3.II.18G

State the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.

A circular cylinder of radius $a$ is immersed in unbounded inviscid fluid of uniform density $\rho$. The cylinder moves in a prescribed direction perpendicular to its axis, with speed $U$. Use cylindrical polar coordinates, with the direction $\theta=0$ parallel to the direction of the motion, to find the velocity potential in the fluid.

If $U$ depends on time $t$ and gravity is negligible, determine the pressure field in the fluid at time $t$. Deduce the fluid force per unit length on the cylinder.

[In cylindrical polar coordinates, $\nabla \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{r}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta}$.]

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