Semi-invariant solutions of the Navier-Stokes equations

We present new exact solutions and reduced differential systems of the Navier-Stokes equations of incompressible viscous fluid flow. We apply the method of semi-invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier-Stokes equations are, in fact, semi-invariant and that the reduced differential systems we derive using semi-invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi-invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier-Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier-Stokes equations or find their approximations, and physical interpretation.