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.]