Extensão de GENSMAC para escoamentos de fluidos governados pelos modelos integrais Maxwell e K-BKZ

Abstract

The aim of this work is to develop a numerical technique for simulating incompressible, isothermal, free surface (also confined) viscoelastic flows of fluids governed by the integral models of Maxwell and K-BKZ (Kaye-Bernstein, Kearsley and Zapas). The numerical technique described herein is an extension of the GENSMAC method (Tome and McKee, J. Comput. Phys., 110, pp. 171-186, 1994) to the solution of the momentuum and mass conservation equations together with the integral constitutive Maxwell and K-BKZ equations. The governing equations are solved by the finite difference method on a staggered grid using a Marker-and-Cell approach. The fluid is represented by marker particles on the fluid surface only. This provides the visualization and location of the fluid free surface so that the free surface stress conditions can be applied. The Finger tensor B t (t) is computed using the ideias of the deformation fields method (Peters et al. J. Non-Newtonian Fluid Mech., 89, pp. 209-228, 2001) so that it is not necessary to track a fluid particle in order to calculate its deformation history. However, in this work modifications to the deformation fields method are introduced: the past time is discretized using a different formula, the Finger tensor B t (x, t) is obtained by a second order method and the stress tensor τ (x, t) is computed by a second order quadrature formula. The numerical method presented in this work is validated by simulating the flow of a Maxwell fluid in a two-dimensional channel and the numerical solutions of the velocity and the stress components are compared with the respective analytic solutions providing a good agreement. Further, the flow through a 4:1 planar contraction of a specific fluid studied experimentally by Quinzani et al. (J. Non-Newtonian Fluid Mech., 52, pp. 1-36, 1994) was simulated and the numerical results were compared qualitatively and quantitatively with the experimental results and very good agreement was obtained. The Maxwell and the K-BKZ models were applied to simulate the 4:1 planar contraction problem using various Weissenberg numbers and the numerical results were in agreement with those published in the literature. Finally, numerical results of free surface flows using the Maxwell and K-BKZ integral constitutive equations are presented. In particular, the numerical simulation of jet buckling using several Weissenberg numbers and various Reynolds numbers are presented.

Implementation of integral viscoelastic constitutive models in OpenFOAM® computational library

“…3, two major steps are required: (1) define an approximation for the integral and, (2) devise an equation for the evolution in time of the Finger tensor. Although these steps have already been described elsewhere [18,23] for ease of understanding, they are now briefly described.…”

“…Although very good results can be obtained with these solvers, integral models are not yet implemented, therefore, in this work we describe the implementation of a new solver in OpenFOAM® able to deal with integral constitutive models [20,23], based on the upper-convected Maxwell (UCM) model.…”

Solução numérica do modelo PTT para escoamentos viscoelásticos com superfícies livres

“…Também, foram temas de dissertações de mestrado as seguintes extensões no sistema Freeow2D: escoamentos viscoelásticos bidimensionais e também axissimétricos, ambos envolvendo superfícies livres e modelados pela equação constitutiva Oldroyd-B ( [72], [74]), e os escoamentos viscoelásticos bidimensionais com superfícies livres modelados pela equação constitutiva`Upper Convected Maxwell (UCM)' [14] e uido de segunda ordem [83]. Recentemente, Araújo [4] desenvolveu uma técnica de diferenças nitas para simular escoamentos viscoelásticos bidimensionais com superfícies livres usando a equação integral não-linear K-BKZ através do sistema Freeow2D. A mais recente extensão é a inclusão do modelo não-linear PTT neste sistema, cuja metodologia e resultados obtidos serão apresentados em detalhes neste trabalho.…”

Solução numérica do modelo Giesekus para escoamentos com superfícies livres

“…Esse problema é muito utilizado no desenvolvimento de métodos numéricos para a simulação de escoamentos de fluidos não-newtonianos [Araújo 2006,Oliveira e Pinho 1999. O principal interesse na simulação deste tipo de escoamento vem do fato de fluidos viscoelásticos apresentarem comportamento variado quando sujeitos à geometrias complexas, como a contração planar por exemplo.…”