Self-similar solutions of the three dimensional Navier-Stokes equation

TL;DR

This paper derives explicit three-dimensional self-similar solutions for the Navier-Stokes equations using a generalized Ansatz, providing analytical results and comparing them with group theoretical approaches.

Contribution

It introduces a novel three-dimensional self-similar Ansatz for Navier-Stokes equations and derives solutions expressed in terms of Kummer functions.

Findings

Derived explicit self-similar solutions for 3D Navier-Stokes equations.

Provided a geometric interpretation of the Ansatz.

Compared solutions with group theoretical approaches.

Abstract

In this article we will present pure three dimensional analytic solutions for the Navier-Stokes and the continuity equations in Cartesian coordinates. The key idea is the three-dimensional generalization of the well-known self-similar Ansatz of Barenblatt. A geometrical interpretation of the Ansatz is given also. The results are the Kummer functions or strongly related. Our final formula is compared with other results obtained from group theoretical approaches.