Numerical Approximations of Non-Newtonian Fluid Flows with Applications

点击翻译按钮获取中文摘要

This research is focused on numerical approximation of non-Newtonian fluid flows in physical applications. Such fluid flows are abundant in our everyday lives, from the flow of blood in our bodies to the production of polymeric material such as plastics. There are two prototypal problems considered in the project: (i) optimal control for defective boundary conditions, and (ii) non-Newtonian flow within an elastic medium. Blood flow is one of most important examples related to such situations as a non-Newtonian flow interacts with an elastic vessel wall, where only flow rate or mean pressure is specified on each inflow and outflow boundary. The model problems in this research involve either coupled domains representing multi-physics behavior or coupled state-adjoint systems. This increases the numerical complexity as both stress and velocity must be resolved in the domains, and the strong interaction between the governing equations requires solution algorithms that achieve optimal convergence rates while splitting the operators. Additionally, because of the large number of unknowns to be approximated, there is a need to develop efficient solvers for these problems. The proposed research addresses issues on decoupling schemes, and their stability and convergence. The primary contribution of the research is the development of robust numerical schemes for non-Newtonian flows in coupled systems, and analytical and numerical study of optimal control for non-Newtonian flows.There have been extensive studies on multidisciplinary problems involving Newtonian flows, but to date mathematical and numerical investigations of non-Newtonian flows are still far behind. Because of the many important biological and engineering processes involving non-Newtonian fluid flow, there is a great demand for mathematical support in these applications. The proposed research broadens the mathematical basis for the numerical simulation of non-Newtonian fluid flow problems in physical settings. Also the research benefits biomedical and polymer industries by providing improved algorithms for the numerical simulation of important processes.