Geometric methods for fluid-structure interactions

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When a grass or tree leaf bends in the wind, the motion deforms the internal structure of narrow channels and moves the fluid inside. A geologist looking for oil probes the Earth with sounds waves, which move the porous media and fluid inside it in a complex manner. Blood pulsing through our arteries and veins deforms the elastic walls with every heartbeat. Interactions of fluids and structures are everywhere, and many more examples from everyday life can be easily found. Describing interaction of fluids and structures is always challenging. This project will create a unified framework for considering interactions between fluids and structures, using the approach of geometric mechanics. ******The focus of the project will be on the cases when fluids are flowing inside the elastic materials, such as narrow tubes and porous material media. The unifying theme of the project is the use of general geometric ideas, such as the symmetry of space, and methods of analytical mechanics (variational procedure), yielding the derivation and analysis of equations from the first principles. The methods of geometric mechanics allow to treat a wide variety of problems. The first set of problems concerns the mechanics of tubes conveying fluid, a problem which is relevant for engineering (e.g. chemical), and biomedical applications (blood flow). The methods developed in this project will also lead to the development of variational computational methods which guarantee conservation of linear and angular momenta and absence of artificial sources and sinks of mass, forces and torques due to discretization. The second set of problems concerns the dynamics of flexible porous media, such as sponge filled with water, and sheets and rods made of such media. The theory developed in this project will allow to analyze the motion of fluid-filled sheets and rods, and in particular, compute internal dissipation under the motion, which is difficult to compute without geometric methods. Finally, we apply Hamel's theory of mechanics to our problems. This method, based on choosing the most convenient velocities for the problem at hand, will further simplify the analysis and allow to find the most convenient velocity coordinates for the complex problems considered here. ***We are also incorporating geometric elasticity models for describing the growth of glioma in the brain, with the focus on its mechanical effects, given that the volume of the brain is constrained and the composition the brain matter around glioma changes. Understanding the effects of mechanics and additional stresses on glioma growth may eventually contribute to treatment recommendations. In addition, since the methods we develop here are general, and are based on fundamental principles of symmetry, they can be applied to a wide variety of practical problems coming from different physics, forming the background for studies beyond the scope of the 5-year project described here.