A mass conserving mixed stress-strain rate finite element method for non-Newtonian fluid simulations

Abstract

Many non-Newtonian models assume a non-linear relation between the deviatoric stress tensor τ and the rate-of-strain tensor ε(u), which is not necessarily given in explicit form. Therefore the requirement on a finite element method is the capability to capture the behaviour of the non-linear constitutive relation.Inspired by the work of [GLS19, GLS20] and assuming incompressible, stationary, is other- mal, laminar flow, we present a new mixed finite element method by introducing a variable for the rate-of-strain tensor ε, such that the embedding of a general implicit constitutive relation of the form G(τ,ε) := 0 is very natural. Thus making it suitable for the simula- tion of a broader range of non-Newtonian fluids.We prove solvability of the new discrete variational formulation in a two-dimensional Newtonian setting by showing continuity of the bilinear forms, coercivity on the kernel and the discrete Ladyzhenskaya–Babuska–Brezzi condition. By construction our newly introduced mixed finite element approximates the velocity u in an exactly divergence free matter. This fact results in a property known as pressure robustness.Ultimately, we perform some non-Newtonian numerical experiments in a two-dimensional channel and illustrate the achieved L2-errors in comparison to various other standard mixed finite elements.