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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.

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    $\begingroup$ Why don't the examples where $v_r = v_\theta = 0$ help you? Those are the same assumptions you have listed in your question... $\endgroup$
    – tpg2114
    Oct 3, 2018 at 20:03
  • $\begingroup$ in the examples they are not zero,but the $vφ$ is zero so i have a reversed situation $\endgroup$ Oct 3, 2018 at 20:29
  • $\begingroup$ Could you provide a schematic drawing for the problem? From your description, it's difficult to see how the spheres are located w.r.t. each other. $\endgroup$
    – Deep
    Oct 4, 2018 at 6:00
  • $\begingroup$ @Deep You have 2 concentric spheres of different radii rotating about a common axis passing through their centers. There is a viscous liquid filling the space between the spheres. $\endgroup$ Oct 4, 2018 at 12:02
  • $\begingroup$ @ChesterMiller I see now. $\endgroup$
    – Deep
    Oct 5, 2018 at 5:13

2 Answers 2

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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.

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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

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