A rigid circular cylinder of radius $a$ executes small amplitude oscillations with velocity $U(t)$ in a direction perpendicular to its axis, while immersed in an inviscid fluid of density $\rho$ contained within a larger concentric fixed cylinder of radius $b$. Gravity is negligible. Neglecting terms quadratic in the amplitude, determine the boundary condition on the velocity on the inner cylinder, and calculate the velocity potential of the induced flow.

With the same approximations show that the difference in pressures on the surfaces of the two cylinders has magnitude

$$\rho \frac{\mathrm{d} U}{\mathrm{~d} t} \frac{a(b-a)}{b+a} \cos \theta$$

where $\theta$ is the azimuthal angle measured from the direction of $U$.