The self similar vortex dipole is a structure for which an exact analytic description has not yet been obtained.  In this project we considered the Navier Stokes equations with an unusual non-linear constraint called the "well mixed hypothesis", which states that the contours of the stream function must coincide with the contours of vorticity. This allowed us to linearize the governing equations and find an exact, self similar solution representing a vortex dipole.
 

For many years, it was accepted that the momentum flux in any jet is nearly constant with downstream distance. Within the past thirty years, this assumption has been questioned due to the entrainment of the ambient fluid. In axisymmetric jets emerging from an orifice in a conical wall, steady state solutions for the velocity profiles have been found (Schneider 1985).  While solutions for the velocity profiles have been found for laminar and turbulent flows, a stability analysis on the flow has not been performed. The goal of our research is to determine if transient vortex instabilities in the jet’s boundary layer help to explain the change in momentum flux as well as the transition from laminar to turbulent flow.
 

Polymer-penetrant systems exhibit different behavior from that explained by the standard diffusion equation:

These systems come up in membrane production, protective sealants, VLSI chip etching, drug delivery, and other areas.  Some of the unusual behavior can be explained by a viscoelastic stress ó.  Because the diffusion of the penetrant through the polymer is observed to be abnormal due to the stress, a new model is proposed for the polymer system involving two partial differential equations: 

The goal of this project is to find traveling wave solutions that will profile the change in concentration of the system.  The polymer film is set up as a one dimensional system where concentration of penetrant and stress vary with position and time.  Finding traveling wave solutions will be essential to the understanding of the partial differential equations.