Nonclassical symmetry reductions of the three-dimensional incompressible Navier-Stokes equations

The nonclassical reduction method as pioneered by Bluman and Cole (J.Meth. Mech. 18 1025-42) is used to examine symmetries of the full three-dimensional, unsteady, incompressible Navier-Stokes equations of fluid mechanics. The procedure, when applied to a system of partial differential equations, yields reduced sets of equations with one fewer independent variables. We find eight possibilities for reducing the Navier-Stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables. Some of these reductions are derivable using the Lie-group method of classical symmetries but the remainder are genuinely nonclassical. Further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the Navier-Stokes equations by the nonclassical method.