This dissertation deals with flows impacted by wall roughness and/or uncertain flow-domain geometry. Specifically, it focuses on stochastic analysis of fluid flows in domains whose rough surfaces are modeled as random fields. More broadly, this work addresses some of the unresolved theoretical and practical questions concerning differential equations defined on random domains. It has significant impact on geophysical and biological flows, and can be extended to other areas where surface roughness affects fluid flows, such as nanoscale devices.
In the first part of this thesis, we present the background and the mathematical tools used in our study. They were presented in a manner to help engineers, engineering students and practitionners to grasp the concepts.
The second part of this work discusses the stochastic modeling of Stokes flow in a channel with rough walls. The adopted approach consists of regarding the rough surface as a random field characterized by its statistical moments, a mapping of the stochastic domain of definition onto a deterministic domain, and stochastic homogenization of the resulting differential equations with random coefficients. This enables one to obtain closed-form expressions for the effective or apparent viscosity in terms of the statistical moments characterizing the wall roughness to fluid viscosity, and the Poiseuille number. The most important consequence of this analysis is a rigorous explanation of why Stokes flows drastically change their behavior depending on whether the flow takes place in a micro or macro channel. The results were validated using Comsol multiphysics software to simulate flow through domain bounded at the top by smooth wall and at the bottom by a sinusoidal wall with various amplitudes and different periods.
The third part deals with the application of the proposed approach to technology and life science. In technology, we investigate the impacts of the roughness of the boundary surfaces on the average flow thermal properties.
In the fourth and final part, we dicuss our findings and describe the future direction of this work.
This elucidates the mechanical effects that take place at the stochastic solid/fluid interface in biological systems (blood/endothelial lining). This result has important implications for biology, physiology and medicine, and micro/nano technology because these interfaces are incredibly complex and difficult to quantify deterministically.