An Immersed Boundary-Lattice Boltzmann Method for Swimming Sperms

This work presents an immersed boundary–lattice Boltzmann method for free-swimming sperms in an incompressible viscous fluid. This method involves sperm model, fluid solver and fluid– structure interaction (FSI). The sperm is simplified as a nonlinear beam. To drive the swimming motion, an internal moment is introduced into the beam equation. The fluid dynamics is obtained by solving the lattice Boltzmann equation. The FSI is handled by the immersed boundary method. Two numerical examples, one single sperm swimming in quiescent viscous fluid and two sperms in tandem arrangement, are presented to illustrate the efficiency of the present method. Introduction During their journey up the oviducts, only the few strongest sperms (out of hundreds of millions) will ever reach their destination. The winners have to swim in the right direction over distances that are around 1,000 times their own length in an extremely viscous fluid. During this journey, the sperms are exposed to different currents, and thus may utilise the hydrodynamic interaction benefit from other sperms around them or the oviduct walls so that they have a better chance to reach their destination. The underlying flow physics is complex and still has not been fully understood. Therefore, a numerical model for free-swimming sperms is a demanding task. Among a variety of numerical methods (see Ref. [4] for a brief review), the immersed boundary method (IBM), first developed by Peskin [9], is extremely efficient for sperm swimming. An IBM was developed by Fauci and McDonald [6] to study the fluid dynamics of sperm motility near both rigid and elastic walls. Dillon et al. [5] developed a model to consider the motility of sperm flagella and cilia based on a common axonemal structure. More recently, Qin et al. [11] numerically studied the motion of a small swimmer in a viscous fluid by using the IBM. In their model, the force of the interaction between the swimmer and the fluid was calculated with a feedback law. With a prescribed traveling wave beating, the swimmer was propelled by the interaction force according to Newton’s second law. Similar dynamic model has been extensively used for fish propulsion simulation [4, 13, 17, 21]. In the present work, we develop an immersed boundary–lattice Boltzmann method (IB-LBM) for free-swimming sperms in an incompressible viscous fluid. Mathematical formulation and numerical method Sperm dynamics The geometrically nonlinear motion for the sperm is described as