SOME EXACT-SOLUTIONS OF THE 2-DIMENSIONAL NAVIER-STOKES EQUATIONS

We apply the symmetry approach to the study of the two-dimensional Nav ier-Stokes equations. It turns out that if one adds an external force, this has to satisfy the irrotationality condition. Exploiting the sym metry algebra (which is infinite dimensional) associated with the equa tions under consideration, special classes of exact solutions are obta ined. 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 correspo nds 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 a nd in some cases it presents a vortex-like behavior. The energy densit y for these solutions is proportional to 1/r(2). Finally, examples of boundary conditions invariant under the transformations of the symmetr y variables are displayed. (C) 1998 Elsevier Science Ltd. All rights r eserved.