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Reza Malek-Madani

Rigid Rotation

All three flows illustrated on this page are incompressible. They have various values of spin, however. All three have the same particle paths (phase portraits) but have vastly different flow characteristics.

1. Rigid Rotation

The following is the evolution of three parcels of fluid which were originally disks with their centers located on the negative x-axis. The evolution is governed by the system of linear differential equations 

x' = y, y' = - x.

This flow has a vorticity of -2 k, which accounts for the uniform rate of clockwise rotation. A "log" placed in this flow will experience a rigid rotation, but no stretching or compression. 


The next picture shows the evolution of the same three disks when the system of differential equations is 

x' = y/(x^2 + y^2)^1/2, y' = -x/(x^2 + y^2)^1/2.


This flow's vorticity is -1/r k, where r = (x^2+y^2)^1/2. As is shown in the picture, a "log" placed in this flow will experience an extensive amount of stretching and thinning out. Since the flow is incompressible, the area of each parcel remains invariant.

3. A Line Vortex

The system of differential equations in this instance is

x' = y/(x^2 + y^2), y' = -x/(x^2 + y^2).</P>

Its vorticity is 0 outside of the origin, however, as the graph below shows, there is quite a bit of spinning in this flow. This model is typically used as a first building block to develop mathematical models of tornados and hurricanes.


Finally,the above three pictures are put together to give a sense of the time and space scales in the three flows.

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