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C3: Spinodal decomposition of polymer-solvent systems

The goal of the project is to obtain stable and consistent descriptions of flow dynamics on multiple scales in a class of systems exhibiting highly complex non-equilibrium dynamics, namely phase-separating polymer solutions. This is done by combining (i) the derivation, analysis, and simulation of macroscopic two-fluid models describing the dynamics of viscoelastic phase separation, (ii) the mesoscopic simulation of viscoelastic phase separation by extension of a coupled Lattice-Boltzmann / Molecular Dynamics method, and (iii) the calibration of the macroscopic models to results from mesoscopic simulations by means of parameter estimation and inverse problems methodology.


Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method. Part I: a nonlinear scheme
Lukáčová-Medviďová, M.; Mizerová, H.; Notsu, H.; Tabata, M.
ESAIM Math. Model. Numer. Anal. 51 (5), 1637–1661. (2017);
URL: https://www.esaim-m2an.org/

We present a nonlinear stabilized Lagrange–Galerkin scheme for the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi Pitkäranta’s stabilization method for the conforming linear elements, which yields an efficient computation with a small number of degrees of freedom. We prove error estimates with the optimal convergence order without any relation between the time increment and the mesh size. The result is valid for both the diffusive and non-diffusive models for the conformation tensor in two space dimensions. We introduce an additional term that yields a suitable structural property and allows us to obtain required energy estimate. The theoretical convergence orders are confirmed by numerical experiments. In a forthcoming paper, Part II, a linear scheme is proposed and the corresponding error estimates are proved in two and three space dimensions for the diffusive model.

An improved dissipative coupling scheme for a system of Molecular Dynamics particles interacting with a Lattice Boltzmann fluid
Nikita Tretyakov, Burkhard Dünweg
Computer Physics Communications 216, 102-108 (2017);
doi:10.1016/j.cpc.2017.03.009

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
M. Lukacova-Medvidova, H. Mizerova, H. Notsu, M. Tabata
Mathematical Modelling and Numerical Analysis , (2017);
doi:10.1051/m2an/2017032

This is the second part of our error analysis of the stabilized Lagrange–Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi–Pitkäranta’s stabilization method for the conforming linear elements, which leads to an efficient computation with a small number of degrees of freedom. In this paper, Part II, we apply a semi-implicit time discretization which yields the linear scheme. We concentrate on the diffusive viscoelastic model, i.e. in the constitutive equation for time evolution of the conformation tensor diffusive effects are included. Under mild stability conditions we obtain error estimates with the optimal convergence order in two and three space dimensions. The theoretical convergence orders are confirmed by numerical experiments.

Global existence result for the generalized Peterlin viscoelastic model
Maria Lukacova - Medvidova, Hana Mizerova, Sarka Necasova, Michael Renardy
SIAM J. Math. Anal., 1-14 (2017);
URL: https://www.siam.org/journals/sima.php

We consider a class of differential models of viscoelastic fluids with diffusive stress. These constitutive models are motivated by Peterlin dumbbell theories with a nonlinear spring law for an infinitely extensible spring. A diffusion term is in- cluded in the constitutive model. Under appropriate assumptions on the nonlinear constitutive functions, we prove global existence of weak solutions for large data. For creeping flows and two-dimensional flows, we prove global existence of a classical solution under stronger assumptions.

Energy-stable numerical schemes for multiscale simulations of polymer-solvent mixtures
M. Lukacova-Medvidova, B. Duenweg, P. Strasser, N. Tretyakov
in Mathematical Analysis of Contimuum Mechanics and Industrial Applications II , Editor: Patrick van Meurs, Masato Kimura, Hirofumi Notsu, Chapter Chap5: Interface Dynamics , Pages 1-12, Springer International Publishing AG/ Eds. Patrick van Meurs, Masato Kimura, Hirofumi Notsu (2017);
URL: https://link.springer.com/chapter/10.1007/978-981-10-6283-4_13

We present a new second order energy dissipative numerical scheme to treat macroscopic equations aiming at the modeling of the dynamics of complex polymer-solvent mixtures. These partial differential equations are the Cahn-Hilliard equation for diffuse interface phase fields and the Oldroyd-B equations for the hydrodynamics of the polymeric mixture. A second order combined finite volume / finite difference method is applied for the spatial discretization. A complementary approach to study the same physical system is realized by simulations of a microscopic model based on a hybrid Lattice Boltzmann / Molecular Dynamics scheme. These latter simulations provide initial conditions for the numerical solution of the macroscopic equations. This procedure is intended as a first step towards the development of a multiscale method that aims at combining the two models.

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory
Nikita Tretyakov, Periklis Papadopoulos, Doris Vollmer, Hans-Jürgen Butt, Burkhard Dünweg, Kostas Ch. Daoulas
The Journal of Chemical Physics 145 (13), 134703 (2016);
doi:10.1063/1.4963792

Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid
Maria Lukacova-Medvidova, Hirofumi Notsu, Bangwei She
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 81, 523-557 (2016);
URL: wileyonlinelibrary.com
doi:10.1002/fld.4195

In this paper, we propose new energy dissipative characteristic numerical methods for the approximation of diffusive Oldroyd-B equations that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behavior on a series of numerical experiments. Using both the diffusive model and the logarithmic transformation of the elastic stress, we are able to obtain methods that converge as mesh parameter is refined.

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