# Flow on the surface of a cylinder

1. May 8, 2014

### Matt atkinson

1. The problem statement, all variables and given/known data
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 $$\vec{U}$$
2. Relevant equations

3. 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.

Last edited: May 8, 2014
2. May 8, 2014

### Matt atkinson

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