Diploma and Master Theses (authored and supervised):
"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,
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
Created from the Publication Database of the Vienna University of Technology.