We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.
Mots-clés : Convergence of finite-element schemes, existence of weak solutions, polymeric flow model, two-phase flow, diffuse interface models, Navier–Stokes equations, Fokker–Planck equations, Cahn–Hilliard equations, FENE
@article{M2AN_2018__52_6_2357_0,
author = {Metzger, Stefan},
title = {On convergent schemes for two-phase flow of dilute polymeric solutions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2357--2408},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018042},
mrnumber = {3905186},
zbl = {1421.35251},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018042/}
}
TY - JOUR AU - Metzger, Stefan TI - On convergent schemes for two-phase flow of dilute polymeric solutions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2357 EP - 2408 VL - 52 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018042/ DO - 10.1051/m2an/2018042 LA - en ID - M2AN_2018__52_6_2357_0 ER -
%0 Journal Article %A Metzger, Stefan %T On convergent schemes for two-phase flow of dilute polymeric solutions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2357-2408 %V 52 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018042/ %R 10.1051/m2an/2018042 %G en %F M2AN_2018__52_6_2357_0
Metzger, Stefan. On convergent schemes for two-phase flow of dilute polymeric solutions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2357-2408. doi : 10.1051/m2an/2018042. http://www.numdam.org/articles/10.1051/m2an/2018042/
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