A scheme for identifying the regions of fastest mixing of a blob of ink in a flow induced by a system of point vortices has been developed.
A point vortex is a 2 dimensional mathematical singularity that induces a velocity field. If a point is at a distance 'r' from the center of the vortex then the velocity induced at that point will be proportional to 1/r.
If N vortices in a plane are considered and are initially located at some positions then the each of the vortices will move under the influence of the other N-1 vortices. In such a flow if one considers a blob of ink and wants to place it such that it mixes in the flow. To do this we compute a quantity called the Lyapunov exponent and determine which region will mix efficiently. A plot of the same is given below for a specific initial configuration of vortices.
In this all the regions having a zero lyapunov exponent (the dark blue regions) will not mix at all. On the other hand if a blob is placed in the regions having a positive Lyapunov exponent, the ink will mix. There are simulations that illustrate the above statements. In the following the blue circles are the point vortices and the white blob is the simulated blob of ink.
Click here to see an animation for the case where there is no mixing.
Click here to see an animation for the case where there is high mixing.
This work was done at the Dept. of Aerospace Eng. IIT-Madras by Prabhu Ramachandran under the Guidance of Mr. S.C. Rajan (Assistant Professor). All the simulations were performed on the Linux Operating System.
