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I'm looking for an analytical solution of the Navier-Stokes equation with the following boundary conditions: a liquid is held inside a spherical shell, which is rotating at a constant rate, and is forcing the surface of the liquid to rotate at the same rate. I'm particularly interested in steady-state solution reached after rotation for a long time.

Has anyone seen this problem in the literature before? Thanks

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    $\begingroup$ Let the $z$ axis be the axis of rotation. $$ u(x,y,z) = \begin{pmatrix} y \\ -x \\ 0\end{pmatrix}$$ is a steady-state solution to the incompressible Navier-Stokes equation that co-rotates with your spherical shell. The pressure is $\frac12 (x^2 + y^2)$. $\endgroup$
    – Willie Wong
    Commented May 5, 2016 at 13:31
  • $\begingroup$ @WillieWong That is so simple I dismissed it off hand :) Given that this is the long-term solution, I guess the better question would be how quickly is it dynamically approached from the time the shell begins rotating. I imagine that would be a much more difficult problem to solve analytically though... $\endgroup$
    – tom
    Commented May 6, 2016 at 2:29
  • $\begingroup$ Thinking further, the 2D problem (equivalent to a fluid in a rotating cylindrical shell) is very easy to solve, and the time for homogenisation is on the order of $r^2/\nu$, where $r$ is the cylinder's radius and $\nu$ is the kinematic viscosity. It seems reasonable that we'd expect a similar time scale for a rotating sphere also. For liquid water, this gives us roughly a 1 second homogenisation time for a droplet of 1mm radius. It seems a lot can happen inside rain-drops, but not inside droplets of diameter less $500 \mu m$ or so. $\endgroup$
    – tom
    Commented May 6, 2016 at 5:08

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