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.

Persistent and anti-persistent motion in bounded and unbounded space: resolution of the first-passage problem

“…By relaxing the assumption that all lateral movement directions are uniformly distributed, we introduce the possibility of looping patterns into the movement. We present this as a discrete alternative to the model of 'chiral' persistent motion [30,[61][62][63], where an angular bias is introduced into the turning angle, which is of interest as a model of a charged particle subjected to soft scattering by aligned magnetic domains [30] and particles driven by microscopic torques such as active colloids [64], spermatozoa [65], and chlamydomonas [66]. While chiral motion appears at first to lend itself to a continuous space representation, it is known that even the two-dimensional case leads to an analytical intractable Fokker-Planck equation [30], meaning occupation probabilities and further transport quantities are difficult to obtain.…”

Near-field hydrodynamic interactions determine travelling wave directions of collectively beating cilia

“…Metachronal coordination has been shown to improve the flow generation and energetic efficiency of cilia [5,6] and is found across a wide range of species, from swimming protozoa [7,8], Volvox algal colonies [9,10] and coral larvae [11] to ependymal surfaces in brain ventricles [12] and respiratory epithelia [13]. Theoretical models [5,[14][15][16][17][18][19][20][21] and experimental studies [10,22] have demonstrated that hydrodynamic interactions can explain the synchronization of two or more cilia or flagella. Other studies also point to the role of intracellular basal linkages [23][24][25], probably acting in synergy with fluid-mediated coupling.…”