The investigator develops new numerical methods for the solution of problems involving the interaction of flexible boundaries and the incompressible fluids in which they are immersed. Typical applications of such problems are found in biology and physiology. The approaches pursued include Lagrangian and Eulerian methods based on impulse variables, which combine elements from other numerical techniques such as vortex and projection methods. A hybrid particle method is developed for high Reynolds number flows. In this case the accumulated effect of immersed boundary forces is accounted for through the evolution of impulse variables. The viscous effects are modeled using a deterministic method for the diffusion of vorticity using vortex monopoles. Finite-difference grid-based methods are also developed for the solution of the these problems in moderate Reynolds number flows. Making use of the Lagrangian representation of the flexible boundaries, the investigator and his collaborators use ideas from the Lagrangian impulse method to describe the evolution of the immersed boundaries within the framework of the grid solution. The researchers highlight improvements to existing numerical methods for this type of problems.
The numerical solution of fluid flow problems with thin flexible moving boundaries is motivated partly by the wide range of potential applications in biology and physiology. For example, the membranes of the inner ear, the walls of the heart or lungs, muscle tissue, swimming jellyfish and eels can be modeled as thin membranes embedded in a fluid. These examples are part of everyday life and yet many of their technical aspects are not fully understood. The development and availability of reliable numerical simulations that could help our understanding of such commonly occurring natural phenomena is of great importance. Numerical methods can be used to conduct improved computer simulations of new heart valve designs; computer simulations of collapsible tubes with a fluid inside, such as arteries weakened by disease, can be studied; efficiency in swimming motions of a single organism as well as the apparently synchronized patterns of groups of animals can be analyzed; new propulsion mechanisms can be devised. Undulatory motion, for example, is present in organisms of a wide range of sizes because it is the preferred swimming mode of spermatozoa as well as eels and snakes. The mathematical solution of such problems lies at the crossroads of natural sciences, mathematical modeling and scientific computing.
