J. R. Blake

A note on the image system for a stokeslet in a no-slip boundary

The velocity and pressure fields for Stokess flow due to a point force (‘stokeslet’) in the vicinity of a stationary plane boundary are analysed, using Fourier transforms, to obtain the image system required to satisfy the no-slip condition on the boundary. The image system, which is illustrated by diagrams, is found to consist of a stokeslet equal in magnitude but opposite in sign to the initial stokeslet, a stokes-doublet and a source-doublet, the displacement axes for the doublets being in the original direction of the force. The influence of the wall on the near and far fields is discussed. In the far field it is found that a stokeslet aligned parallel to the wall produces a stokes-doublet far-field, whereas a stokeslet normal to the wall produces a combination of a source-doublet and a stokes-quadrupole far-field. Although results can be alternatively derived by the method of Lorentz (7) using a reciprocal theorem, the present method yields much more clearly the form of the image system.

Transient Cavities Near Boundaries. Part 1: Rigid Boundary

The growth and collapse of transient vapour cavities near a rigid boundary in the presence of buoyancy forces and an incident stagnation-point flow are modelled via a boundary-integral method. Bubble shapes, particle pathlines and pressure contours are used to illustrate the results of the numerical solutions. Migration of the collapsing bubble, and subsequent jet formation, may be directed either towards or away from the rigid boundary, depending on the relative magnitude of the physical parameters. For appropriate parameter ranges in stagnation-point flow, unusual ‘hour-glass’ shaped bubbles are formed towards the end of the collapse of the bubble. It is postulated that the final collapsed state of the bubble may be two toroidal bubbles/ring vortices of opposite circulation. For buoyant vapour cavities the Kelvin impulse is used to obtain criteria which determine the direction of migration and subsequent jet formation in the collapsing bubble.

A spherical envelope approach to ciliary propulsion

In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an ‘envelope’ covering the ends of the numerous cilia of the microscopic organism. This approximation may be made in the case when the cilia are close together, as can occur in the case of the symplectic metachronal wave (i.e. the wave travels in the same direction as the effective beat). For simplicity, a spherical model has been chosen, and the analysis which follows is a correction to Lighthills (1952) paper on squirming motions of a nearly spherical organism. The velocity and efficiency compared to the work done in pushing an inert organism are obtained, and compared to that of a ciliated organism.

The final stage of the collapse of a cavitation bubble close to a rigid boundary

The final stage of the collapse of a laser-produced cavitation bubble close to a rigid boundary is studied both experimentally and theoretically. The temporal evolution of the liquid jet developed during bubble collapse, shock wave emission and the behavior of the “splash” effect are investigated by using high-speed photography with up to 5 million frames/second. For a full understanding of the bubble–boundary interaction, numerical simulations are conducted by using a boundary integral method with an incompressible liquid impact model. The results of the numerical calculations provided the pressure contours and the velocity vectors in the liquid surrounding the bubble as well as the bubble profiles. The comparisons between experimental and numerical data are favorable with regard to both bubble shape history and translational motion of the bubble. The results are discussed with respect to the mechanism of cavitation erosion.

Transient cavities near boundaries Part 2. Free surface

Calculations of the growth and collapse of transient vapour cavities near a free surface when buoyancy forces may be important are made using the boundary-integral method described in Part 1. Bubble shapes, particle paths, pressure contours and centroid motion are used to illustrate the calculations. In the absence of buoyancy forces the bubble migrates away from the free surface during the collapse phase, yielding a liquid jet directed away from the free surface. When the bubble is sufficiently close to the free surface, the nonlinear response of the free surface produces a high-speed jet (‘spike’) that moves in the opposite direction to the liquid jet and, in so doing, produces a stagnation point in the fluid between the bubble and the free surface. For sufficiently large bubbles, buoyancy forces may be dominant, so that the bubble migrates towards the free surface with the resulting liquid jet in the same direction. The Kelvin impulse provides a reasonable estimate of the physical parameter space that determines the migratory behaviour of the collapsing bubbles.

A model for the micro-structure in ciliated organisms

Improved models for the movement of fluid by cilia are presented. A theory which models the cilia of an organism by an array of flexible long slender bodies distributed over and attached at one end to a plane surface is developed. The slender bodies are constrained to move in similar patterns to the cilia of the microorganisms Opalina, Paramecium and Pleurobrachia . The velocity field is represented by a distribution of force singularities (Stokes flow) along the centre-line of each slender body. Contributions to the velocity field from all the cilia distributed over the plane are summed, to give a streaming effect which in turn implies propulsion of the organism. From this we have been able to model the mean velocity field through the cilia sublayer for the three organisms. We find that, in a frame of reference situated in the organism, the velocity near the surface of the organism is very small – up to one half the length of the cilium – but it increases rapidly to near the velocity of propulsion from then on. This is because of the beating pattern of the cilia; they beat in a near rigid-body rotation during the effective (‘power’) stroke, but during the recovery stroke move close to the wall. Backflow (‘reflux’) is found to occur in the organisms exhibiting antiplectic metachronism (i.e. Paramecium and Pleurobrachia ). The occurrence of gradient reversal, but not backflow, has recently been confirmed experimentally (Sleigh & Aiello 1971). Other important physical values that are obtained from this analysis are the force, bending moment about the base of a cilium and the rate of working. It is found, for antiplectic metachronism, that the force exerted by a cilium in the direction of propulsion is large and positive during the effective stroke whereas it is small and negative during the recovery stroke. However, the duration of the recovery stroke is longer than the effective stroke so the force exerted over one cycle of a ciliary beat is very small. The bending moment follows a similar pattern to the component of force in the direction of propulsion, being larger in the effective stroke for antiplectic metachronism. In symplectic metachronism (i.e. Opalina ) the force and bending moment are largest in magnitude when the bending wave is propagated along the cilium. The rate of working indicates that more energy is consumed in the effective stroke for Paramecium and Pleurobrachia than in the recovery stroke, whereas in Opalina it is found to be large during the propagation of the bending wave.

Mechanics of ciliary locomotion.

CONTENTS

Human sperm accumulation near surfaces: a simulation study

A hybrid boundary integral/slender body algorithm for modelling flagellar cell motility is presented. The algorithm uses the boundary element method to represent the ‘wedge-shaped’ head of the human sperm cell and a slender body theory representation of the flagellum. The head morphology is specified carefully due to its significant effect on the force and torque balance and hence movement of the free-swimming cell. The technique is used to investigate the mechanisms for the accumulation of human spermatozoa near surfaces. Sperm swimming in an infinite fluid, and near a plane boundary, with prescribed planar and three-dimensional flagellar waveforms are simulated. Both planar and ‘elliptical helicoid’ beating cells are predicted to accumulate at distances of approximately 8.5–22 μm from surfaces, for flagellar beating with angular wavenumber of 3π to 4π. Planar beating cells with wavenumber of approximately 2.4π or greater are predicted to accumulate at a finite distance, while cells with wavenumber of approximately 2π or less are predicted to escape from the surface, likely due to the breakdown of the stable swimming configuration. In the stable swimming trajectory the cell has a small angle of inclination away from the surface, no greater than approximately 0.5°. The trapping effect need not depend on specialized non-planar components of the flagellar beat but rather is a consequence of force and torque balance and the physical effect of the image systems in a no-slip plane boundary. The effect is relatively weak, so that a cell initially one body length from the surface and inclined at an angle of 4°–6° towards the surface will not be trapped but will rather be deflected from the surface. Cells performing rolling motility, where the flagellum sweeps out a ‘conical envelope’, are predicted to align with the surface provided that they approach with sufficiently steep angle. However simulation of cells swimming against a surface in such a configuration is not possible in the present framework. Simulated human sperm cells performing a planar beat with inclination between the beat plane and the plane-of-flattening of the head were not predicted to glide along surfaces, as has been observed in mouse sperm. Instead, cells initially with the head approximately 1.5–3 μm from the surface were predicted to turn away and escape. The simulation model was also used to examine rolling motility due to elliptical helicoid flagellar beating. The head was found to rotate by approximately 240° over one beat cycle and due to the time-varying torques associated with the flagellar beat was found to exhibit ‘looping’ as has been observed in cells swimming against coverslips.

Acoustic cavitation: the fluid dynamics of non–spherical bubbles

In acoustic cavitation the spatial variation and time–dependent nature of the acoustic pressure field, whether it is a standing or propagating wave, together with the presence of other bubbles, particles and boundaries produces gradients and asymmetries in the flow field. This will inevitably lead to non–spherical bubble behaviour, often of short duration, before break–up into smaller bubbles which may act as nuclei for the generation of further bubbles. During the collapse phase, high temperatures and pressures will occur in the gaseous interior of the bubble. This paper concentrates on the non–spherical bubble extension to the earlier spherical–bubble studies for acoustic cavitation by exploiting the techniques that had previously been used to model incompressible hydraulic cavitation phenomena. Bubble behaviour near an oscillating boundary, jet impact and damage to boundaries, bubble interactions, bubble clouds and bubble behaviour near rough surfaces are considered. In many cases the key manifestation of the asymmetry is the development of a high–speed liquid jet that penetrates the interior of the bubble. Jetting behaviour can lead to high pressures, high strain rates (of importance to break–up of macromolecules) and toroidal bubbles, all of which can enhance mixing. In addition it may provide a mechanism for injecting the liquid into the hot bubble interior. Many practical applications such as cleaning, enhanced rates of chemical reactions, luminescence and novel metallurgical processes may be associated with this phenomenon.

Collapsing cavities, toroidal bubbles and jet impact

The present study is aimed at clarifying some of the factors which affect the formation and direction of a liquid jet in a collapsing cavity and the pressures induced on a nearby rigid boundary. The flow can be accurately represented by a velocity potential leading to the use of boundary integral methods to compute bubble collapse. For configurations with axial symmetry, the jet motion and that of the bubble centroid are along the axis of symmetry. Examples are presented for bubbles close to a rigid surface and to a free surface. These are followed through to the toroidal stage after jet penetration. When there is no axis of symmetry, fully three–dimensional computations show that the buoyancy force can cause the jet to move parallel to a vertical rigid boundary, thus negating its damaging effect. The computational study is extended to model cavitation bubble growth and collapse phases in a forward stagnation point flow as a model of reattachment of a boundary layer; a region where severe cavitation damage is often observed. The Kelvin impulse is introduced to aid a better understanding of the mechanics of bubble migration and jet direction in the examples presented. Finally a comparison between the spherical and axisymmetric theories is made for an oscillating bubble in a periodic pressure field; this being of particular interest to current studies in acoustic cavitation and sonoluminescence.