The three-dimensional coarse-graining formulation of interacting elastohydrodynamic filaments and multi-body microhydrodynamics.

Abstract

Elastic filaments are vital to biological, physical and engineering systems, from cilia driving fluid in the lungs to artificial swimmers and micro-robotics. Simulating slender structures requires intricate balance of elastic, body, active and hydrodynamic moments, all in three dimensions. Here, we present a generalized three-dimensional (3D) coarse-graining formulation that is efficient, simple-to-implement, readily extendable and usable for a wide array of applications. Our method allows for simulation of collections of 3D elastic filaments, capable of full flexural and torsional deformations, coupled non-locally via hydrodynamic interactions, and including multi-body microhydrodynamics of structures with arbitrary geometry. The method exploits the exponential mapping of quaternions for tracking 3D rotations of each interacting element in the system, allowing for computation times up to 150 times faster than a direct quaternion implementation. Spheres are used as a 'building block' of both filaments and solid microstructures for straightforward and intuitive construction of arbitrary three-dimensional geometries present in the environment. We highlight the strengths of the method in a series of non-trivial applications including bi-flagellated swimming, sperm-egg scattering and particle transport by cilia arrays. Applications to lab-on-a-chip devices, multi-filaments, mono-to-multi flagellated microorganisms, Brownian polymers, and micro-robotics are straightforward. A Matlab code is provided for further customization and generalizations.