An improved slender-body theory for Stokes flow
TL;DR: In this article, the authors examined the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects, and showed that the boundary condition is satisfied up to an error term of O(e2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centreline.
Abstract: The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (e) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(e2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(e2). The O(e2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.
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TL;DR: The biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming, tens of micrometers and below are reviewed, with emphasis on the simple physical picture and fundamental flow physics phenomena in this regime.
Abstract: Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming, tens of micrometers and below. At this scale, inertia is unimportant and the Reynolds number is small. Our emphasis is on the simple physical picture and fundamental flow physics phenomena in this regime. We first give a brief overview of the mechanisms for swimming motility, and of the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming such as resistance matrices for solid bodies, flow singularities and kinematic requirements for net translation. Then we review classical theoretical work on cell motility, in particular early calculations of swimming kinematics with prescribed stroke and the application of resistive force theory and slender-body theory to flagellar locomotion. After examining the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers and the optimization of locomotion strategies. (Some figures in this article are in colour only in the electronic version) This article was invited by Christoph Schmidt.
2,274 citations
Cites methods from "An improved slender-body theory for..."
...An improvement of the method was later proposed by accurately taking into account end effects and a prolate spheroidal cross-section, with an accuracy of order (a/λ)(2) log(a/λ) [106]....
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TL;DR: The physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies, are reviewed and the hydrodynamic aspects of swimming are addressed.
Abstract: Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.
1,220 citations
Cites methods from "An improved slender-body theory for..."
...The data compare very well with the slender body theories by Lighthill (1976) and Johnson (1980), respectively, and the regularized Stokeslet approach by Cortez et al. (2005)....
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TL;DR: In this article, the authors review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies, including synchronization and the concerted beating of flagella and cilia.
Abstract: Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.
983 citations
TL;DR: In this paper, the authors employ a nonlocal slender body theory that yields an integral equation along the filament centerline, relating the force exerted on the body to the filament velocity.
358 citations
Cites background or result from "An improved slender-body theory for..."
...For details on the derivation, see [14,21,23]....
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...Johnson [21] showed that with this specific choice of the radius (rðsÞ 1⁄4 2e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðL sÞ p ), formula (2) is uniformly accurate all the way out to, and including, the end points of the filament....
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...These two results agree to Oðe(2)Þ to the exact result of Chang and Wu [7] for an ellipsoid, the base shape upon which our slender body theory is based 2, see [21]....
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...This accuracy holds also for a filament with free ends, if the ends are tapered [21]....
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...Assuming that the filament does not reapproach itself, and that the radius of the filament is given by rðsÞ 1⁄4 2e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðL sÞ p , so that rðL=2Þ 1⁄4 eL, a non-local slender body approximation [14,21] of the velocity of the filament centerline is given by...
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TL;DR: The biological structure of the motile sperm appendage, the flagellum, is described and placed in the context of the mechanics underlying the migration of mammalian sperm through the numerous environments of the female reproductive tract.
Abstract: Mammalian spermatozoa motility is a subject of growing importance because of rising human infertility and the possibility of improving animal breeding. We highlight opportunities for fluid and continuum dynamics to provide novel insights concerning the mechanics of these specialized cells, especially during their remarkable journey to the egg. The biological structure of the motile sperm appendage, the flagellum, is described and placed in the context of the mechanics underlying the migration of mammalian sperm through the numerous environments of the female reproductive tract. This process demands certain specific changes to flagellar movement and motility for which further mechanical insight would be valuable, although this requires improved modeling capabilities, particularly to increase our understanding of sperm progression in vivo. We summarize current theoretical studies, highlighting the synergistic combination of imaging and theory in exploring sperm motility, and discuss the challenges for future observational and theoretical studies in understanding the underlying mechanics.
332 citations
Cites background or methods from "An improved slender-body theory for..."
...An alternative representation is based on modeling the flagellum as a slender ellipsoid with cross-sectional radius ā(s 21 − s 2)1/2, with s1 and ā constant; then one has ghyd = −ā2(s 21 − s 2)f hyd/[4μ], as derived by Johnson (1980) and used in sperm modeling by Smith et al. (2009b)....
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...In detail, resistive-force theory can be interpreted as a logarithmically accurate local approximation, which treats the ratio of the flagellum radius to its bending radius of curvature as a small parameter ( Johnson 1980)....
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...This early work has since motivated the development of slenderbody theory for Stokes flow (Lighthill 1976, Johnson 1980) and has influenced later advances such as the boundary integral method (Youngren & Acrivos 1975) and regularized stokeslets (Cortez 2001)....
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01 Jan 1973
TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Abstract: Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.
5,427 citations
TL;DR: In this article, the Stokeslet strength density of a rigid body is estimated to be independent of the body shape and is of order μUe, where U is a measure of the undisturbed velocity and e = (log 2l/R0)−1.
Abstract: A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUe, where U is a measure of the undisturbed velocity and e = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of e2 are presented, together with iterative relations which allow their determination to the order of any power of e. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of e, and, for a cylindrical body, to the order of e3.
965 citations
TL;DR: In this article, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.
Abstract: 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.
759 citations
TL;DR: In this paper, a solid long slender body is placed in a fluid undergoing a given undisturbed flow, and the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio of the cross-sectional radius to body length.
Abstract: A solid long slender body is considered placed in a fluid undergoing a given undisturbed flow. Under conditions in which fluid inertia is negligible, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio of the cross-sectional radius to body length. Specific examples are given for the resistance to translation of long slender bodies for cases in which the body centre-line is curved as well as for those for which the centre-line is straight.
694 citations
TL;DR: In this article, the Stokeslet is associated with a singular point force embedded in a Stokes flow and other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them.
Abstract: The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they
include the Stokeson and its derivatives, called the roton and stresson.
These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyperbolic
profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.
484 citations