Exact solutions to the three-dimensional Navier–Stokes equations based on the stream function approach

Using the stream function approach, we derive three-dimensional exact solutions to the incompressible, unsteady Navier–Stokes equations. These solutions are generated by Beltrami vector fields and satisfy generalized impermeability boundary conditions on quadric surfaces. We also briefly consider exact solutions of the Navier–Stokes equations with time-dependent viscosity. Using a dynamical systems approach, we study the existence of chaotic streaklines and invariant manifolds of these solutions.