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Mesoscopic Models for Viscoelastic flows: Coupling Finite Element and Monte Carlo Methods

Published/Copyright: October 16, 2009
Monte Carlo Methods and Applications
From the journal Volume 8 Issue 1

Monte Carlo Methode and AppL, Vol. 8, No. l, pp.73 - 81 (2002)© VSP2002Mesoscopic Models for Viscoelastic flows :Coupling Finite Element and Monte Carlo Methods *tJohn Bonvin and Marco PicassoDepartement de MathematiquesEcole Polytechnique Federale de Lausanne1015 Lausanne, SwitzerlandE-mail: marco.picasso@epfl.chAbstract — Mesoscopic models are considered for dilute Solutions of polymer chains. The fluidis assumed to be a Newtonian solvent with non interacting polymer chains. The polymer chainsare modelled by dumbbells, that is two beads connected with an elasting spring. Stochasticdifferential equations govern the elongations of the springs. The extra-stress due to the polymerchains is then defined s a moment of these elongations. A finite element method is used for thespace discretization and a classical Euler scheme is used for time discretization. The extra-stressdue to the polymer chains is then computed by means of a Monte-Carlo method. A variancereduction method is proposed. Numerical results are compared with experiments in the frameof the 4:1 planar contraction flow.l IntroductionNumerical Simulation of viscoelastic flows is of great importance for processes involvingplastics, paints, food or biological flows. The macroscopic modelling of viscoelastic flowsgenerally consists in supplementing the mass and momentum equations with a rheologicalconstitutive equation relating the velocity and the non-Newtonian part of the stress. Thisconstitutive equation can be either differential or integral. Alternative models consist inintroducing some of the molecular aspects of the fluid. This approach leads to mesoscopic(or kinetic) models. When considering polymeric liquids, two descriptions are available.If the liquid is a dilute solution of polymers, the interaction between the polymer chainscan be neglected and chain-like models are adequate. On the other side, if the liquid is apolymer melt, then reptation models should be used. The interested reader is referred to[3, 12] for more details. Due to the constant increase of Computer resources, these kineticmodels can nowadays be solved numerically, see for instance [11, 9, 8, 5]. In this paper,we describe a numerical model to compute the flow of a dilute solution of polymers.A dilute solution of polymers is considered in a domain Ω of R2. As usual, the totalstress is split into the sum of a Newtonian contribution 2τ7β€(ιι) -pl and a non-Newtonian"Project Supported by the Swiss National Science Foundationtpresented at the International Conference on Monte Carlo and Probabilistic Methods for PartialDifferential Equations, Monte Carlo (Monaco), July, 3-5, 200073
Published Online: 2009-10-16
Published in Print: 2002

Walter de Gruyter

Articles in the same Issue

  1. Monte Carlo difference schemes for the wave equation
  2. Monte Carlo Simulation of a Random Field Model for Transport
  3. Using Genetic Operators to Speed up Markov Chain Monte Carlo Integration
  4. Mesoscopic Models for Viscoelastic flows: Coupling Finite Element and Monte Carlo Methods
  5. Rate of Weak Convergence of the Euler Approximation for Diffusion Processes with Jumps
  6. Some geometric properties of (0, m, 2)-nets in base b ≥ 2
  7. Editorial Board
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https://doi.org/10.1515/mcma.2002.8.1.73

Articles in the same Issue

  1. Monte Carlo difference schemes for the wave equation
  2. Monte Carlo Simulation of a Random Field Model for Transport
  3. Using Genetic Operators to Speed up Markov Chain Monte Carlo Integration
  4. Mesoscopic Models for Viscoelastic flows: Coupling Finite Element and Monte Carlo Methods
  5. Rate of Weak Convergence of the Euler Approximation for Diffusion Processes with Jumps
  6. Some geometric properties of (0, m, 2)-nets in base b ≥ 2
  7. Editorial Board
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