Paper 4, Section II, B

Write down the velocity potential for a line source flow of strength $m$ located at $(r, \theta)=(d, 0)$ in polar coordinates $(r, \theta)$ and derive the velocity components $u_{r}, u_{\theta}$.

A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius $a(a<d)$ centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at $\left(a^{2} / d, 0\right)$ and another at the origin, of a strength to be determined.

Use Bernoulli's law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the $x$-direction and of magnitude

$\frac{m^{2} \rho}{2 \pi^{2}} \int_{0}^{2 \pi} \frac{a d^{2} \sin ^{2} \theta \cos \theta}{\left(a^{2}+d^{2}-2 a d \cos \theta\right)^{2}} d \theta$

where $\rho$ is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.

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