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Archive for April 16th, 2012

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

Posted by Peeter Joot on April 16, 2012

[Click here for a PDF of this post with nicer formatting.]

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.

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