A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow

TL;DR

This paper introduces a semismooth Newton method for non-Newtonian fluid models with implicit constitutive relations, enabling efficient numerical approximation of Bingham flow without variational inequalities.

Contribution

The paper develops a novel semismooth Newton approach that simplifies the treatment of implicit constitutive relations in non-Newtonian fluids, ensuring convergence and mesh-independent performance.

Findings

Method effectively approximates Bingham flow

Convergence of stresses is proven for the regularized system

Nonlinear iterations are mesh-independent

Abstract

We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; this class of constitutive relations captures for instance the models of Bingham and Herschel-Bulkley. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the nonlinear iterations.