Flow on the surface of a cylinder

[Matt atkinson]

Homework Statement

An infinite cylinder is moving at constant velocity $$\vec{U}$$ in a stationary background flow. On the surface of the sphere no fluid penetrates, so that $$\vec{U} \cdot \vec{n} = \vec{u} \cdot \vec{n}$$. Where $$\vec{n}$$ is the vector normal to the surface of the cylinder. At the instant the axis of the cylinder coincides with the origin, the velocity potential in cylindrical polar coordinates is given by;
$$\phi=-\frac{U a^2 cos(\theta)}{r}$$
where a is the radius of the cylinder.
i) find the velocity field $$\vec{u}$$
ii) prove the relavent boundry condition for the background flow.
iii) Find $$\vec{U}$$

Homework Equations

The Attempt at a Solution

i) basical it is grad $$\phi$$ I got $$\vec{u} =\frac{U a^2 cos(\theta)}{r^2} \vec{r}+\frac{U a^2 sin(\theta)}{r^2} \vec{\theta}$$
ii) not sure what boundry i tried solving U dot n = u dot n
with n as r_hat, but didnt get anywhere.

 

[Matt atkinson]

for ii) I proved that $$\nabla \times \vec{u} = 0$$ which shows its irrotational but I am not positive if that's the correct thing to do