Mathematical modelling, analysis and computation of some complex and nonlinear flow problems

Abstract

This thesis consists of two parts: (I) modelling, analysis and computation of sweat transport in textile media; (II) unconditional convergence and optimal error analysis of the Galerkin FEM for nonlinear parabolic equations. The first part of the thesis is concerned with heat and sweat transport in porous textile media, which can be viewed as a nonisothermal, multiphase and multicomponent flow with complex phases changes. We present a more precise formulation of the condensation/evaporation process with a truncated Hertz-Knudsen equation, which makes the model applicable in the general dry-wet case. We introduce a flux type boundary condition for the fiber absorption equation to describe the absorption process in a wet environment more precisely, while the previous models with a simple saturated condition may not be realistic. Numerical simulations are performed to compare with experimental data, with both finite difference methods and finite element methods. Several practical cases are simulated for clothing assemblies with the human thermoregulation system. Moreover, we provide optimal error estimates for an uncoupled finite difference method in one-dimensional space and a splitting finite element method in three-dimensional space. The error analysis relies on some interesting skills used in PDEs analysis and physical features in modelling. The physical process of heat and sweat transport is governed by a system of nonlinear, degenerate and strongly coupled parabolic equations in general. However, mathematical analysis for these models is very limited due to the lack of reasonable link between modelling in engineering and analysis in mathematics. We prove existence of weak solutions for the dynamic models with complex phase changes. The proof is based on the nature of gas convections in the mass equations and energy equation, with physically realistic assumptions. The analysis presented in this thesis may be applied to the multicomponent heat and mass transport models in many other areas, and it also provides a fundamental tool for theoretical analysis of numerical methods. The second part of the thesis is concerned with unconditional convergence and optimal error analysis of the Galerkin/mixed finite element method for nonlinear parabolic equations, with commonly-used linearized semi-implicit schemes for the time discretization. To illustrate our method, we study the time-dependent nonlinear Joule heating equations and the equations of incompressible miscible flow in porous media, respectively. Optimal L2 error estimates are obtained without any time step restriction, while all the previous works required certain conditions for the time stepsize. Theoretical analysis is based on more precise analysis of a corresponding time-discrete partial differential equations. The approach used in this paper is applicable for more general nonlinear evolution equations and many other linearized semi-implicit (or implicit) time discretizations.for which previous works often require certain restrictions on the time stepsize τ.

Date of Award	3 Oct 2012
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Weiwei SUN (Supervisor)