Rigorous investigation of the Navier–Stokes momentum equations and correlation tensors

Abstract

An analytical solution to the Navier–Stokes momentum equations for a compressible flow with volume and small shear viscosities as well as external friction is presented while the dynamic viscosity is set to zero. The demonstrated methodology holds in d dimensions. However, in this study, the three-dimensional case is considered in detail. The analytical solution blows up at finite times T, which is determined by a cubic relation if the initial flow velocity is not divergence-free. The existence of T is a necessary and sufficient condition for implementing a singularity. Nonetheless, for external friction μe>T−1, all analytical expressions are smooth while the averaged expressions are smooth for all times.