Sistema de leis de balanço em problemas de dinâmicas de fluidos: modelagem matemática e aproximação numérica

Author(s)

Abel Alvarez Bustos

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2015-07-23

Examining board members

Eduardo Cardoso de Abreu; Frederico da Cunha Furtado; Grigori Chapiro; Aparecido Jesuino de Souza; Maria Cristina de Castro Cunha

• Advisor: Eduardo Cardoso de Abreu

Abstract

In this thesis, we are concerned with the limit behaviour of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems of conservation laws, particularly the question of stability and singular limits of such solutions to the zero relaxation time. Relaxation is important in many physical situations, as such, in kinetic theory, gases not in local thermodynamic equilibrium, elasticity with memory (hysteresis), multiphase and phase transition and linear and nonlinear waves. Although the mathematical theory of nonlinear balance law with relaxation has presented significant progress on well-posedness linked to extended thermodynamics and kinetic theory, a complete understanding for systems larger than $2\times2$ about how solutions evolve from a given initial data and their regularity and asymptotic behaviour remains elusive, mainly for weak solutions of hyperbolic systems. Thus, due to the complexity inherent to this class of models, there are few solutions for such relaxation balance laws by means of analytical methods. Then, abstract analysis as well as practical computing via approximation algorithms are both significant mathematical tools to tackle as well as to get further insights to enlarge the knowledge for systems of balance laws. Therefore, it was also developed a new unsplitting finite volume methods, which in turn is locally conservative by formal construction. This method was able to corroborate the new solutions for Euler systems with a non-monotonic character as well as to reproduce correct qualitatively solutions of the Euler models with high friction regime and gravity, recently published in the literature. Indeed, the novel unsplitting approximation algorithms were also used to address injection problems of nitrogen and steam in porous media. Another crucial viewpoint pursued in this thesis is the comparison between two methodologies to tackle the issue of solving balance laws with relaxation source terms: one methodology is based by assuming that the physical phenomenon is under thermodynamic equilibrium (instantaneous equilibrium), which is modelled by systems of conservation laws, and the other methodology is based in the relaxation of such equi\-li\-brium, which in turn gives rise to the use of systems of balance laws in the modelling of the relaxation process, for instance, in the modelling of phase transition. At this moment a natural questions is: how different are these both solutions obtained by means of two approaches? In this regard, a more stringent -- and more fundamental -- question is: how is the behaviour of such solutions during the relaxation process and how is its limit? In order to better understand these methodologies we will consider two distinct mathematical formalisms. In thesis, we give an example of modelling using this novel methodology for the injection of nitrogen and steam in porous media. We were not able to give assertive answers to the above questions, but a solid starting point is a thorough study of the one-dimensional case for a concrete problem, which is done in this thesis. We believe we have a very interesting (and promising) field of work ahead of us, which we intend to continue studying in order to better understand abstract and numerical analysis for these important questions that remains elusive. This thesis is a small attempt to get new insights in this direction (AU)

• FAPESP's process: 11/23628-0 - Systems of Balance Laws Problems in Fluid Dynamics in Porous Media: Mathematical Modeling and Numerical Approximation
• Grantee: Abel Alvarez Bustos
• Support Opportunities: Scholarships in Brazil - Doctorate