Exact spherically symmetric solutions of pressureless Navier-Stokes equations with density-dependent viscosity

Abstract

The spherically symmetric solutions of the pressureless Navier-Stokes equations with density-dependent viscosity in N -dimensional space are examined in this work. The method of variable separation is applied to explore some reduced equations which are associated with the spherically symmetric solutions. Some novel exact spherically symmetrical solutions with velocity u in the nonlinear form of the spatial variable x are presented. In particular, the exact solutions with u(x, t) = c(t)|x| α x (α = 0) are derived. This method might be generalized to examine the exact spherically symmetric solutions for some other nonlinear equations, such as the Euler equations and multidimensional compressible Navier-Stokes equations.