Self-similar analytic solution of the two dimensional Navier-Stokes equation with a non-Newtonian type of viscosity

TL;DR

This paper derives a self-similar analytic solution for the 2D Navier-Stokes equations with a non-Newtonian viscosity, extending previous Newtonian models and providing insights into non-Newtonian fluid behavior.

Contribution

It introduces a multi-dimensional self-similar Ansatz for non-Newtonian fluids, specifically using the Ladyzenskaya model, and compares results to Newtonian cases.

Findings

Analytic self-similar solutions for non-Newtonian fluids derived.

Comparison shows differences from Newtonian fluid behavior.

Geometrical interpretation of the solution discussed.

Abstract

We investigate the two dimensional incompressible Navier-Stokes(NS) and the continuity equations in Cartesian coordinates and Eulerian description for non-Newtonian fluids. As a non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multi-dimensional generalization of the well-known self-similar Ansatz, which was already used for non-compressible and compressible viscous flow studies. The geometrical interpretations of the trial function is also discussed. We compare our recent results to the former Newtonian ones.