Existence of global weak solutions to an inhomogeneous Doi model for active liquid crystals - Sieber, Oliver (Journal of Differential Equations, 2023)
In this paper, we consider an inhomogeneous Doi model which was introduced by E and Zhang (2006) [17]. We extend their model, which couples a Smoluchowski equation to a Navier-Stokes type equation, for active particles by introducing an additional stress tensor. Exploiting the energetic and entropic structure of the system, we establish the existence of global-in-time weak solutions in two and three space dimensions for both passive and active particles. In particular, our result holds for minimal regularity assumptions on the initial data and without restrictions on the Reynolds and Deborah number.
CRIS-Team Scopus Importer:2023-01-27
Keywords: Active material; Doi model; Existence of weak solutions; Global entropy solution; Liquid crystals; Navier-Stokes equation

On convergent schemes for a two-phase Oldroyd-B type model with variable polymer density - Sieber, Oliver (Journal of Numerical Mathematics, 2020)
This paper is concerned with a diffuse-interface model that describes two-phase flow of dilute polymeric solutions with a variable particle density. The additional stresses, which arise by elongations of the polymers caused by deformations of the fluid, are described by Kramers stress tensor. The evolution of Kramers stress tensor is modeled by an Oldroyd-B type equation that is coupled to a Navier-Stokes type equation, a Cahn-Hilliard type equation and a parabolic equation for the particle density. We present a regularized finite element approximation of this model, prove that our scheme is energy stable and that there exist discrete solutions to it. Furthermore, in the case of equal mass densities and two space dimensions, we are able to pass to the limit rigorously as the regularization parameters and the spatial and temporal discretization parameters tend towards zero and prove that a subsequence of discrete solutions converges to a global-in-time weak solution to the unregularized coupled system. To the best of our knowledge, this is the first existence result for a two-phase flow model of viscoelastic fluids with an Oldroyd-B type equation. Additionally, we show that our finite element scheme is fully practical and we present numerical simulation
Keywords: Two-phase flow; Cahn-Hilliard equation; Oldroyd-B model; Finite element approximation; Existence of weak solutions; convergence analysis