Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids

Abstract

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.