Mathematical Modelling of Natural Phenomena

Abstract

We propose a hierarchy of mathematical models for the numerical simulation of active thin structures in a viscous fluid and its application to mucociliary transport. Our aim is to simulate large forests of cilia and analyze the collective dynamics arising in the flow, as well as their impact on the efficiency of the mucus transport. In a 3d model we describe the cilia individually and study their joint actions on the fluid. The model is built upon a 3d Stokes problem with singular source terms that represent the action of the 1d cilia on the fluid, including the background flow (making the problem nonlocal). Surface tension between the periciliary layer and the mucus is taken into account. From the 3d model we also derive a 1d space averaged model, describing the dynamics of the mean velocity of the mucus that is propelled by the cilia, hence allowing lower computational costs and still providing useful characterization of the efficiency of the transport. Mathematical properties of the models (existence and uniqueness of solutions in suitable functional spaces) are analyzed. Numerical simulations highlight the influence of critical parameters on the efficiency of the mucociliary transport in the case of dense forests of cilia.