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Publication# Lattice Boltzmann method for the simulation of viscoelastic fluid flows

Abstract

The simulation of flows of viscoelastic fluids is a very challenging domain from the theoretical as well as the numerical modelling point of view. In particular, all the existing methods have failed to solve the high Weissenberg number problem (HWNP). It is therefore clear that new tools are necessary. In this thesis we propose to tackle the problem of the simulation of viscoelastic fluids presenting memory effects, which is the first attempt of applying the lattice Boltzmann method (LBM) to this field for non-trivial flows. A theoretical development of the discrete models corresponding to the equations of mass, momentum conservation and of the constitutive equation is presented as well as the particular treatment of the associated boundary conditions. We start by presenting a simplified case where no memory but shear-thinning or shear-thickening effects are present : simulating the flow of generalized Newtonian fluids. We test the corresponding method against two-dimensional benchmarks : the 2D planar Poiseuille and the 4:1 contraction flows. Then we propose a new model consisting in solving the constitutive equations that account for memory effects, by explicitly including an upper-convected derivative, using the lattice Boltzmann method. In particular, we focus on the polymer dumbbell models, with infinite or finite spring extension (Oldroyd-B and FENE-P models). Using our model, we study the periodic (simplified) 2D four-roll mill and the 3D Taylor-Green decaying vortex cases. Finally, we propose an approach for simulating flat walls and show the applicability of the method on the 2D planar Poiseuille case. Two of the advantages of the proposed method are the ease of implementation of new viscoelastic models and of an algorithm for parallel computing.

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Newtonian fluid

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of change of the fluid's velocity vector. A fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow.

Viscoelasticity

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain.

Constitutive equation

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.