Fluid-elastic structure interaction with the Navier slip boundary condition

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CanicDMS-1613757 The investigator studies the interaction between viscous, incompressible fluids such as water or blood, and elastic structures, such as human heart valves, modeled using a non-classical boundary condition called the Navier slip condition. This condition allows the fluid to slip over a surface, which is known to occur with hydrophobic surfaces, with human tissue constructs that have rough surfaces, and with shark's skin. Very recently it was shown that using the standard no-slip condition to study contact between structures (e.g., rigid balls) immersed in a viscous, incompressible fluid, contact can never occur in finite time. Using sophisticated mathematics, recent results also show that if the Navier slip condition is used to model contact of immersed bodies, allowing fluid to slip in between the structures leads to the structures touching each other in finite time. These recent results all address contact between rigid structures. This project goes a step further to study the interaction between fluids and elastic structures when the Navier slip boundary condition is used to model the physics of the problem. The aim is to resolve questions related to the modeling of closure of human heart valves interacting with blood flow, and flows over rough elastic surfaces such as tissue constructs interacting with blood. A series of novel mathematical and computational methods is proposed to understand these complex physical and physiological problems. Students and postdocs are involved in the project. The mathematical techniques to study these problems are based on a partitioned fluid-structure interaction algorithm. The investigator develops a novel approach to studying existence of weak solutions to this class of problems by first semi-discretizing the multi-physics coupled problem in time, and then using an operator splitting strategy to split the fluid from structure sub-problems. Particular care is used to deal with the well-known added mass effect that may lead to instabilities. The splitting approach developed here to deal with the Navier slip condition promises to provide the correct splitting strategy, which leads to the uniform energy estimates. With the construction of compactness arguments based on Simon's and Ehrling Lemmas, the existence of a weak solution is obtained by calculating the limits of the approximate sequences converging in time to a weak solution. This analysis can be used as a base for a computational algorithm in the calculation of solutions to this class of nonlinear, moving-boundary problems. The result of this work promises to contribute to the understanding of the so called "no-collision paradox," referring to the impossibility of a contact of smooth rigid bodies immersed in an incompressible, viscous fluid, modeled with the no-slip condition.