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.
Highlights
The mechanics of flexible filaments on the microscale underpin much of biology, from the propulsive flagella of motile bacteria and spermatozoa to nodal cilia, the latter hypothesised to be responsible for the breaking of left–right symmetry in mammals (Gray 1928; Berg & Anderson 1973; Smith, Montenegro-Johnson & Lopes 2019)
For the case of a straight uniform filament aligned parallel to a planar boundary, utilising the approach of § 2.2 we compute the hydrodynamic drag on the slender body as it moves parallel to the wall along its tangent at unit non-dimensional velocity, comparing the solutions given by the methods of regularised Stokeslet segments, free-space resistive force theory (RFT) and wall-corrected RFT, solving separately (2.12) and (2.31) for force density
Good agreement near the boundary can be seen between the wall-corrected resistive force theory (W-RFT) of Katz et al and our implementation of the method of regularised Stokeslet segments, the former as previously validated in the limit h/L → 0 with high-accuracy boundary element methods by Ramia et al (1993)
Summary
The mechanics of flexible filaments on the microscale underpin much of biology, from the propulsive flagella of motile bacteria and spermatozoa to nodal cilia, the latter hypothesised to be responsible for the breaking of left–right symmetry in mammals (Gray 1928; Berg & Anderson 1973; Smith, Montenegro-Johnson & Lopes 2019). The consideration of soft deformable sensors has already motivated extensive studies of attached filaments (Roper et al 2006; Guglielmini et al 2012), as has the characterisation of attached filament forces for understanding the drag induced by slender appendages (Pozrikidis 2011; Curtis et al 2012; Simons et al 2014). Such appendages range from the primary cilium to carbon nanotube mats, with an extensive review of the field presented by du Roure et al (2019), which notes that both theoretical and numerical developments are very much still required in this field.
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