A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells

We propose a numerical procedure to study closure approximations for FENE dumbbells in terms of chosen macroscopic state variables, enabling to test straightforwardly which macroscopic state variables should be included to build good closures. The method involves the reconstruction of a polymer distribution related to the conditional equilibrium of a microscopic Monte Carlo simulation, conditioned upon the desired macroscopic state. We describe the procedure in detail, give numerical results for several strategies to define the set of macroscopic state variables, and show that the resulting closures are related to those obtained by a so-called quasi-equilibrium approximation \cite{Ilg:2002p10825}

Numerical approximation of corotational dumbbell models for dilute polymers

We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. In the case of a corotational drag term we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system

Direct numerical simulation of complex viscoelastic flows via fast lattice-Boltzmann solution of the Fokker–Planck equation

Micro–macro simulations of polymeric solutions rely on the coupling between macroscopic conservation equations for the fluid flow and stochastic differential equations for kinetic viscoelastic models at the microscopic scale. In the present work we introduce a novel micro–macro numerical approach, where the macroscopic equations are solved by a finite-volume method and the microscopic equation by a lattice-Boltzmann one. The kinetic model is given by molecular analogy with a finitely extensible non-linear elastic (FENE) dumbbell and is deterministically solved through an equivalent Fokker–Planck equation. The key features of the proposed approach are: (i) a proper scaling and coupling between the micro lattice-Boltzmann solution and the macro finite-volume one; (ii) a fast microscopic solver thanks to an implementation for Graphic Processing Unit (GPU) and the local adaptivity of the lattice-Boltzmann mesh; (iii) an operator-splitting algorithm for the convection of the macroscopic viscoelastic stresses instead of the whole probability density of the dumbbell configuration. This latter feature allows the application of the proposed method to non-homogeneous flow conditions with low memory-storage requirements. The model optimization is achieved through an extensive analysis of the lattice-Boltzmann solution, which finally provides control on the numerical error and on the computational time. The resulting micro–macro model is validated against the benchmark problem of a viscoelastic flow past a confined cylinder and the results obtained confirm the validity of the approach

Multiscale stochastic simulation of transient complex flows

The thesis reports new multiscale simulation methods to predict rheological properties of complex fluids. Dilute polymer solutions, polymer melts and fibre suspensions with a Newtonian matrix are the main interests of this study. In the present multiscale approach, the stress contributed by polymers or suspended fibres is determined by the Brownian configuration field (BCF) method using kinetic models whereas integrated radial basis function (IRBF) based numerical methods are used to approximate field variables and their derivatives and to discretise governing equations. The macro-micro multiscale system is linked together by a stress formula by kinetic models. The IRBF-BCF based multiscale method is first applied to simulate dilute polymer solutions modelled by bead-spring chains (BSCs), incorporating finitely extensible nonlinear elastic springs, hydrodynamic interaction and excluded volume effects. Then, the simulation method is further developed for polymer melt systems, in which the entanglement of polymer molecules is described by Doi-Edwards, Curtiss-Bird, reptating rope and double reptation models. The numerical stability of the method, which is generally known as a challenging problem in the simulation of polymer melts, is enhanced owing to the combination of the IRBF method and the BCF idea. As an illustration of the method, the start-up Couette flow and the flow over a cylinder in a channel are investigated for both dilute polymer solutions and polymer melts. A new multiscale approach is also developed to simulate the rheological characteristics of fibre suspensions in both dilute and non-dilute regimes. The approach is a combination of the IRBF scheme, the discrete adaptive viscoelastic stress splitting (DAVSS) formulation and the BCF idea. The macroscopic conservation equations described in stream function-vorticity formulation are solved using the 1D-IRBF scheme combined with the DAVSS technique. The evolution equation for fibre configuration fields governed by the Jeffery equation for dilute fibre suspensions or the Folgar-Tucker equation for non-dilute fibre suspensions is explicitly advanced in time using the BCF approach. The fibre stress is determined based one fibre configuration fields using the Lipscomb and Phan-Thien–Graham models for dilute and non-dilute fibre suspensions, respectively. The method is verified with the simulation of flows of fibre suspensions between two parallel plates, flows through a circular tube, the 4:1 and 4.5:1 axisymmetric contraction flows, and the 1:4 axisymmetric expansion flows. Numerical experiments confirm the present method efficiency based on both the enhanced convergence rate of the solution and the stability of a stochastic process