J. Ellmenreich:

"A mass conserving mixed stress-strain rate Finite Element Method for Non-Newtonian fluid simulations";

Supervisor: J. Schöberl, P. Lederer; Institut für Analysis und Scientific Computing, 2021; final examination: 2021-12-08.

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, isother-

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 adyzhenskaya-Babuˇska-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.

Non-Newtonian Fluids; Mixed Finite Elements

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