A Fluid-Structure Interaction Model of Ciliary Beating

Abstract

The coupled system of a viscous, incompressible fluid and a single, force-generating organism is difficult to analyze, even when the kinetics of an organism’s waveform is taken as given. In the past decades, the efforts to describe quantitatively the fluid dynamics of spermatozoa and ciliary propulsion have been very successful. Since the Reynolds number is quite small and inertial effects can be neglected, the linear Stokes flow assumption has been used to investigate the hydrodynamic consequences of flagellar undulations [3]. These investigations have been both analytical and computational. Resistive-force theory, initially developed by Gray and Hancock [12], makes use of the linear Stokes flow assumption, and constructs the flow field by means of distributions of fundamental singularities. Lighthill later improved this theory by incorporating slender body approximations [19], since the diameter of a flagellum is much smaller than its length. More detailed hydrodynamic analysis, such as refined slender body theory and boundary element methods, have produced excellent simulations of both two- and three-dimensional flagellar propulsion in an infinite fluid domain or in a domain with a fixed wall [17], [18], [16], [22]. In addition, ciliary motion, where the effect of a single plane surface at the base of the cilia is included, is studied by Gueron and Liron [14], [15], wherein they also present a thorough survey of flagellar hydrodynamics.

The work of the first author was supported in part by NSF grant DMS-9805501.

The work of the second author was supported in part by NSF grant DMS 9805492 and the NSF Group Infrastructure Grant DMS 9709754.