6. Extensions of the Splitting Scheme

This chapter explores the extensions of the splitting scheme method to various complex fluid-structure interaction (FSI) models, highlighting its versatility and practical applications. It begins by examining 3D-2D fluid-structure systems, where incompressible viscous fluids in three dimensions interact with two-dimensional elastic structures like plates, membranes, or shells. The chapter discusses the challenges and solutions in achieving constructive existence results for weak solutions in these systems. It also delves into the analysis of nonlinear elastic structures, such as St. Venant-Kirchhoff elastic structures and Koiter shells, demonstrating how the splitting scheme method can be extended to these models. The chapter further explores the Navier slip condition, which allows for a mismatch between the tangential components of the structure and fluid velocities, providing a more realistic representation of certain FSI dynamics. Additionally, it covers multilayered structures in FSI, which are highly relevant in biomedical contexts, and FSI with mesh-supported elastic structures, motivated by advancements in biomedical engineering. The chapter concludes with a discussion on the practical implications of these models, particularly in the study of stents and their performance in arterial walls. Throughout, the chapter provides detailed mathematical frameworks and practical examples, making it a valuable resource for professionals seeking to apply these models in real-world scenarios.