I'm trying to solve a simple problem where I have an INVISCID fluid between two cylinders and they are rotating with some angular velocity $\Omega_1$ and $\Omega_2.$ The cylinders have radii $R_1$ and $R_2$ where the latter is larger than the former. I'm using polar coordinates and the Euler equations.
I'm trying to show that the Euler equations give a steady solution
$$\textbf{U} = V(r) \textbf{e}_\theta$$
where $V(r)$ is just an arbitrary function of $r$. I am having trouble satisfying boundary conditions for this problem since the fluid is inviscid.
Would the conditions
$$u_\theta = R_1 \Omega_1, r=R_1 \\ u_\theta = R_2 \Omega_2, r=R_2$$
apply here?
Using the Euler equation
$$\frac{D \textbf{u}}{Dt} = -\nabla p$$
where $p=p(r)$
$$\frac{V_\theta^2}{r^2} = \frac{1}{\rho}\frac{\partial p}{\partial r}$$
How can I solve for the velocity? The viscous term is zero so it removes the possibility to use the $\theta$ momentum equation.
