Some exact solutions of the two-dimensional Navier-Stokes equations

We apply the symmetry approach to the study of the two-dimensional Navier-Stokes equations. It turns out that if one adds an external force, this has to satisfy the irrotationality condition. Exploiting the symmetry algebra (which is infinite dimensional) associated with the equations under consideration, special classes of exact solutions are obtained. We have a solution which can be expressed in terms of parabolic cylinder functions. For a certain choice of the parameters involved we find a solution related to the error function. This solution corresponds to the laminar motion of a fluid in which the flow is in parallel planes and uniform over each plane, the direction being everywhere the same. Another class of solutions is connected with Bessel functions and in some cases it presents a vortex-like behavior. The energy density for these solutions is proportional to 1/r(2). Finally, examples of boundary conditions invariant under the transformations of the symmetry variables are displayed. (C) 1998 Elsevier Science Ltd. All rights reserved.