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# Archive for April 16th, 2012

## Flow between infinite moving inner cylinder and outer cylinder (3D visualizations).

Posted by peeterjoot on April 16, 2012

We previously looked at the problem of an infinite cylinder of radius $R_1$ is moving with velocity $v$ parallel to its axis. It is places inside another cylinder of radius $R_2$. The axes of the two cylinders coincide. The fluid is incompressible, with viscosity $\mu$ and density $\rho$, the flow is assumed to be stationary, and no external pressure gradient is applied, and found the solution to have the form

\begin{aligned}\boxed{w = \frac{G}{4 \mu} (R_2^2 - r^2) + \left(v - \frac{G}{4 \mu} (R_2^2 - R_1^2)\right)\frac{\ln r/R_2}{\ln R_1/R_2}.}\end{aligned} \hspace{\stretch{1}}(3.27)

This system with such simple geometry lends itself well to 3D visualization of the velocity profiles

An animation of this can be found at http://youtu.be/OiJTopWx7L8. The Mathematica notebook that generated this is available at phy454/twoCylinders3D.cdf. The 3D visualization portion of that notebook has also been made available as part of the wolfram demonstrations project.