The 3D 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 generalised 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 three-dimensional 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 micro-structures 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.

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

“…We offer a simple and intuitive method for simulating fully 3D interacting elastic filaments coupled via non-local multi-body microhydrodynamics (Matlab code provided [ 47 ]). By using asymptotic integration of momentum balance along each coarse-grained segment [ 12 ], we avoid challenging numerical treatment of high-order fluid–structure interaction PDEs and the numerically stiff Lagrange multipliers required to enforce the inextensibility constraint.…”

“…We use Matlab solver ode15s to handle numerical integration, but note that any ODE solver can be used, highlighting the simplicity of the matrix representation. A Matlab implementation is provided here [ 47 ] to serve as a basis for rapid customization and further generalizations. We hope that the simplicity of the matrix system allows researchers to use a numerical platform of their choice for quick implementation.…”

“…Data accessibility. The Matlab codes used to run the simulations presented in this paper are available on github via the link: https://github.com/ polymaths-lab/3d-coarse-graining-elastohydrodynamics or on Zenodo at: https://zenodo.org/record/7940817 [47].…”

“…The Matlab codes used to run the simulations presented in this paper are available on github via the link: or on Zenodo at: [ 47 ].…”