Dk. Ludlow et al., Similarity reductions and exact solutions for the two-dimensional incompressible Navier-Stokes equations, STUD APPL M, 103(3), 1999, pp. 183-240
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 red
uced to their conventional stream function form, and it is shown that there
exist essentially five reductions into lower-order partial differential eq
uations. 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 ne
w exact solutions of the governing equations. Some of our solutions admit p
hysical interpretations, and many of them contain well-known Navier-Stokes
solutions as special examples.