Filament mechanics in a half-space via regularised Stokeslet segments

Abstract

We present a generalisation of efficient numerical frameworks for modelling fluid-filament interactions via the discretisation of a recently-developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively-simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet segments. This highlights that resistive force theories do not accurately quantify filament dynamics in a range of circumstances, even with analytical corrections for the boundary. However, there is the notable and important exception of movement in a plane parallel to the boundary, where accuracy is maintained. In particular, this justifies the judicious use of resistive force theories in examining the mechanics of filaments and monoflagellate microswimmers with planar flagellar patterns moving parallel to boundaries. We proceed to apply the numerical framework developed here to consider how filament elastohydrodynamics can impact drag near a boundary, analysing in detail the complex responses of a passive cantilevered filament to an oscillatory flow. In particular, we document the emergence of an asymmetric periodic beating in passive filaments in particular parameter regimes, which are remarkably similar to the power and reverse strokes exhibited by motile 9+2 cilia. Furthermore, these changes in the morphology of the filament beating, arising from the fluid-structure interactions, also induce a significant increase in the hydrodynamic drag of the filament. †

Slender phoretic theory of chemically active filaments

“…The second use is in dynamic fluid-structure interaction problems for flexible chemically active filaments. There have been a number of recent advances in efficient simulation elastohydrodynamics combining local drag theory or slender body theory for Stokes flow with elastic beam theory (Moreau, Giraldi & Gadêlha 2018;Hall-McNair et al 2019;Schoeller et al 2019;Walker et al 2019). Taking Hall-McNair et al (2019) as an example, a typical simulation took 0.0007 s per time step, with a single beat of the active filament model (the minimum for interesting dynamics) taking approximately 20 s to resolve, i.e.…”

“…SBT has provided the basis for numerous insights in bioactive flows, for instance cilia-driven symmetry-breaking flow in vertebrates (Smith, Montenegro-Johnson & Lopes 2019), mucociliary clearance (Smith, Gaffney & Blake 2008) and sperm motility (Gaffney et al 2011), amongst others. Indeed, SBT continues to be an invaluable tool for dynamic fluid-filament interaction simulations (Hall-McNair et al 2019;Schoeller et al 2019;Walker et al 2019) where boundary element methods would prove prohibitively costly.…”

Active cloaking in Stokes flows via reinforcement learning

“…Liu, Breuer & Powers 2013;Pimponi et al 2016) and the regularized Stokeslets approach (e.g. Cortez, Fauci & Medovikov 2005;Walker et al 2019). Formulation of the boundary element method for Stokes flows is based on integrating a distribution of Stokeslets over the surface of each swimmer (see, e.g.…”

Regularised non-uniform segments and efficient no-slip elastohydrodynamics

“…(2019) and Walker et al. (2019) to include improved non-local hydrodynamics, applied to the model biological problem of flagellar efficiency (Neal et al. 2020), and extended to motion in three dimensions (Walker, Ishimoto & Gaffney 2020 b ).…”

“…2019; Walker et al. 2019), with circular cross-sections invariably assumed. The general ansatz of (1.1) is also commonly used in conjunction with the hydrodynamic no-slip condition, although is evaluated not on the surface of the body, but on the filament centreline.…”