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 & 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. † Email: e.lauga@damtp.cam.ac.uk arXiv:1806.04192v1 [physics.flu-dyn] 11 Jun 2018 † The integration factor is obtained by recognising that ∂sS(s, θ) =t(s) + ∂s[ρ(s)êρ(s, θ)], ∂ θ S(s, θ) = ρ(s)∂ θêρ (s, θ), andt × ∂ θêρ = −êρ and performing a Taylor expansion in .

developed swimming strategies that do not apply to the macroscopic world due to the high viscous drag experienced by the body at a length scale where the inertial forces are negligible. [5,6] The ratio between these forces defines the Reynolds number Re, which is on the order of 10 −4 for microorganisms. [5] Two major mechanisms used by micro-organism to swim and stir surrounding fluids are the beating of the cilia and rotating of the helical flagella, which are filaments capable of active bending deformation or rotation. [5][6][7][8][9][10] These filaments driven by internally generated forces work with viscous and elastic forces to generate propulsive thrusts. [11][12][13] The physical balance between elastic forces and viscous drag determines the swimming speed, or pumping frequency and performance. [14] To our best knowledge, a microswimmer based on curling of a 2D spiral is yet to be found in nature, though helical-shaped micro-organisms are abundant, such as spirilla and spirochetes. Curling is a way to store elastic energy and is widely used in engineering, such as piezoelectric devices, [15] energy storage in clock springs, or 3D displacement in artificial muscles. [16] In this report, we present a novel type of microswimmer, i.e., a spiralshaped microgel that is capable of rotating by nonreciprocal deformations. The body of the microswimmer is ribbon-shaped and consists of a cross-linked poly(N-isopropylacrylamide) (PNIPAM) hydrogel laden with gold nanorods (AuNRs). AuNRs are engineered to absorb photon energy in the near infrared and generate localized heat that triggers volume changes of the surrounding hydrogel. [17][18][19] Coated on one surface with a thin metal layer, the bilayer hydrogel ribbon is able to swell and curl in water along its length to form a 2D spiral. [20][21][22] Upon heating, the solubility decrease of the polymer network causes the hydrogel layer to shrink, so that the spiral unwinds. We note here that the bending rigidity of the ribbon and its stiffness are greatly affected by the water content in it. The light-induced temperature-jumps give rise to mechanical response over a few hundred milliseconds, where the rate-limiting factor is the mass transport within the gel. [22][23][24][25][26][27][28] We hypothesize that photothermal heating creates a transient state, where an imbalance exists between the stresses defining the local and the mean curvature of the ribbon. [21] These elastic restoring forces of bending and stretching are counterbalanced by the viscous drag of the surrounding fluid, which sets the rotor in motion. Since the material is relatively soft, the curling dynamics depends strongly on the dissipation mechanisms. [21,29,30] Unlike previous studies that focused on the end equilibrium states of the hydrogel volume-phase transition, we study the response dynamics upon short-lived stimuli here, an aspect thatThe current understanding of motility through body shape deformation of micro-organisms and the knowledge of fluid flows at the microscale provides ample example...

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.While one tends to think of biological cells as stubby, their environment is in fact rich with filamentous structures. Inside cells, polymeric filaments of microtubules, actin, and intermediate filaments fill the eukaryotic cytoplasm [1] and provide it with its mechanical structure [2]. Outside cells, the motion of flagella and cilia allows cells to generate propulsive forces [3][4][5] and induces flows critical to human health [6,7].In all cases, these biological filaments are immersed in a viscous fluid in which they move at low Reynolds number, be it due to their polymerisation, to fluctuations and thermal forces, or to the action of molecular motors [8]. At low Reynolds number, the flows induced locally by the motion of filaments relative to a background fluid have a slow spatial decay as ∼ 1/r [9,10]. In situations where filaments are close to each other, we thus expect nonlocal hydrodynamic interactions to be important [11].Integrating long-ranged hydrodynamic interactions between filaments has long been recognised as a challenging problem, and one where the theoretical approach has consisted of either full numerical simulations or very simplified analysis. A variety of computational methods have been developed to tackle it including slender-body theory [12][13][14], boundary elements to implement boundary integral formulations [15], the immersed boundary method [16,17], regularised flow singularities [18] and particlebased methods [19,20].While these computational approaches allow to address complex geometries and dynamics, the difficulty of integrating long-range hydrodynamic interactions has prevented analytical approaches from providing insight beyond simplified setups. The two most common approaches in biophysics consist in replacing the dynamics in three dimensions by a two-dimensional problem for which the analysis may be easier to carry out [21,22], or by focusing on far-field hydrodynamic interactions and ignoring the geometrical details of near-field hydrodynamics, a popular approach to study synchronisation of flagella and cilia [23][24][25][26][27][28][29] Fig. 6. In each event, the head became the tail (at least once). For an angle change of 0°, the cell exhibited successive reversals; this was rare. For an angle change of 180°, the cell backed up without changing the orientation of its long axis (the head became th...

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.a)

Abstract. Motivated by recent experimental measurements, the passive diffusion of the bacterium 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 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.

Since their development in 2001, regularised stokeslets have become a popular numerical tool for low-Reynolds number flows since the replacement of a point force by a smoothed blob overcomes many computational difficulties associated with flow singularities , SIAM J. Sci. Comput. 23, 1204. The physical changes to the flow resulting from this process are, however, unclear. In this paper, we analyse the flow induced by general regularised stokeslets. An explicit formula for the flow from any regularised stokeslet is first derived, which is shown to simplify for spherically symmetric blobs. Far from the centre of any regularised stokeslet we show that the flow can be written in terms of an infinite number of singularity solutions provided the blob decays sufficiently rapidly. This infinite number of singularities reduces to a point force and source dipole for spherically symmetric blobs. Slowly-decaying blobs induce additional flow resulting from the non-zero body forces acting on the fluid. We also show that near the centre of spherically symmetric regularised stokeslets the flow becomes isotropic, which contrasts with the flow anisotropy fundamental to viscous systems. The concepts developed are used to identify blobs that reduce regularisation errors. These blobs contain regions of negative force in order to counter the flows produced in the regularisation process, but still retain a form convenient for computations. a

Many microorganisms and artificial microswimmers use helical appendages in order to generate locomotion. Though often rotated so as to produce thrust, some species of bacteria such Spiroplasma, Rhodobacter sphaeroides and Spirochetes induce movement by deforming a helical-shaped body.Recently, artificial devices have been created which also generate motion by deforming their helical body in a non-reciprocal way (Mourran et al. Adv. Mater. 29, 1604825, 2017). Inspired by these systems, we investigate the transport of a deforming helix within a viscous fluid. Specifically, we consider a swimmer that maintains a helical centreline and a single handedness while changing its helix radius, pitch and wavelength uniformly across the body. We first discuss how a deforming helix can create a non-reciprocal translational and rotational swimming stroke and identify its principle direction of motion. We then determine the leading-order physics for helices with small helix radius before considering the general behaviour for different configuration parameters and how these swimmers can be optimised. Finally, we explore how the presence of walls, gravity, and defects in the centreline allow the helical device to break symmetries, increase its speed, and generate transport in directions not available to helices in bulk fluids.

We derive a set of equations that describes the shape and behavior 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 extends a previous similar derivation by Svidzinsky and Fetter [Phys. Rev. A 62, 063617 (2000)], and enables a comparison with recent numerical results.