Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor

TL;DR

This paper introduces a novel numerical scheme for simulating transient flows of non-Newtonian fluids with non-monotone constitutive relations, addressing multiple stress values for a given shear rate through reformulation and proving solution existence.

Contribution

The paper presents a reformulated system and a numerical scheme that ensures solution existence for complex non-monotone fluid flow equations, with practical numerical solutions demonstrated.

Findings

The scheme successfully computes flow simulations with non-monotone relations.

The reformulation handles multiple stress values for a given shear rate.

Existence of solutions is proven at the discrete level.

Abstract

We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are reformulated as a system for the triple pressure-velocity-apparent viscosity, where the apparent viscosity is given by a scalar implicit equation. We prove that the proposed numerical scheme has---on the discrete level---a solution, and using the proposed scheme we numerically solve several flow problems.