Variational Methods for Fluid-Structure Interactions

This chapter gives an introduction to the variational methods recently developed in fluid-structure interaction, by focusing on the dynamics of flexible tubes conveying fluid. This is a topic of high importance for biomedical and industrial applications, such as arterial or lung flows, and problems involving high-speed motion of gas in flexible pipes. Our goal is to derive a variational approach to fluid-structure interaction with the aim of developing the corresponding variational numerical methods. Variational approaches and corresponding discretizations are advantageous for fluid-structure interactions, since they possess excellent long term energy behavior, exact conservation of momenta, and yield consistent models for complex mechanical systems. We present a model for the three-dimensional evolution of tubes with expandable walls conveying fluid, that can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We show how this model reduces to previously derived models under specific assumptions. We present particular solutions of the model, such as propagation of a shock wave in an elastic tubes via the Rankine–Hugoniot conditions. Finally, we develop the variational discretization of the model, based on the discretization of the back-to-labels map, for the cases of spatial and spatio-temporal discretizations.