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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.

Schematic of Problem

whiteiverson
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    Those boundary conditions hold in the viscous case. With the Euler equations you can impose $u_r = 0$ if the cylinder walls are impermeable. Specifying the no-slip condition generally over-determines the problem. – RRL Jan 16 '19 at 20:27
  • To make some progress here you should show your attempt to solve the PDE up to applying boundary conditions or ask about this on Physics.SE. – RRL Jan 16 '19 at 20:32
  • I don't understand how to impose $u_r = 0$ in this problem? – whiteiverson Jan 16 '19 at 21:50
  • Well you already satisfied it by assuming unidirectional flow with $u_r = 0$ everywhere. Why are you trying to solve the Euler equation for this configuration? With full Navier-Stokes including the viscous terms you get a second-order linear differential equation with solution $u_{\theta}(r) = Ar + B/r$. $A$ and $B$ are determined by the no-slip boundary conditions. I don't see this as well posed for the Euler equations. – RRL Jan 16 '19 at 22:10
  • You have only a trivial solution of rigid rotation $u_\theta = \Omega r$ which can satisfy $\frac{u_\theta^2}{r^2} = \frac{1}{\rho} \frac{\partial p}{\partial r}$ with a constant pressure gradient, but that won't satisfy the boundary conditions. – RRL Jan 16 '19 at 22:13
  • Yes for the viscous case I understand how to calculate the velocity profile. You are saying it must be a rigid body rotation for the inviscid fluid? – whiteiverson Jan 16 '19 at 22:17
  • It is if there is a constant radial pressure gradient, but I don't see how to get that from rotating cylinders at different speeds. Again -- what is the motivation here? Some information about the physical situation seems to missing. – RRL Jan 16 '19 at 22:20
  • I'm trying to solve in the case where the fluid is viscous and inviscid. – whiteiverson Jan 16 '19 at 22:38

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