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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an/2018042/
[1] , and , Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 1150013. | DOI | MR | Zbl
[2] , , , , and , Quantitative comparison of Taylor flow simulations based on sharp-interface and diffuse-interface models. Int. J. Numer. Methods Fluids 73 (2013) 344–361. | DOI
[3] and , Formulation of a new class of fractional-step methods for the incompressible MHD equations that retains the long-term dissipativity of the continuum dynamical system. Fields Inst. Commun. 10 (1996) 1–24. | MR | Zbl
[4] and , Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. SIAM J. Math. Anal. 33 (2001) 847–859. | DOI | MR | Zbl
[5] and , Existence of global weak solutions to some regularized kinetic models for dilute polymers. Multiscale Model. Simul. 6 (2007) 506–546. | DOI | MR | Zbl
[6] and , Existence and equilibration of global weak solutions to kinetic models for dilute polymers. I: Finitely extensible nonlinear bead-spring chains. M3AS 21 (2011) 1211–1289. | MR | Zbl
[7] and , Finite element approximation of kinetic dilute polymer models with microscopic cut-off, ESAIM: M2AN 45 (2011) 39–89. | DOI | Numdam | MR | Zbl
[8] and , Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. ESAIM: M2AN 46 (2012) 949–978. | DOI | Numdam | MR | Zbl
[9] and , The mathematical theory of finite element methods. Springer (2002). | DOI | MR | Zbl
[10] , and , On modeling and simulation of electrokinetic phenomena in two-phase flow with general mass densities. SIAM J. App. Math. 72 (2012) 1899–1925. | DOI | MR | Zbl
[11] , The finite element method for elliptic problems. North-Holland (1978). | MR
[12] , Stationary stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I. Linearized equations. SIAM J. Math. Anal. 20 (1989) 74–97. | DOI | MR | Zbl
[13] and , A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model. ESAIM: M2AN 47 (2013) 1433–1464. | DOI | Numdam | MR | Zbl
[14] and , Finite element methods for Navier-Stokes equations. In Vol. 5 of Springer Series in Computational Mathematics. Springer (1986). | DOI | MR | Zbl
[15] , On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51 (2013) 3036–3061. | DOI | MR | Zbl
[16] , and , On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities. CiCP 19 (2016) 1473–1502. | DOI | MR | Zbl
[17] and , Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257 Part A (2014) 708–725. | DOI | MR | Zbl
[18] and , On micro-macro-models for two-phase flow with dilute polymeric solutions – modeling and analysis. Math. Model. Methods Appl. Sci. 26 (2016) 823–866. | DOI | MR | Zbl
[19] and , Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math 87 (2000) 113–152. | DOI | MR | Zbl
[20] and , Splitting schemes for a Navier-Stokes-Cahn-Hilliard model for two fluids with different densities. J. Comput. Math. 32 (2014) 643–664. | DOI | MR | Zbl
[21] and , Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. | DOI | MR | Zbl
[22] and , Multiscale modelling of complex fluids: a mathematical initiation, in Multiscale modeling and simulation in science, edited by , and . In Vol. 66 of Lecture Notes in Computational Science and Engineering. Springer (2009) 49–137. | DOI | MR | Zbl
[23] , Well-posedness for the fene dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61 (2008) 1685–1714. | DOI | MR | Zbl
[24] , Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Invent. Math. 191 (2013) 427–500. | DOI | MR | Zbl
[25] , Diffuse interface models for complex flow scenarios: modeling, analysis and simulations. Ph.D. thesis, Friedrich-Alexander-Universitát Erlangen-Nürnberg (2017).
[26] , An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer Methods Partial Differ Equ. 29 (2013) 584–618. | DOI | MR | Zbl
[27] , Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. (IV) 146 (1987) 65–96. | DOI | MR | Zbl
[28] , Navier-Stokes equations: theory and numerical analysis. Reprint. with corr. AMS Chelsea Publ (2001). | MR | Zbl
[29] , Galerkin finite element methods for parabolic problems. Springer (1984). | MR
[30] . Interpolation theory, function spaces, differential operators. North-Holland Publ. (1978). | MR | Zbl
[31] and , Gewöhnliche Differentialgleichungen. Springer Verlag (1991).
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