The Stokesian hydrodynamics of flexing, stretching filaments

Shelley, Michael; Ueda, Tomoya

doi:10.1016/s0167-2789(00)00131-7

Cited by 71 publications

(89 citation statements)

References 26 publications

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“…The first integral in (20) can be evaluated even when δ = 0 because its singular behavior has been explicitly extracted. It is easy to see that in the limit δ → 0, the expression in (20) converges to the one obtained in [16]:…”

Section: 4mentioning

confidence: 99%

“…In computations, although (21) does not contain singularities, the function is nearly singular (or spiky) so that there is a computational advantage to using (20) instead of (21).…”

Section: Ricardo Cortez and Michael Nicholasmentioning

confidence: 99%

“…Therefore the equation can be evaluated at a valid point, say s = 1/2, and the result should be valid for any point on the filament. The forces f (t) and parametrization X (t) are periodic functions but one last modification is necessary because the function 1/ √ t 2 + δ 2 , which appears in the integrand of (20), is not periodic in t. In order to replace it with a periodic function, we use the identity…”

Section: 6mentioning

confidence: 99%

“…We use this formulation, rather than (20), for periodic filaments. We note that in this case, the local terms (outside the integral in (23)) can also be written as 2 .…”

Section: Periodic Filaments Equationmentioning

confidence: 99%

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Slender body theory for Stokes flows with regularized forces

Cortez

Nicholas

2012

CAMCoS

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Existing slender body theories for the dynamics of a thin tube in a Stokes flow differ in the way the asymptotic errors depend on a small parameter defined as the radius of the body over its length. Examples are the theory of Lighthill, that of Keller and Rubinow, and that of Johnson. Slender body theory is revisited here in the more general setting of forces which are localized but smoothly varying within a small neighborhood of the filament centerline, rather than delta distributions along the centerline. Physically, this means that the forces are smoothly distributed over the cross-section of the body. The regularity in the forces produces a final expression that has built-in smoothing which helps eliminate instabilities encountered in computations with unsmoothed formulas. Consistency with standard theories is verified in the limit as the smoothing parameter vanishes, where the original expressions are recovered. In addition, an expression for the fluid velocity at locations off the slender body is derived and used to compute the flow around a filament.

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Section: 4mentioning

confidence: 99%

“…In computations, although (21) does not contain singularities, the function is nearly singular (or spiky) so that there is a computational advantage to using (20) instead of (21).…”

Section: Ricardo Cortez and Michael Nicholasmentioning

confidence: 99%

Section: 6mentioning

confidence: 99%

“…We use this formulation, rather than (20), for periodic filaments. We note that in this case, the local terms (outside the integral in (23)) can also be written as 2 .…”

Section: Periodic Filaments Equationmentioning

confidence: 99%

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Slender body theory for Stokes flows with regularized forces

Cortez

Nicholas

2012

CAMCoS

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“…For more complex environments, these drag laws remain mostly unknown. Under isotropic drag, we have ξ ⊥ = ξ [20][21][22][23]. Here, we keep their values general in the calculations, and show that the final swimming speed of the two-ring swimmer is independent of these coefficients.…”

mentioning

confidence: 95%

Extensibility enables locomotion under isotropic drag

Pak

Lauga

2011

Physics of Fluids

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Anisotropic viscous drag is usually believed to be a requirement for the low Reynolds number locomotion of slender bodies such as flagella and cilia. Here we show that locomotion under isotropic drag is possible for extensible slender bodies. After general considerations, a two-ring swimmer and a model dinoflagellate flagellum are studied analytically to illustrate how extensibility can be exploited for self-propulsion without drag anisotropy. This new degree of freedom could be useful for some complex swimmer geometries and locomotion in complex fluid environments where drag anisotropy is weak or even absent.Due to the absence of inertial forces, low-Reynolds number locomotion is subject to interesting mathematical and physical constraints [1][2][3]. In particular, locomotion by time-reversible strokes is ruled out by Purcell's scallop theorem [4,5]. To escape these constraints, microorganisms swim by either propagating deformation waves along slender appendages, termed flagella, or rotating them. Anisotropic viscous drag is believed to be the fundamental property enabling drag-based propulsion of slender filaments [3,6,7]. It is a classical result that for a slender filament moving in an unbounded Newtonian fluid, the Stokes drag is almost twice when moving perpendicular than parallel to its axis [3,8]. This drag anisotropy allows propulsive forces to be created perpendicularly to the deformation of the filament. Under isotropic drag, it is generally accepted that locomotion of this kind would be impossible [6,7,[9][10][11].Unlike in Newtonian flows, drag laws in more complex media, and their consequences on locomotion, remain largely unexplored. Theoretical studies, via Brinkman models, suggest that porosity enhances drag anisotropy [12,13], explaining, e.g., the increase in propulsion speed of C. elegans in a granular medium. Recent experiments also measured and characterized granular drag in beds of glass beads and granular media [14][15][16][17], which have been applied to study locomotion in sand [18]. Besides the fluid medium, the geometry of a swimming body also plays a role in the drag law. Some flagella, such as those of Ochromonas, possess rigid projections termed mastigonemes, protruding into the fluid [1]. In these geometries the viscous drag in the longitudinal direction of the flagellum is increased, possibly resulting in a more isotropic drag. In situations where drag anisotropy is weak or even absent, what are the alternative mechanisms, if any, offered by physics to achieve locomotion? In this letter, we point out a new degree of freedom enabling inertialess swimming, namely extensibility. Using a general derivation and two simple geometrical models, we demonstrate that the periodic stretching and contraction of a filament allow self-propulsion even under isotropic drag. * Corresponding author. Email: elauga@ucsd.eduWe start by considering the general calculation of Becker et al. [7] showing that drag anisotropy is required for the propulsion of inextensible swimmers. Here we revisit their der...

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Theoretical Justification and Error Analysis for Slender Body Theory

Mori

Ohm

Spirn

2019

Comm Pure Appl Math

149

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Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well-posed boundary value problem for the Stokes equations in R 3 . Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.

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The Stokesian hydrodynamics of flexing, stretching filaments

Cited by 71 publications

References 26 publications

Slender body theory for Stokes flows with regularized forces

Slender body theory for Stokes flows with regularized forces

Extensibility enables locomotion under isotropic drag

Theoretical Justification and Error Analysis for Slender Body Theory

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