UC Berkeley

This thesis will study fluid-structure interaction (FSI), which describes the coupled multiphysical dynamical interaction between fluids and deformable structures. From modeling the flow of blood in compliant elastic arteries to modeling biomedical prostheses and large-scale structures such as wings, bridges, and dams, FSI is prevalent in science, making the rigorous analysis of such coupled fluid-structure systems important for continued technological development and progress in engineering. While prototypical models of FSI involving incompressible, viscous, Newtonian fluids interacting with elastic structures have been well-studied in the literature, the types of FSI models found in present-day real-life applications have unique and interesting features that require new mathematical methods for their analysis. The goal of this thesis will be to develop new tools for studying new complex FSI models of practical importance that extend past work on prototypical models of FSI. Motivated by real-life applications, we will study stochastic FSI systems involving coupled FSI dynamics under the additional influence of random noise in time, and fluid-poroelastic structure interaction (FPSI) which describes FSI systems in which the structure is poroelastic and hence admits fluid flow through its pores. In the study of stochastic FSI, we establish well-posedness for two models: (1) a reduced model where the full stochastic fluid-structure dynamics can be reduced to a single stochastic equation known as the stochastic viscous wave equation and (2) a fully coupled stochastic FSI system involving linear coupling between a Stokes flow through a channel and the stochastically forced elastic walls of the channel, where the full system is described by a stochastic system of PDEs. Next, we study deterministic nonlinearly coupled FPSI and consider a model in which a multilayered poroelastic structure consisting of a thin plate and a thick poroelastic medium, modeled by the Biot equations, interacts with an incompressible fluid modeled by the Navier-Stokes equation. We study well-posedness and consistency of this nonlinearly coupled FPSI model, which is especially challenging since the fluid and poroelastic structure domains are time-dependent and a priori unknown. To the best of our knowledge, the results in this thesis represent the first well-posedness results for stochastic fluid-structure systems and nonlinearly coupled FPSI with moving domains.