Slender Body Interactions for Low Reynolds Numbers—Part  I:  Body-Wall Interactions

Barta, E; Liron, N.

doi:10.1137/0148058

Cited by 28 publications

(23 citation statements)

References 23 publications

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“…In doing so he also created a set of equations that were accurate to order (r f / ) 2 and showed that f (s) determined the force per unit length to order r 2 f log(r f / )/ 2 . Slender body theories have proved crucial in understanding the hydrodynamics and rheology of filaments (Tornberg & Shelley 2004;Tornberg & Gustavsson 2006), the swimming of microorganisms (Barta & Liron 1988;Myerscough & Swan 1989;Smith et al 2009;Koens & Lauga 2014), and the behaviour of active systems (Guo et al 2018).…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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 .

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Section: Introductionmentioning

confidence: 99%

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

2018

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“…The above results suggest that there are configurations where it is important to take hydrodynamic interactions into account in order to provide a more accurate prediction of the rotational velocities than the classical RF theory. As far as we know, theoretical efforts have been reported to address the hydrodynamic interactions between a slender body and a wall (Mestre 1973;Russel et al 1977; Barta & Liron 1988) or two walls (Katz 1974). Based on the classical RF theory, we hereby make an attempt by utilising a theoretical framework Man et al (2016) recently developed for two interacting weakly deformable filaments that are in parallel in their relaxed state (see figure 6(a)).…”

Section: Theoretical Frameworkmentioning

confidence: 99%

Rotation of a low-Reynolds-number watermill: theory and simulations

Zhu

Stone

2018

J. Fluid Mech.

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Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven "watermill" at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally-spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill's instantaneous rotational velocity as a function of its rod number N , position and orientation. When N 4, the RF theory predicts that the watermill's rotational velocity is independent of N and its orientation, implying the full rotational symmetry (of infinity order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on N and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.

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“…This parametrisation allows the rigid-body hydrodynamics of a ribbon torus to be found from Eq. (19). Furthermore f 2 is neglected for all these calculations as it always of order b or higher.…”

Section: B Rigid-body Motion Of a Ribbon Torusmentioning

confidence: 99%

“…Johnson's SBT has been very successful in capturing the hydrodynamics of slender filaments in a variety of settings [10,[15][16][17][18][19] and can be used to determine the hydrodynamics of thin prolate ellipsoids [12,16] and slender tori [8] analytically. The use of slender-body theory, combined with accurate experimental measurements, has significantly improved our understanding of the motion of swimming microorganisms [10,[20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions to slender-ribbon theory

Koens

Lauga

2017

Phys. Rev. Fluids

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The low-Reynolds number hydrodynamics of slender ribbons is accurately captured by slenderribbon 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. a lmk42@cam.ac.uk b e.lauga@damtp.cam.ac.uk 1 arXiv:1707.05119v1 [physics.flu-dyn]

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Slender Body Interactions for Low Reynolds Numbers—Part I: Body-Wall Interactions

Cited by 28 publications

References 23 publications

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

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

Rotation of a low-Reynolds-number watermill: theory and simulations

Analytical solutions to slender-ribbon theory

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