11. FPSI with Nonlinear Coupling

This chapter delves into the intricate world of nonlinearly coupled fluid-poroelastic structure interaction (FPSI) systems, focusing on the interaction between an incompressible fluid modeled by the Navier-Stokes equations and a multilayered poroelastic structure consisting of a thin reticular plate and a thick Biot poroelastic material. The chapter addresses the mathematical challenges posed by the nonlinearly coupled nature of the system and the geometric nonlinearities arising from the moving domains. A key highlight is the development of a well-posedness theory for the FPSI problem, involving weak solutions to a regularized FPSI problem. The chapter also establishes a weak-classical consistency result, demonstrating that weak solutions to the regularized problem converge to smooth solutions of the original problem as the regularization parameter tends to zero. The text provides a detailed analysis of the Biot equations, the reticular plate equation, and the Navier-Stokes equations on moving domains, along with the coupling conditions that define the interaction between the fluid and the poroelastic structure. The chapter concludes with a discussion of the practical importance of these mathematical results and their applications to real-life engineering and science problems, particularly in the design of bioartificial organs and the study of blood flow in arteries.