Abstract

A few exact solutions to the incompressible, laminar Navier-Stokes and induction equations are examined. To begin with, velocity fields with a simplified structure, such as similarity flows are derived. Magnetic fields with a similar spatial reduction are then sought. If these grow kinematically, then they can be followed exactly into the nonlinear regime without approximation. An example is von-Karman flow between two rotating discs, a configuration resembling the VKS experiment. The resulting nonlinear system reduces to 4 coupled ODEs. Although some of the “dynamos” thus found are axisymmetric, they are saved from Cowling’s theorem by the unbounded domain. It is shown that some of these solutions involve energy being generated locally, rather than being advected in from infinity.  This work involved collaborations with Ali Arslan, Raquel Vaz, and Leszek Zabielski