Stokes flow between 2 spheres

Question

The problem is given like this:

Given two rotating spheres with constant angular velocity $\Omega_1$ and $\Omega_2$ around the vertical axis and pressure on the borders of spheres is $p_1$ and $p_2$ ,filled with fluid between them, using Navier-Stokes equations, find the physical velocity components in spherical coordinates (no gravity, no slip condition).

Assume (${v_r} = 0,\quad {v_\theta } = 0,\quad {v_\varphi } = f(r)\sin > \theta$)

It's a Stokes flow problem, and I need to find the function $f(r)$. The equations I derived look like this:

\begin{align} \frac{{\partial p}}{{\partial r}} &= {\rho _0}\sin^{2}\left( \theta \right)\frac{{{f^2}\left( r \right)}}{r}\tag{1}\\ \frac{{\partial p}}{{\partial \theta }} &= 0\tag{2}\\ \frac{{\partial p}}{{\partial \varphi }} &= \, - \mu \sin\left( \theta \right)\frac{{\left( { - 2f\left( r \right) + {r^2}f''\left( r \right)} \right)}}{r}\tag{3} \end{align}

I have no idea how to solve this. Is there any extra equations I can use to find $f(r)$? When I searched for Stokes flow around a sphere, I keep finding examples where the ${v_r},{v_\theta }$ is something given in form of a steam function but ${v_\varphi }$ is zero.

Answer 1

I don't confirm your 2nd and 3rd equations. I get:

$$\frac{\partial p}{\partial \theta}=\rho f^2\sin{\theta}\cos{\theta}$$ $$\frac{\partial p}{\partial \phi}=\frac{\mu \sin^2{\theta}}{r}\left[\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)-2f\right]$$ By symmetry, p is not a function of $\phi$. Therefore, $$\frac{d}{d r}\left(r^2\frac{d f}{d r}\right)-2f=0$$

This is a Stokes (low Reynolds number) flow, and the radial- and latitudinal equations contain only inertial terms. So they can be neglected. The only relevant equation is the one in the $\phi$ direction.

Answer 2

Maybe you have a look into "Analytical investigation of the compressible navier-stokes-equations for low mach- and reynolds-numbers" by Michael Viehl from 1989. It should help. Best wishes Michael Viehl