Modeling Elastic Shells Immersed in Fluid

We describe a numerical method to simulate an elastic shell immersed in a viscous incompressible fluid. The method is developed as an extension of the immersed boundary method using shell equations based on the Kirchhoff-Love and the planar stress hypotheses. A detailed derivation of the shell equations used in the numerical method is presented. This derivation as well as the numerical method, use techniques of differential geometry in an essential way. Our main motivation for the development of this method is its use in the construction of a comprehensive three-dimensional computational model of the cochlea (the inner ear). The central object of study within the cochlea is the “basilar membrane”, which is immersed in fluid and whose elastic properties rather resemble those of a shell. We apply the method to a specific example, which is a prototype of a piece of the basilar membrane and study the convergence of the method in this case. Some typical features of cochlear mechanics are already captured in this simple model. In particular, numerical experiments have shown a traveling wave propagating from the base to the apex of the model shell in response to external excitation in the fluid.