The self-propulsion of microscopic organisms through liquids
24 Mar 1953-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 217, Iss: 1128, pp 96-121
TL;DR: In this article, it is shown that the propulsion of a single filament which forms itself into a single wave is very near to that of an infinite filament with the same wave motion, where the velocity field can be described in terms of singularities situated at the centre of the sphere.
Abstract: Since the Reynolds number of motion of microscopic organisms through liquids, defined as L ρ V /μ, where L is the length of the organism, V the velocity with which it moves, ρ the density of the liquid and μ the viscosity, is small, propulsion is due predominantly to the viscous forces, the effect of the inertial forces being negligible. The best-known problem that neglects all inertial forces is Stokes’s solution for the slow steady fluid motion past a sphere, in which the velocity field can be described in terms of singularities situated at the centre of the sphere. The movement of microscopic organisms is determined by placing distributions of these singularities inside the surface of the organism and satisfying all boundary conditions. The motions that are considered are restricted to organisms which propagate some kind of disturbance along filaments of circular cross-section with small radius. The first problem to be considered is that of an infinite thin filament along which are propagated plane waves of lateral displacement. Formulae for the velocity of propulsion are obtained for (i) the limiting case of zero radius and (ii) the case when the amplitude of the displacement is small compared to the wave-length. Computations have been carried out to estimate the propulsion in the case of small non-zero filament radius when the amplitude is larger than that allowed for in case (ii) above. It is also shown that the propulsion of a finite filament which forms itself into a single wave is very near to that of an infinite filament with the same wave motion. The second problem is that of an infinite filament along which any general three-dimensional disturbance is propagated. The movement is then deduced for the propagation of a spiral wave along an infinite filament, and also for the propagation of longitudinal waves along a finite filament.
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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 background or methods from "The self-propulsion of microscopic ..."
...Applying this approach to a sine wave with wavelength λ, Gray and Hancock found [31, 32] ξ|| = 2πη ln(2λ/a) − 12 , (32) ξ⊥ = 2ξ||....
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...The idea, termed slender-body theory and pioneered by Hancock [30], is to take advantage of the slenderness of the filaments and replace the solution for the dynamics of the three-dimensional of the filament surface by that of its centerline using an appropriate...
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...Applying this approach to a sine wave with wavelength λ leads to [30, 12]...
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...We then discuss the classic contributions of Taylor [30], Hancock [31] and Gray [32], who all but started the field more than 50 years ago (section 5); we also outline many of the subsequent works that followed....
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...We then discuss the classic contributions of Taylor [29], Hancock [30] and Gray [31], who all but started the field more than 50 years ago (§5); we also outline many of the subsequent works that followed....
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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 background from "The self-propulsion of microscopic ..."
...istive force theory, when the friction anisotropy is chosen appropriately. This yields the friction anisotropy ?= k= 1:81 0:07. The swimming of sperm has also been analyzed by slender-body theory (Hancock, 1953; Johnson and Brokaw, 1979; Lighthill, 1976) (taking into account the hydrodynamic interactions of dierent parts of the deformed agellum as in Sec.III.A for slender rods) by Higdon (1979). Results ag...
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...Resistive force theory (Gray and Hancock, 1955; Lighthill, 1976) as well as slender body theory (Hancock, 1953; Johnson and Brokaw, 1979; Lighthill, 1976) have been applied to describe propulsion of rotating helical flagella....
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...The swimming of sperm has also been analyzed by slender-body theory (Hancock, 1953; Johnson and Brokaw, 1979; Lighthill, 1976) (taking into account the hydrodynamic interactions of different parts of the deformed flagellum as in Sec....
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TL;DR: Although the amplitude may change as a wave passes along the tail, the propulsive properties of the latter may be expected to be closely similar to those of a tail generating waves of the same average amplitude.
Abstract: 1. The general theory of flagellar propulsion is discussed and an expression obtained whereby the propulsive speed of a spermatozoon can be expressed in terms of the amplitude, wave-length and frequency of the waves passing down the tail of a spermatozoon of Psammechinus miliaris . 2. The expression obtained is applicable to waves of relatively large amplitude, and allowance is made for the presence of an inert head. 3. The calculated propulsive speed is almost identical with that derived from observational data. Unless the head of a spermatozoon is very much larger than that of Psammechinus , its presence makes relatively little difference to the propulsive speed. Most of the energy of the cell is used up in overcoming the tangential drag of the tail. 4. Although the amplitude may change as a wave passes along the tail, the propulsive properties of the latter may be expected to be closely similar to those of a tail generating waves of the same average amplitude.
1,035 citations
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: This review restricts this review primarily to a summary of present understanding of the low-Reynolds-number flows associated with microorganism propulsion and the hydromechanics of ciliary systems.
Abstract: Since the Annual Review of Fluid Mechanics first published a review on microorganism locomotion by Jahn & Votta (1972) considerable progress has been made in the understanding of both the biological and the fluid-mechanical processes involved not only in microorganism locomotion but also in other fluid systems utilizing cilia. Much of this knowledge and research, which has been built on the solid foundation of the pioneering work of Sir James Gray (1928, 1968) and Sir Geoffrey Taylor (1951, 1952a,b), has been reported extensively elsewhere, particularly by Gray (1928, 1968), Sleigh (1962), Lighthill (1975), and Wu, Brokaw & Brennen (1975). The subject is now sufficiently broad that it precludes any exhaustive treatment in these few pages. Rather, we restrict this review primarily to a summary of present understanding of the low-Reynolds-number flows associated with microorganism propulsion and the hydromechanics of ciliary systems. In this introductory section we wish to put such fluid-mechanical studies in biological perspective. Section 2 outlines the present status of low-Reynolds-number slender-body theory, and we discuss the application of this theory to biological systems in the final sections.
908 citations
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TL;DR: In this article, the mean velocity of translation is a t most of the order of the square of the amplitude of the deformations of a spherical deformable body and the mean power required to obtaii a given mean velocity is twenty times that given by Stokes' formula for the uniform motion of a rigid sphere under an external force.
Abstract: A spherical deformable body can swim, at very small Reynolds numbers, by performing small oscillations of shape. However, the mean velocity of translation is a t most of the order of the square of the amplitude of the deformations. Three examples of swimming motions, in &h of which the mapimum surface strain is 1/3, are illustrated in Figures 1, 2 and 3. Even in the moat efficient of the three (Fig. 2), the mean power required to obtaii a given mean velocity is twenty times that given by Stokes' formula for the uniform motion of a rigid sphere under an external force. This ratio varies as the inverse square of the maximum surface strain.
710 citations