arXiv subjects

Lyndon Koens

Publications and source records attributed to Lyndon Koens.

28 records · Page 2Linked to original sources

arXiv2018

The boundary integral formulation of Stokes flows includes slender-body theory

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller and Rubinow (1976) and Johnson (1979) using the singularity method. Hence we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore when derived from either the single-layer or double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.

physics.flu-dyn↗

arXiv2017

Analytical solutions to slender-ribbon theory

The low-Reynolds number hydrodynamics of slender ribbons is accurately captured by slender-ribbon theory, an asymptotic solution to the Stokes equation which assumes that the three length scales characterising the ribbons are well separated. We show in this paper that the force distribution across the width of an isolated ribbon located in a infinite fluid can be determined analytically, irrespective of the ribbon's shape. This, in turn, reduces the surface integrals in the slender-ribbon theory equations to a line integral analogous to the one arising in slender-body theory to determine the dynamics of filaments. This result is then used to derive analytical solutions to the motion of a rigid plate ellipsoid and a ribbon torus and to propose a ribbon resistive-force theory, thereby extending the resistive-force theory for slender filaments.

physics.flu-dyn↗

arXiv2017

The non-Gaussian tops and tails of diffusing boomerangs

Experiments involving the two-dimensional passive diffusion of colloidal boomerangs tracked off their centre of mobility have shown striking non-Gaussian tails in their probability distribution function [Chakrabarty et al., Soft Matter 12, 4318 (2016)]. This in turn can lead to anomalous diffusion characteristics, including mean drift. In this paper, we develop a general theoretical explanation for these measurements. The idea relies on calculating the two-dimensional probability densities at the centre of mobility of the particle, where all distributions are Gaussian, and then transforming them to a different reference point. Our model clearly captures the experimental results, without any fitting parameters, and demonstrates that the one-dimensional probability distributions may also exhibit strongly non-Gaussian tops. These results indicate that the choice of tracking point can cause a considerable departure from Gaussian statistics, potentially causing some common modelling techniques to fail.

cond-mat.soft↗

arXiv2016

Hydrodynamic interactions between nearby slender filaments

Cellular biology abound with filaments interacting through fluids, from intracellular microtubules, to rotating flagella and beating cilia. While previous work has demonstrated the complexity of capturing nonlocal hydrodynamic interactions between moving filaments, the problem remains difficult theoretically. We show here that when filaments are closer to each other than their relevant length scale, the integration of hydrodynamic interactions can be approximately carried out analytically. This leads to a set of simplified local equations, illustrated on a simple model of two interacting filaments, which can be used to tackle theoretically a range of problems in biology and physics.

physics.flu-dyn↗

arXiv2016

Microscale Flow Dynamics of Ribbons and Sheets

Numerical study of the hydrodynamics of thin sheets and ribbons presents difficulties associated with resolving multiple length scales. To circumvent these difficulties, asymptotic methods have been developed to describe the dynamics of slender fibres and ribbons. However, such theories entail restrictions on the shapes that can be studied, and often break down in regions where standard boundary element methods are still impractical. In this paper we develop a regularised stokeslet method for ribbons and sheets in order to bridge the gap between asymptotic and boundary element methods. The method is validated against the analytical solution for plate ellipsoids, as well as the dynamics of ribbon helices and an experimental microswimmer. We then demonstrate the versatility of this method by calculating the flow around a double helix, and the swimming dynamics of a microscale "magic carpet".

physics.flu-dyn↗

arXiv2016

Rotation of slender swimmers in isotropic-drag media

The drag anisotropy of slender filaments is a critical physical property allowing swimming in low-Reynolds number flows, and without it linear translation is impossible. Here we show that, in contrast, net rotation can occur under isotropic drag. We first demonstrate this result formally by considering the consequences of the force- and torque-free conditions on swimming bodies and we then illustrate it with two examples (a simple swimmers made of three rods and a model bacterium with two helical flagellar filaments). Our results highlight the different role of hydrodynamic forces in generating translational vs.~rotational propulsion.

physics.flu-dyn↗

arXiv2016

Slender-ribbon theory

Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is bent into a circle it produces a Möbius strip. Significant effort has gone into determining the structural shapes of ribbons but less is know about their behavior in viscous fluids. In this paper we determine, asymptotically, the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows. The derivation, reminiscent of slender-body theory for filaments, assumes that the length of the ribbon is much larger than its width, which itself is much larger than its thickness. The final result is an integral equation for the force density on a mathematical ruled surface, termed the ribbon plane, located inside the ribbon. A numerical implementation of our derivation shows good agreement with the known hydrodynamics of long flat ellipsoids, and successfully captures the swimming behavior of artificial microscopic swimmers recently explored experimentally. We also study the asymptotic behavior of a ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how accurately the hydrodynamics of a ribbon can be effectively captured by that of a slender filament. Our asymptotic results provide the fundamental framework necessary to predict the behavior of slender ribbons at low Reynolds numbers in a variety of biological and engineering problems.

physics.flu-dyn↗

arXiv2014

The passive diffusion of Leptospira interrogans

Motivated by recent experimental measurements, the passive diffusion of the bacterium \textit{Leptospira interrogans} is investigated theoretically. By approximating the cell shape as a straight helix and using the slender-body-theory approximation of Stokesian hydrodynamics, the resistance matrix of \textit{Leptospira} is first determined numerically. The passive diffusion of the helical cell is then obtained computationally using a Langevin formulation which is sampled in time in a manner consistent with the experimental procedure. Our results are in excellent quantitative agreement with the experimental results with no adjustable parameters.

physics.flu-dyn↗

arXiv2012

Vibrations of a Columnar Vortex in a Trapped Bose-Einstein Condensate

We derive a governing equation for a Kelvin wave supported on a vortex line in a Bose-Einstein condensate, in a rotating cylindrically symmetric parabolic trap. From this solution the Kelvin wave dispersion relation is determined. In the limit of an oblate trap and in the absence of longitudinal trapping our results are consistent with previous work. We show that the derived Kelvin wave dispersion in the general case is in quantitative agreement with numerical calculations of the Bogoliubov spectrum and offer a significant improvement upon previous analytical work.

cond-mat.quant-gas↗

arXiv2012

Perturbative behaviour of a vortex in a trapped Bose-Einstein condensate

We derive a set of equations that describe the shape and behaviour of a single perturbed vortex line in a Bose-Einstein condensate. Through the use of a matched asymptotic expansion and a unique coordinate transform a relation for a vortex's velocity, anywhere along the line, is found in terms of the trapping, rotation, and distortion of the line at that location. This relation is then used to find a set of differential equations that give the line's specific shape and motion. This work corrects a previous similar derivation by Anatoly A. Svidzinsky and Alexander L. Fetter [Phys. Rev. A \textbf{62}, 063617 (2000)], and enables a comparison with recent numerical results.

cond-mat.quant-gas↗