Similarity reductions and exact solutions for the two-dimensional incompressible Navier-Stokes equations

We study similarity reductions and exact solutions of the (2+1)-dimensional incompressible Navier-Stokes equations using the direct method originally developed by Clarkson and Kruskal [37]. The Navier-Stokes equations are reduced to their conventional stream function form, and it is shown that there exist essentially five reductions into lower-order partial differential equations. Furthermore, we study the possibilities for reducing each of these five forms to ordinary differential equations, some of which can be solved analytically, while others necessitate numerical treatment, In particular we exhibit several new reductions that are not obtained using the classical Lie group method of infinitesimal transformations, and thus we generate new exact solutions of the governing equations. Some of our solutions admit physical interpretations, and many of them contain well-known Navier-Stokes solutions as special examples.