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Doctoral Thesis
Full name
Débora de Oliveira Medeiros
Knowledge Area
Date of Defense
São Carlos, 2022
Cuminato, José Alberto (President)
Cruz, Daniel Onofre de Almeida
Oliveira, Taygoara Felamingo de
Souza, Leandro Franco de
Title in English
Numerical analysis of finite difference schemes for constitutive equations in viscoelastic fluid flows
Keywords in English
Finite differences method
Generalized Lie derivative
Viscoelastic flows
Abstract in English
In this work, we present a study of numerical methods for the solution of incompressible fluid flows, with emphasis on viscoelastic effects. The upper-convected derivative term is rewritten, using the definition of the generalized Lie derivative in a Lagrangian framework, providing a new numerical scheme for viscoelastic fluid flows. The mathematical modeling involves the Navier-Stokes equations and a system of equations that define the contribution of the polymer stress tensor. The numerical formulation combines a finite difference discretization, in the MAC context, with a projection method and the reformulation of the constitutive equation. We carried out theoretical analyses of the proposed methods, convergence studies of simple problems, and applications to the solution of complex fluid flows. The numerical results agree with the theory developed, present results fairly comparable with other numerical methods from the literature, and allowing a discussion about the numerical instabilities of high Weissenberg number problems.
Title in Portuguese
Análise numérica de esquemas de diferenças finitas para equações constitutivas em escoamentos de fluidos viscoelásticos
Keywords in Portuguese
Derivada generalizada de Lie
Escoamentos viscoelásticos
Método de diferenças finitas
Abstract in Portuguese
Neste trabalho, apresentamos um estudo de métodos numéricos para a solução de escoamentos de fluidos incompressíveis, com ênfase nos efeitos viscoelásticos. O termo da derivada convectada superior é reescrito usando a definição da derivada generalizada de Lie em uma estrutura Lagrangiana, fornecendo um novo esquema numérico para escoamentos de fluidos viscoelásticos. A modelagem matemática envolve as equações de Navier-Stokes e um sistema de equações que definem a contribuição viscoelástica do tensor tensão extra. A formulação numérica combina uma discretização de diferenças finitas, no contexto MAC, com um método de projeção e a reformulação da equação constitutiva. Realizamos análises teóricas dos métodos propostos, estudos de convergência de problemas simples e aplicações na solução de escoamentos de fluidos complexos. Os resultados numéricos concordam com a teoria desenvolvida, apresentam bons resultados quando comparado com outros métodos numéricos da literatura e permitem uma discussão sobre as instabilidades numéricas de problemas de alto número de Weissenberg.
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