Efficient implementation of elastohydrodynamics via integral operators

Abstract

The dynamics of geometrically non-linear flexible filaments play an important role in a host of biological processes, from flagella-driven cell transport to the polymeric structure of complex fluids. Such problems have historically been computationally expensive due to numerical stiffness associated with the inextensibility constraint, as well as the often non-trivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of nonlocal interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a nonlocal bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse-graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant non-dimensional parameters. From these experiments, insight is gained into how filament-filament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm-and sperm-like swimmers for different governing parameters. The MATLAB R implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows.

Doing more with less: The flagellar end piece enhances the propulsive effectiveness of human spermatozoa

“…Following [19,20], we parameterize the filament by arclength s, with s = 0 corresponding to the head-flagellum joint and s = L to the distal end of the flagellum, and apply force and moment free boundary conditions at s = L to get…”

“…The problem is numerically discretized as described by Hall-McNair et al [20], accounting for nonlocal hydrodynamics via the method of regularized stokeslets [22]. This framework is modified to take into account the presence of the head via the nearest-neighbor discretization of Gallagher & Smith [24].…”

Filament mechanics in a half-space via regularised Stokeslet segments

“…We now proceed to combine the work of Moreau et al (2018) and Cortez (2018) to give an efficient scheme for solving non-local planar elastohydrodynamics, similar in concept to the piecewise-constant force density approach of Hall-McNair et al (2019) but with continuous piecewise-linear force discretisation and the additional inclusion of an infinite planar boundary. As formulated by Moreau et al (2018), we integrate the pointwise conditions of force and moment balance on the filament, given by…”

“…Both the calculation of force densities from kinematic data and the solution of full elastohydrodynamics were implemented in MATLAB, the latter utilising the inbuilt stiff ODE solver ode15s (Shampine & Reichelt 1997). Prior to the recent work of Hall-McNair et al (2019) the solution of non-local elastohydrodynamics has required significant computational work and minimal timestep for the solution of this stiff problem (Ishimoto & Gaffney 2018;Olson et al 2013), and we replicate in our implementation the low computational cost associated with the integrated elasticity equations of Moreau et al (2018). In particular, typical simulations of a cantilevered filament in background flow, explored in detail in sections 3.4 and 3.5, have a typical runtime of 10s, where N = 40 and we simulate over 10 periods of oscillation of the background flow on modest hardware (Intel R Core TM i7-6920HQ CPU).…”

“…Numerical simulations that attempt to move past slender body theory are frequently plagued by extensive numerical stiffness, such as the regularised Stokeslet simulations of Ishimoto & Gaffney (2018) and Olson et al (2013). The recent advance of Moreau et al (2018) and its subsequent extension by Hall-McNair et al (2019) have sought to address this stiffness, with their methodologies significantly reducing the computational cost associated with filament-fluid interactions via the use of integrated force and moment balance equations, but have not considered even the simplest confined geometry that is an infinite planar wall. Hence our first objective will be to generalise these improved frameworks to dynamics in a half-space bounded by a wall, adapting the recent regularised Stokeslet segment approach of Cortez (2018) for drag calculations and elastohydrodynamics.…”