Volume 52, Number 6, November-December 2018
|Page(s)||2357 - 2408|
|Published online||01 February 2019|
On convergent schemes for two-phase flow of dilute polymeric solutions
Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
* Corresponding author: email@example.com
Accepted: 25 June 2018
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.
Mathematics Subject Classification: 35Q30 / 35Q35 / 35Q84 / 65M12 / 65M60 / 76A05 / 82D60 / 76T99
Key words: 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
© EDP Sciences, SMAI 2019
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