Nonlinear dynamics of micro rotors and mixing of fluid in a confined domain

We present some results on the two dimensional mixing of a fluid confined to a circular domain at low Reynolds number through the use of microrotors. The microrotors are modeled as rotlets, which are a singularity solution of the Stokes equation. The microrotors are free to move under the influence of other rotors and the image singularities which arise to satisfy the boundary conditions on the circular boundary. Two specific cases are explored, the mixing due to one rotor and the mixing due to a pair of rotors. In the former case computations show that the fluid in the domain is not mixed well, while in the latter case the fluid could be mixed well for certain initial configurations of the rotors.

The two rotlet case produces a rich array of dynamics of the rotlets themselves. The boundary plays an important role, by making the dynamics of the two rotlets nonintegrable. This role of the boundary is identified as a cause of the mixing. Poincare maps, Lyapunov exponents and variance calculations are employed to quantify mixing. The stretching and folding of initially coherent blobs of fluid tracers into thin striations and consequent mixing are easily visualized.