Stochastic shear thickening fluids: Strong convergence of the Galerkin approximation and the energy equality

TL;DR

This paper studies a stochastic PDE modeling shear thickening non-Newtonian fluids, proving strong convergence of Galerkin approximations and establishing the energy equality for the velocity field.

Contribution

It demonstrates the strong convergence of Galerkin schemes and confirms the energy equality for stochastic shear thickening fluids, a novel result in this context.

Findings

Galerkin approximation converges strongly to the velocity field

Energy equality holds for the stochastic shear thickening fluid model

Applicable for p in the specified shear thickening range

Abstract

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree p-1 of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case $p\in [1+{\frac{d}{2}},{\frac{2d}{d-2}})$, where d is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.