Cavitation and bubble dynamics: the Kelvin impulse and its applications

Cavitation and bubble dynamics have a wide range of practical applications in a range of disciplines, including hydraulic, mechanical and naval engineering, oil exploration, clinical medicine and sonochemistry. However, this paper focuses on how a fundamental concept, the Kelvin impulse, can provide practical insights into engineering and industrial design problems. The pathway is provided through physical insight, idealized experiments and enhancing the accuracy and interpretation of the computation. In 1966, Benjamin and Ellis made a number of important statements relating to the use of the Kelvin impulse in cavitation and bubble dynamics, one of these being 'One should always reason in terms of the Kelvin impulse, not in terms of the fluid momentum…'. We revisit part of this paper, developing the Kelvin impulse from first principles, using it, not only as a check on advanced computations (for which it was first used!), but also to provide greater physical insights into cavitation bubble dynamics near boundaries (rigid, potential free surface, two-fluid interface, flexible surface and axisymmetric stagnation point flow) and to provide predictions on different types of bubble collapse behaviour, later compared against experiments. The paper concludes with two recent studies involving (i) the direction of the jet formation in a cavitation bubble close to a rigid boundary in the presence of high-intensity ultrasound propagated parallel to the surface and (ii) the study of a 'paradigm bubble model' for the collapse of a translating spherical bubble, sometimes leading to a constant velocity high-speed jet, known as the Longuet-Higgins jet.

Modelling the fluid mechanics of cilia and flagella in reproduction and development

Cilia and flagella are actively bending slender organelles, performing functions such as motility, feeding and embryonic symmetry breaking. We review the mechanics of viscous-dominated microscale flow, including time-reversal symmetry, drag anisotropy of slender bodies, and wall effects. We focus on the fundamental force singularity, higher-order multipoles, and the method of images, providing physical insight and forming a basis for computational approaches. Two biological problems are then considered in more detail: 1) left-right symmetry breaking flow in the node, a microscopic structure in developing vertebrate embryos, and 2) motility of microswimmers through non-Newtonian fluids. Our model of the embryonic node reveals how particle transport associated with morphogenesis is modulated by the gradual emergence of cilium posterior tilt. Our model of swimming makes use of force distributions within a body-conforming finite-element framework, allowing the solution of nonlinear inertialess Carreau flow. We find that a three-sphere model swimmer and a model sperm are similarly affected by shear-thinning; in both cases swimming due to a prescribed beat is enhanced by shear-thinning, with optimal Deborah number around 0.8. The sperm exhibits an almost perfect linear relationship between velocity and the logarithm of the ratio of zero to infinite shear viscosity, with shear-thickening hindering cell progress.

Fluid mechanics of nodal flow due to embryonic primary cilia

Breaking of left-right symmetry is crucial in vertebrate development. The role of cilia-driven flow has been the subject of many recent publications, but the underlying mechanisms remain controversial. At approximately 8 days post-fertilization, after the establishment of the dorsal-ventral and anterior-posterior axes, a depressed structure is found on the ventral side of mouse embryos, termed the ventral node. Within the node, 'whirling' primary cilia, tilted towards the posterior, drive a flow implicated in the initial left-right signalling asymmetry. However, the underlying fluid mechanics have not been fully and correctly explained until recently and accurate characterization is required in determining how the flow triggers the downstream signalling cascades. Using the approximation of resistive force theory, we show how the flow is produced and calculate the optimal configuration to cause maximum flow, showing excellent agreement with in vitro measurements and numerical simulation, and paralleling recent analogue experiments. By calculating numerical solutions of the slender body theory equations, we present time-dependent physically based fluid dynamics simulations of particle pathlines in flows generated by large arrays of beating cilia, showing the far-field radial streamlines predicted by the theory.

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous 'singularity models' is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a 'posterior tilt,' and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 mum/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system.

A model of tracer transport in airway surface liquid

This study is concerned with reconciling theoretical modelling of the fluid flow in the airway surface liquid with experimental visualisation of tracer transport in human airway epithelial cultures. The airways are covered by a dense mat of cilia of length approximately 6 microm beating in a watery periciliary liquid (PCL). Above this there is a layer of viscoelastic mucus which traps inhaled pathogens. Cilia propel mucus along the airway towards the trachea and mouth. Theoretical analyses of the beat cycle (smith et al., 2006b; Fulford and Blake, 1986) predict small transport of PCL compared with mucus, based on the assumption that the epithelium is impermeable to fluid. However, an experimental study (Matsui et al., 1998) indicates nearly equal transport of PCL and mucus. Building on existing understanding of steady advection-diffusion in the ASL (Blake and Gaffney, 2001; Mitran, 2004), numerical simulation of an advection-diffusion model of tracer transport is used to test several proposed flow profiles and to test the importance of oscillatory shearing caused by the beating cilia. A mechanically derived oscillatory flow with very low mean transport of PCL results in relatively little 'smearing' of the tracer pulses. Other effects such as mixing between the PCL and mucus, and significant transport in the upper part of the PCL above the cilia tips are tested and result in still closer transport, with separation between the tracer pulses in the two layers being less than 9%. Furthermore, experimental results may be replicated to a very high degree of accuracy if mean transport of PCL is only 50% of mucus transport, significantly less than the mean PCL transport first inferred on the basis of experimental results.

A viscoelastic traction layer model of muco-ciliary transport

A new mathematical model of the transport of mucus and periciliary liquid (PCL) in the airways by cilia is presented. Mucus is represented by a linearly viscoelastic fluid, the mat of cilia is modelled as an 'active porous medium.' The propulsive effect of the cilia is modelled by a time-dependent force acting in a shear-thinned 'traction layer' between the mucus and the PCL. The effects of surface and interface tension are modelled by constraining the mucus free surface and mucus-PCL interface to be flat. It is assumed that the epithelium is impermeable to fluid. Using Fourier series, the system is converted into ODEs and solved numerically. We calculate values for mean mucus speed close to those observed by Matsui et al. [J. Clin. Invest., 102(6):1125-1131, 1998], (approximately 40 microm s(-1)). We obtain more detail regarding the dynamics of the flow and the nonlinear relationships between physical parameters in healthy and diseased states than in previously published models. Pressure gradients in the PCL caused by interface and surface tension are vital to ensuring efficient transport of mucus, and the role of the mucus-PCL interface appears to be to support such pressure gradients, ensuring efficient transport. Mean transport of PCL is found to be very small, consistent with previous analyses, providing insight into theories regarding the normal tonicity of PCL.

Enhanced efficiency of feeding and mixing due to chaotic flow patterns around choanoflagellates

The motion of particles and feeding currents created by micro-organisms due to a flagellum are considered. The calculations are pertinent to a range of sessile organisms, but we concentrate on a particular organism, namely Salpingoeca amphoridium (SA) (a choanoflagellate), due to the availability of experimental data (Pettitt, 2000). These flow fields are characterized as having very small Reynolds numbers, which implies that viscous forces dominate over inertial ones consistent with using the Stokes flow equations. The flow generated by the flagellum is modelled via the consideration of a point force known as a stokeslet. The interaction between the boundary, to which the organism is attached, and its flagellum leads to toroidal eddies, which serve to transport particles towards the micro-organism, promoting filtering of nutrients by the microvilli which constitute the cell's collar (the filtering mechanism in SA). It is our conjecture that the interaction of multiple toroidal eddies will lead to chaotic advection and hence enhance the domain of feeding for these organisms. The degree of mixing in the region around SA is investigated using chaotic and statistical measures to study the influence the flagellum has on the surrounding fluid. The three-dimensional particle paths around such organisms are also considered with the aim of showing that the plane within which they are situated is an attractor.

Hydrodynamics of filter feeding

A fluid mechanical model is developed for the filtering mechanism in mussels that enables estimates to be made of the pressure drop through the gill filaments due to (i) the latero-frontal filtering cilia, (ii) the lateral (pumping) cilia and through the non-ciliated zone at the ventral end of the filament. Calculations indicate that the lateral cilia can generate a sufficient pressure change to 'pump' water through the gill filaments. The velocity profile across the filaments indicates that a backflow can occur in the centre of the channel. Normally the latero-frontal cilia would damp out this backflow but in the case when all the cilia are upright, two 'standing' eddies may form at the mouth of the channel, forcing the incoming water to the side.

Computerised audit for colorectal cancer

An audit has been performed of cases of colorectal cancer presenting over an 8-year period. The results of 242 patients are discussed with emphasis placed on the process of surgical audit. In particular, the ease of data handling by a computer database system is stressed. The figures produced on age, sex, presentation, diagnosis, treatment and outcome have allowed a more detailed and fruitful discussion of our practice at the monthly audit meeting.

The propulsion of mucus by cilia

The presence of cilia on epithelia of the respiratory tract was reported more than 150 yr ago, and the two-layer model of mucus transport was put forward more than 50 yr ago. However, it is only in the last 10 yr or so that the motion of mucus-propelling cilia of the mammalian respiratory system has been adequately described, and fluid dynamic studies have developed far enough to allow descriptions of the mechanisms by which ciliary movement is coupled to mucus transport. In this review, scientific developments on the study of cilia and mucus, and interactions between them, are drawn together to further understanding of mucociliary clearance mechanisms of the respiratory tract. The study of the cilia incorporates a discussion of the internal mechanics and biochemistry of the ciliary axoneme, the physical principles of the beat pattern, and the (weak) metachronal coordination of cilia in the lung. Mucus rheology plays a central role in mucociliary transport with the rheologic properties of the mucus determining the effective functioning of this clearance mechanism. Theoretical models provide information on the mechanical principles of the beat pattern as well as providing reliable estimates of the transport rates. Although airflow is not thought to contribute to mucus transport in the normal state, high frequency ventilation and coughing may make significant contributions.

Muco-ciliary transport in the lung

A two-layer Newtonian fluid model for muco-ciliary transport in the lung is developed where the viscosity of the upper mucous layer is very much greater than the viscosity of the lower periciliary layer. Theory is presented for both cases when the cilia penetrate, and do not penetrate, the very viscous mucous layer. Calculations suggest that, in normal circumstances, it is not essential for the cilia to penetrate the mucus to provide positive transport. However, it does suggest that there is a weak optimal penetration depth of the cilia of between 10-20% of the cilium length. In the case of high ciliary inactivity (e.g. 90% inactive), penetration of cilia into the mucus is essential for normal transport rates suggesting the mucociliary system may be deliberately overdesigned to cater for a whole range of pathological circumstances.

Dermoid cyst: a rare tumour of the appendix

Dermoid cyst is a rare but well recognised cause of a mass in the abdomen. As a cause of a palpable mass in the right iliac fossa, dermoid and epidermoid cysts of the caecum have hitherto been described. We here report a case of a dermoid cyst arising from the vermiform appendix.

A model of ovum transport

A theoretical fluid dynamical model of ovum transport in the oviduct incorporating transport mechanisms due to ciliary activity, muscular activity and an applied pressure drop across the oviduct is developed. Theory suggests that the cilia provide the steady component of ovum transport whereas muscular activity results in highly oscillatory motion. If muscular activity is to provide transport in a pro-uterine direction, a coordinated sequence of muscular activity with a strong pro-uterine bias is needed. Changes in pressure are highest in the narrowest sections. The highly convoluted rugae may allow "leakback" around the ovum so relieving the pressure drop across the ovum in narrower sections of the oviduct.

Endoscopic review of patients who have had gastric surgery

Seventy one patients who had had operations on their stomachs over 15 years previously were examined by endoscopy and multiple mucosal biopsy sampling. Sixty six had histologically proved gastritis (56 chronic atrophic gastritis, 10 superficial gastritis), 38 intestinal metaplasia, and 11 epithelial dysplasia. In three cases the epithelial dysplasia was severe (carcinoma in situ). One patient had an infiltrating carcinoma and another, whose biopsy appearances were reported as severe dysplasia, developed a carcinoma of the stomach eight months later. All patients having undergone gastric surgery more than five years previously should be screened endoscopically and any found to have moderate dysplasia subjected to regular endoscopic screening thereafter. Patients with severe dysplasia (carcinoma in situ) should be considered for radical surgery.

Relationship between primary breast tumor receptor status and patient survival

For a minimum of 3- months, 250 women who underwent mastectomy for primary breast cancer have been followed up. ER status has had a pronounced effect upon disease-free interval and survival: in patients with node involvement ER-positive (ER+) tumors carry a better prognosis. Of patients with ER-positive primary tumors, 43% underwent objective response to their secondaries for a minimum period of six months. This compares with a response of only 18% for ER-negative (ER-) tumors. In patients who had previously received endocrine therapy and on relapse were treated with cytotoxic chemotherapy, objective response rate to chemotherapy was better in patients in whom the primary tumor had been ER-, but not significantly so.

Fluid flow in the human foetal lung: a theoretical model

A theoretical model is developed to predict pressure changes and velocity profiles within the foetal lung during its sporadic bursts of activity. Because of the small volume flow rates and relatively high frequencies, the linearised, unidirectional Navier-Stokes equations are used to calculate these values. About 70% of the pressure drop occurs in the first four generations and is an order of magnitude higher than the equivalent Poiseuille pressure drop. Velocity profiles, pressure falls within each generation together with the total pressure drop at different times during the cycle are illustrated.