a)^{2}\right)+\frac{q}{4 \pi} \ln \left(x^{2}+(y+a)^{2}\right)$

has no normal flow across the line $y=0$. What fluid flow does this represent in the unbounded plane? What flow does it represent for fluid occupying the domain $y>0$ ?

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