| course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
21 |
|
Paper 1, Section II, A |
2012 |
Consider inviscid, incompressible fluid flow confined to the $(x, y)$ plane. The fluid has density $\rho$, and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential $\phi(r)$ of a point source of strength $m$, in terms of the distance $r$ from the source.
Two point sources each of strength $m$ are located at $\boldsymbol{x}{+}=(0, a)$ and $\boldsymbol{x}{-}=(0,-a)$. Find the velocity potential of the flow.
Show that the flow in the region $y \geqslant 0$ is equivalent to the flow due to a source at $\boldsymbol{x}_{+}$and a fixed boundary at $y=0 .$
Find the pressure on the boundary $y=0$ and hence determine the force on the boundary.
[Hint: you may find the substitution $x=a \tan \theta$ useful for the calculation of the pressure.]