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Register a new userAsymptotic theory of hydrodynamic interactions between slender filaments
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Added by
Maria T\\u{a}tulea-Codrean
Publication date
2021
fields
Physics
and research's language is
English
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Hydrodynamic interactions (HIs) are important in biophysics research because they influence both the collective and the individual behaviour of microorganisms and self-propelled particles. For instance, HIs at the micro-swimmer level determine the attraction or repulsion between individuals, and hence their collective behaviour. Meanwhile, HIs between swimming appendages (e.g. cilia and flagella) influence the emergence of swimming gaits, synchronised bundles and metachronal waves. In this study, we address the issue of HIs between slender filaments separated by a distance larger than their contour length (d>L) by means of asymptotic calculations and numerical simulations. We first derive analytical expressions for the extended resistance matrix of two arbitrarily-shaped rigid filaments as a series expansion in inverse powers of d/L>1. The coefficients in our asymptotic series expansion are then evaluated using two well-established methods for slender filaments, resistive-force theory (RFT) and slender-body theory (SBT), and our asymptotic theory is verified using numerical simulations based on SBT for the case of two parallel helices. The theory captures the qualitative features of the interactions in the regime d/L>1, which opens the path to a deeper physical understanding of hydrodynamically governed phenomena such as inter-filament synchronisation and multiflagellar propulsion. To demonstrate the usefulness of our results, we next apply our theory to the case of two helices rotating side-by-side, where we quantify the dependence of all forces and torques on the distance and phase difference between them. Using our understanding of pairwise HIs, we then provide physical intuition for the case of a circular array of rotating helices. Our theoretical results will be useful for the study of HIs between bacterial flagella, nodal cilia, and slender microswimmers.
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Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent the flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob therefore removing the divergent behaviour. However it is unclear how best to regularize the singularities to minimize errors. In this paper we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the non-slip boundary conditions on the bodies surface to leading order, with one of the most commonly used regularizations showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs { and we construct one such example blob which could be effectively used to simulate the flow around a slender body
The viscous drag on a slender rod by a wall is important to many biological and industrial systems. This drag critically depends on the separation between the rod and the wall and can be approximated asymptotically in specific regimes, namely far from, or very close to, the wall, but is typically determined numerically for general separations. In this note we determine an asymptotic representation of the local drag for a slender rod parallel to a wall which is valid for all separations. This is possible through matching the behaviour of a rod close to the wall and a rod far from the wall. We show that the leading order drag in both these regimes has been known since 1981 and that they can used to produce a composite representation of the drag which is valid for all separations. This is in contrast to a sphere above a wall, where no simple uniformly valid representation exists. We estimate the error on this composite representation as the separation increases, discuss how the results could be used as resistive-force theory and demonstrate their use on a two-hinged swimmer above a wall.
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