I am not very good at physics, so I don't understand how to solve the following problem with vector calculus.
A perfect incompressible fluid moves steadily under gravity around the outside of a fixed cylinder of radius $a$ and vertical axis $Oz$. The fluid particles move in horizontal circles with centres on $Oz$ and the velocity field of the fluid in cylindrical polar coordinates $(R,\phi ,z)$ is $\mathbf{v} = \frac{a}{R}{\mathbf{e}_\phi }$.
(a) Verify that the equation of continuity is satisfied and show that the motion is irrotational.
(b) The surface of the fluid is open to the atmosphere and the origin is chosen so that, on the free surface, $z = 0$ when $R = a$. The body force is $\mathbf{F} = \mathbf{g}$ and the corresponding potential is $U = gz$. Using Bernouilli’s equation, show that the equation of the free surface is given by $z = \frac{1}{{2g}}(1 - \frac{{{a^2}}}{{{R^2}}})$.
Can you help me translate it so that I can work it out? Thanks.
Part a - just show that $\nabla \cdot \mathbf{v}=0$ (continuity) and that $\nabla \times \mathbf{v}=0$ (irrotational).
Part b - put $\mathbf{v}$ into Bernoulli's equation and then solve for the unknown function which is the equation of the free surface.