This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.

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.

In this paper we examine the dynamics of an initially stable bubble due to ultrasonic forcing by an acoustic wave. A tissue layer is modelled as a density interface acted upon by surface tension to mimic membrane effects. The effect of a rigid backing to the thin tissue layer is investigated. We are interested in ultrasound contrast agent type bubbles which have immediate biomedical applications such as the delivery of drugs and the instigation of sonoporation. We use the axisymmetric boundary integral technique detailed in Curtiss et a l. (J. Comput. Phys., 2013, submitted) to model the interaction between a single bubble and the tissue layer. We have identified a new peeling mechanism whereby the re-expansion of a toroidal bubble can peel away tissue from a rigid backing. We explore the problem over a large range of parameters including tissue layer depth, interfacial tension and ultrasonic forcing.

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.

In hydraulic and naval engineering, cavitation is often associated with intense vortical structures. In this paper, we discuss developments associated with solid spheres and bubbles near a tip-vortex and a ring vortex. The first part of the paper reviews the mathematical and numerical development of vortex filament methods and representation of solid spheres and bubbles. Equations of motion are based on the Biot-Savart law and the Kelvin impulse. The classic studies of the motion of a sphere towards and parallel to the vortex axis is extended to consider bubble motion in the presence of a line vortex. The second study is concerned with bubble-vortex ring interaction for both small and large bubble examples. The multipole approach used in this paper compares favourably with the numerical experimental studies of Chahine (1995), 'Bubble interactions with vortices', in Fluid Vortices, Kluwer Academic Press, for the prediction of bubble shapes.

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.

Left–right symmetry breaking is critical to vertebrate embryonic development; in many species this process begins with cilia-driven flow in a structure termed the ‘node’. Primary ‘whirling’ cilia, tilted towards the posterior, transport morphogen-containing vesicles towards the left, initiating left–right asymmetric development. Recent theoretical models based on the point-force stokeslet and point-torque rotlet singularities, explaining how rotation and surface-tilt produce directional flow are reviewed. Analysis of image-singularity systems enforcing the no-slip condition shows how tilted rotation produces a far-field ‘stresslet’ directional flow, and how time-dependent point-force and time-independent point-torque models are in this respect equivalent. Associated slender-body-theory analysis is reviewed; this approach enables efficient and accurate simulation of three-dimensional time-dependent flow, time-dependence being essential in predicting features of the flow such as chaotic advection, which have subsequently been determined experimentally. A new model for the nodal flow utilising the regularized stokeslet method is developed, to model the effect of the overlying Reichert’s membrane. Velocity fields and particle paths within the enclosed domain are computed and compared with the flow profiles predicted by previous ‘membrane-less’ models. Computations confirm that the presence of the membrane produces flow-reversal in the upper region, but no continuous region of reverse flow close to the epithelium. The stresslet far-field is no longer evident in the membrane model, due to the depth of the cavity being of similar magnitude to the cilium length. Simulations predict that vesicles released within one cilium length of the epithelium are generally transported to the left via a ‘loopy drift’ motion, sometimes involving highly unpredictable detours around leftward cilia. Particles released just above the cilia tips were not predicted to reach to the extreme edges of the node, but rather are returned to the right by the counterflow. Flow to the right and left of the cilia array is of very small magnitude, suggesting that effective transport of particles to the extremities of the node requires cilia to be distributed all the way to the edges. There is no continuous layer of rightward flow close to the epithelium, except for a region close to the posterior edge of the node. Future work will involve investigating issues such as the precise shape of the node and cilia distribution and the effect of advection and diffusion on morphogens, hence explaining more fully the role of fluid mechanics in this vital developmental process.

Laser-generated cavitation bubbles in a thin liquid layer lead to the formation of fast liquid jets at both the free surface to the liquid layer and as an opposing jet within the collapsing bubble. This paper studies this phenomenom from both an analytical perspecitive, using the Kelvin impulse, and through computational techniques based on the boundary integral method. Output includes bubble and jet shapes and the percentage of the impulse and energy in the jet of the collapsing bubble. Calculations indicate that in excess of 30% of the liquid energy and 60% of the impulse can be found in the jet in the examples considered in this paper.

This paper investigates the behaviour of a non-spherical cavitation bubble in an acoustic standing wave. The study has important applications to sonochemistry and in understanding features of therapeutic ultrasound in the megahertz range, extending our understanding of bubble behaviour in the highly nonlinear regime where jet and toroidal bubble formation may be important. The theory developed herein represents a further development of the material presented in Part 1 of this paper (Wang & Blake, J. Fluid Mech. vol. 659, 2010, pp. 191–224) to a standing wave, including repeated topological changes from a singly to a multiply connected bubble. The fluid mechanics is assumed to be compressible potential flow. Matched asymptotic expansions for an inner and outer flow are performed to second order in terms of a small parameter, the bubble-wall Mach number, leading to weakly compressible flow formulation of the problem. The method allows the development of a computational model for non-spherical bubbles by using a modified boundary-integral method. The computations show that the bubble remains approximately of a spherical shape when the acoustic pressure is small or is initiated at the node or antinode of the acoustic pressure field. When initiated between the node and antinode at higher acoustic pressures, the bubble loses its spherical shape at the end of the collapse phase after only a few oscillations. A high-speed liquid bubble jet forms and is directed towards the node, impacting the opposite bubble surface and penetrating through the bubble to form a toroidal bubble. The bubble first rebounds in a toroidal form but re-combines to a singly connected bubble, expanding continuously and gradually returning to a near spherical shape. These processes are repeated in the next oscillation.

The slender body transform, known in abbreviated form as the S-transform, arises naturally from the asymptotic analysis of very viscous flow around slender bodies. This paper considers some of the properties of the S-transform as well as exploiting the special properties associated with Legendre polynomials to generate a range of slender body shapes with fixed drag. The viable parameter space for these examples is also presented.

Recent mathematical fluid dynamics models have shed light into an outstanding problem in reproductive biology: why do spermatozoa cells show a 'preference' for swimming near to surfaces? In this paper we review quantitative approaches to the problem, originating with the classic paper of Lord Rothschild in 1963. A recent 'boundary integral/slender body theory' mathematical model for the fluid dynamics is described, and we discuss how it gives insight into the mechanisms that may be responsible for the surface accumulation behaviour. We use the simulation model to explore these mechanisms in more detail, and discuss whether simplified models can capture the behaviour of sperm cells. The far-field decay of the fluid flow around the cell is calculated, and compared with a stresslet model. Finally we present some new findings showing how, despite having a relatively small hydrodynamic drag, the sperm cell 'head' has very significant effects on surface accumulation and trajectory.

Cilia-driven flow occurs in the airway surface liquid, in the female and male reproductive tracts and enables symmetry-breaking in the embryonic node. Viscoelastic rheology is found in healthy states in some systems, whereas in others may characterise disease, motivating the development of mathematical models that take this effect into account. We derive the fundamental solution for linear viscoelastic flow, which is subsequently used as a basis for slender-body theory. Our numerical algorithm allows efficient computation of three-dimensional time-dependent flow, bending moments, power and particle transport. We apply the model to the large-amplitude motion of a single cilium in a linear Maxwell liquid. A relatively short relaxation time of just 0.032 times the beat period significantly reduces forces, bending moments, power and particle transport, the last variable exhibiting exponential decay with relaxation time. A test particle is propelled approximately one-fifth as quickly along the direction of cilia beating for scaled relaxation time 0.032 as in the Newtonian case, and mean volume flow is abolished, emphasizing the sensitivity of cilia function to fluid rheology. These results may have implications for flow in the airways, where the transition from Newtonian to viscoelastic rheology in the peri-ciliary fluid may reduce clearance.

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.

An analysis is developed for the behaviour of a cloud of cavitation bubbles during both the growth and collapse phases. The theory is based on a multipole method exploiting a modified variational principle developed by Miles [“Nonlinear surface waves in closed basins”, J. Fluid Mech. 75 (1976) 418–448] for water waves. Calculations record that bubbles grow approximately spherically, but that a staggered collapse ensues, with the outermost bubbles in the cloud collapsing first of all, leading to a cascade of bubble collapses with very high pressures developed near the cloud centroid. A more complex phenomenon occurs for bubbles of variable radius with local zones of collapse, with a complex frequency spectrum associated with each individual bubble, leading to both local and global collective behaviour. 2000 Mathematics subject classification: primary 76T10; secondary 76B10.

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 within vitromeasurements 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.

AbstractAn analysis is developed for the behaviour of a cloud of cavitation bubbles during both the growth and collapse phases. The theory is based on a multipole method exploiting a modified variational principle developed by Miles [“Nonlinear surface waves in closed basins”, J. Fluid Mech.75 (1976) 418–448] for water waves. Calculations record that bubbles grow approximately spherically, but that a staggered collapse ensues, with the outermost bubbles in the cloud collapsing first of all, leading to a cascade of bubble collapses with very high pressures developed near the cloud centroid. A more complex phenomenon occurs for bubbles of variable radius with local zones of collapse, with a complex frequency spectrum associated with each individual bubble, leading to both local and global collective behaviour.

Mathematical modelling of the fluid mechanics of mucociliary clearance (MCC) is reviewed and future challenges for researchers are discussed. The morphology of the bronchial and tracheal airway surface liquid (ASL) and ciliated epithelium are briefly introduced. The cilia beat cycle, beat frequency and metachronal coordination are described, along with the rheology of the mucous layer. Theoretical modelling of MCC from the late 1960s onwards is reviewed, and distinctions between 'phenomenological', slender body theory' and recent 'fluid-structure interaction' models are explained. The ASL consists of two layers. an overlying mucous layer and underlying watery periciliary layer (PCL) which bathes the cilia. Previous models have predicted very little transport of fluid in the PCL compared with the mucous layer. Fluorescent tracer transport experiments on human airway cultures conducted by Matsui et al. [Matsui, H., Randell, S.H., Peretti, S.W., Davis, C.W., Boucher, R.C., 1998. Coordinated clearance of periciliary liquid and mucus from airway surfaces. J. Clin. Invest. 102 (6), 1125-1131] apparently showed equal transport in both the PCL and mucous layer. Recent attempts to resolve this discrepancy by the present authors are reviewed, along with associated modelling findings. These findings have suggested new insights into the interaction of cilia with mucus due to pressure gradients associated with the flat PCL/mucus interface. This phenomenon complements previously known mechanisms for ciliary propulsion. Modelling results are related to clinical findings, in particular the increased MCC observed in patients with pseudohypoaldosteronism. Recent important advances by several groups in modelling the fluid-structure interaction by which the cilia movement and fluid transport emerge from specification of internal mechanics, viscous and elastic forces are reviewed. Finally, we discuss the limitations of existing work, and the challenges for the next generation of models, which may provide further insight into this complex and vital system. (C) 2008 Elsevier B.V. All rights reserved.