Under appropriate assumptions for the form of velocity/pressure fields, the Navier-Stokes equations (in some cases) have what are called "exact solutions". The existence of an exact solution does not imply that the flow field can be written in closed form, but rather that the N-S equations can be reduced to a (much) simpler system, crucially without any approximation (see "The Navier-Stokes Equations: A Classification of Flows and Exact Solutions" by Drazin & Riley). The most interesting of these retain nonlinearity in the reduced equations, and in some cases can possess a rich bifurcation structure. The goal in this project is to take one such reduced system and compute (via MATLAB for example) nonlinear time-periodic solutions by direct methods (rather than by time marching). The goal is to explain some preliminary results that point to ("exact") period-doubling cascades (and possibly chaotic) states.

This project will require some familiarity with fluid mechanics, computational methods (e.g finite-difference schemes) and practical approaches to computation (e.g. Matlab).