1.II.17A

A point source of fluid of strength $m$ is located at $\mathbf{x}_{s}=(0,0, a)$ in inviscid fluid of density $\rho$. Gravity is negligible. The fluid is confined to the region $z \geqslant 0$ by the fixed boundary $z=0$. Write down the equation and boundary conditions satisfied by the velocity potential $\phi$. Find $\phi$.

[Hint: consider the flow generated in unbounded fluid by the source $m$ together with an 'image source' of equal strength at $\overline{\mathbf{x}}_{s}=(0,0,-a)$.]

Use Bernoulli's theorem, which may be stated without proof, to find the fluid pressure everywhere on $z=0$. Deduce the magnitude of the hydrodynamic force on the boundary $z=0$. Determine whether the boundary is attracted toward the source or repelled from it.

*Typos? Please submit corrections to this page on GitHub.*