‘Early days’ and ‘turbulence’ are vague terms which cover much ground and the following recollections concern the period 1945–1956, referring mostly to experimental work done in G. I. Taylor's old room at the Cavendish Laboratory. My arrival there can be blamed entirely on George Batchelor. We had both been working in the C.S.I.R. Aeronautical Laboratory in Melbourne, George in the Aerodynamics Section and I in the Instruments Section. My work was devising and building gadgets for use in other sections, for wind-tunnel tests, engine tests and so on. One was a very primitive amplifier for hot-wire signals but I did not use it. At the same time, George had been studying the Taylor papers on the statistical theory of isotropic turbulence, and extending it to axisymmetric turbulence. He had in mind coming to Cambridge at the end of the war as a research student of G. 1. Taylor and asked me to consider doing the same. I had interrupted my research in nuclear physics in 1939, intending to return and finish my scholarship, but I found myself signed on to do experimental work on turbulent flows. Why I was accepted is still a mystery, as my ignorance of fluid dynamics, let alone t,urbulence, was almost total.

Most scientists and engineers associate the name of George Batchelor first with the founding and firm editorship of this Journal, then perhaps with the authorship of two widely influential books. Over the years, a considerable number have, as visitors to the Department in Cambridge which he founded and headed for twenty years, had the opportunity for scientific discussion with him, but a much larger number have had their work greatly influenced by the activities of ‘Euromech’, in which GKB was heavily involved from the start in 1964.

Experiments on the rupture of a thin spherical liquid film – formed from a solution of surfactant – are reported. The mean film thickness, prior to rupture, was measured by an electrical conductivity method: initial film thicknesses were of order 0.3–0.9 μm. For an unruptured film, drainage due to gravity reduced the film thickness; the films ruptured naturally at a thickness of order 0.05–0.09 μm.

When the spherical film was punctured by a needle, a hole was formed, which grew rapidly, bounded by a liquid rim. As the rim moved, it collected the liquid from the film; but the rim was itself unstable, generating droplets continuously. The rim velocity, of order 10 m/s, was measured by cine photography at 2000 frames/s. Measured rim velocities compared well with a simple theoretical result derived from either (i) a force balance on the rim or (ii) an energy balance, which demonstrates that there is continuous energy dissipation due to collision between the moving rim and the elements of the stationary film.

When the moving rim had swept up the whole spherical film, much of the rim had disintegrated into droplets, but the remaining rim finally converged to give an ‘implosion’ generating more droplets. These droplets, together with those generated by fragmentation of the rim in flight, were collected: their number, of order 104, was measured by an image analyser, which also measured mean droplet size, of order 102 μm. The total droplet area was a few per cent of the area of the original spherical film.

Details of the shape of the liquid bridge joining a nascent water drop to its parent body are presented for times before, after and at the instant of bifurcation when the drop is created and also when the secondary droplet is formed. After the instant of bifurcation there is ‘unbalanced’ surface tension which gives an impulse to the rest of the fluid causing strong surface deformations. The major point of this work is to draw attention to the strong up–down asymmetry at each bifurcation point. The geometric similarity at each bifurcation instant supports the conjecture that the flow converges to just one similarity solution of the type described by Keller & Miksis (1983) in which only surface tension and inertia are important. Features of the flow before and after bifurcation are discussed.

Wave formation on a thin liquid layer used for de-icing air-plane wings is investigated by studying the stability of air flow over a liquid-coated flat plate at zero angle of incidence. The ratio of the viscosity of the liquid to that of air is very high (over half a million), and the Reynolds number based on liquid depth and air viscosity is of the order of a few thousand in actual practice. Under these circumstances the analysis gives two formulas, in closed form, for the growth rate and phase velocity of the waves in terms of the wavenumber and other relevant parameters, including the Froude number F representing the gravity effect and a parameter S representing the surface-tension effect. In the calculation, the wavenumber is not restricted in any way.

The wavenumber of the waves that one expects to observe is that for which the growth rate is the maximum. The instability is one in which the viscosity difference between the two fluids (air and liquid) plays the dominant role, and is of the kind found by Yih (1967).

We discuss the motion of a triple line for a fluid spreading on a flat solid surface in conditions of partial wetting: the equilibrium contact angle θe is assumed to be finite but small: 0 < θe [Lt ] 1. We distinguish four regions: (1) a molecular domain of size a (≈ a few Ångströms) very near the triple line, where the continuum description breaks down; (2) a proximal region (of length a/θ2e and height a/θe) where the long-range Van der Waals forces dominate; (3) a central region, where capillary forces and Poiseuille friction are the only important ingredients; (4) a distal region where macroscopic features (related to the size of the droplet, or to gravitational forces) come into play. In regions (2, 3, 4) the flow may be described in a lubrication approximation, and with a linearized form of the capillary forces. We restrict our attention to low capillary numbers Ca and expand the profiles to first order in Ca near the static solution. The main results are: (a) the logarithmic singularity which would have occurred in a simple wedge picture is truncated by the long-range forces, at a fluid thickness a/θe. This effect is more important, at small θe, than the effects of slippage which have often been proposed to remove the singularity, and which would lead to a truncation thickness comparable with the molecular size a; (b) in the central region, the local slope θ(x) grows logarithmically with the distance x from the triple line; (c) one can match explicitly the solutions in the central and distal region: we do this for one specific example: a plate plunging into a fluid with an incidence angle exactly equal to θe. In this case we show that, far inside the distal region, the perturbation of the slope decays like 1/x2.

The theory developed aims to explain erratic motions that have been observed experimentally to be performed by small bubbles in liquids irradiated by sound. An exact relation between dynamic and kinematic integral properties of any bubble is used as the basis for calculating the propulsive effects of deformations from spherical shape. The analysis deals first with arbitrary axisymmetric perturbations, such that the equation of the bubble's surface is representable in terms of zonal spherical harmonics, and then more general deformations are treated. It is shown that self-propulsion is accountable wholly to interactions of surface modes n and n+1 (n = 2, 3,…). The resulting velocity W of the bubble's centroid is found to depend on the relative orientation of the interacting modes, |W| being greatest when they are coaxial but the direction of W having the more sensitive dependence. Supported by the theoretical results, an interpretation of the observed erratic motions is presented finally, and a few experimental observations are noted.

The term ‘electroacoustic’ is used in connection with phenomena that involve the interaction of sound waves and electric fields, such as the voltage difference generated across a transducer by the passage of a sound wave. This paper is concerned with electroacoustic effects in suspensions of electrically charged colloidal particles. The existing methods for calculating electroacoustic effects in suspensions are limited either to the dilute case, or to one particular effect, namely the open-circuit voltage generated between two parallel wire probes by a locally plane sound wave, propagating perpendicular to the wires. In this paper we present a procedure for calculating any electroacoustic effects in suspensions of arbitrary concentration. The only restriction on the method is that the particles must be small compared with the sound wavelength. The procedure involves the solution of a set of differential equations – referred to here as the ‘electroacoustic equations’ – for the macroscopic pressure, velocity and electrical potential in the suspension. The derivation of these equations and the associated boundary conditions form the major part of this paper. General features of the solution are also discussed, and an application is given involving a novel electroacoustic measurement.

Two problems of similar mathematical structure are studied: the thermocapillary motion of bubbles and the electrophoresis of colloidal particles. The thermocapillary motion induced in a cloud of bubbles by a uniform temperature gradient is investigated under the assumptions that the bubbles are all the same size that the surface tension is high enough to keep the bubbles spherical, and that the bubbles are non-conducting. In the electrophoresis problem, the particles, identical spheres having a uniform zeta potential, are suspended in an electrolyte under conditions that make the diffuse charge cloud around each particle small when compared with the particle radius. For both problems, it is shown that in a cloud of n particles surrounded by an infinite expanse of fluid, the velocity of each sphere under creeping flow conditions is equal to the velocity of an isolated particle, unchanged by interactions between the particles. However, when the cloud fills a container, conservation of mass shows that this result cannot continue to hold, and the average translational velocity must be calculated subject to a constraint on the mass flux. The computation requires ‘renormalization’, but it is shown that the renormalization procedure is ambiguous in both problems. An extension of Jeffrey's (1974) second group expansion, together with the constraint of conservation of mass, removes the ambiguity. Finally, it is shown that the average thermocapillary or electrophoretic translational velocity of a particle in the cloud is related to the effective conductivity of the cloud over the whole range of particle volume fractions, provided that the particles are identical, non-conducting and, for the thermocapillary problem, inviscid.

In this paper we consider a uniform gas bubble-liquid mixture rising under buoyancy. When the gas volume flux is decreased, while keeping bubble size constant, a smooth transition is formed between the region of lower concentration by volume and the region of initial concentration. This transition travels through the mixture as a permanent wave. We start by discussing the mechanisms which make possible such a permanent wave. The first is its tendency to steepen at the low concentration side. At the root of this is the decrease of the uniform rise velocity, under buoyancy, with increasing concentration. Associated with the motion of the bubbles is the liquid impulse. It is shown that this increases with increasing concentration, producing a reactive force on the bubbles which counteracts buoyancy and reduces the force available to overcome friction. In the transition a balance between these two effects occurs. The internal structure following from this balance is analysed in detail and it is shown that under certain conditions all its properties can be derived from knowledge of the average rise velocities of bubbles in uniform mixtures as a function of concentration.

Measurements on these are reported subsequently, followed by a discussion of our experiments on transitions of the kind mentioned in which velocity, thickness etc. of the waves have been measured. The data are compared with the results of the analysis. Order-of-magnitude agreement is found but there are differences as well, requiring further research.

The steady sedimentation of a suspension over an inclined surface is analysed by considering the combined effects of settling hindrance, bulk motion and particle resuspension. The coupled momentum and mass balances suggest that a thin high-density sediment layer will form over the inclined surface, reminiscent of the thin thermal boundary layers in the classical problem of natural convection. It is shown that for a given value of the particle volume fraction in the unsettled suspension, a steady flow of the sediment can be maintained only if the angle of inclination exceeds a minimum value. The analysis further predicts the existence of a sharp discontinuity in the particle volume fraction across the suspension–sediment interface along which the bulk velocity has a local maximum. High particle volume fractions within the sediment are predicted when the unsettled suspension is either very dilute or very concentrated. This leads to the formation of relatively large sediment-layer thicknesses which reflect the fact that a large body force is required in these two limiting cases to overcome the viscous resistance to flow near the inclined boundary.

A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker–Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Σ can be computed; Vc and D are required in the cell conservation equation, and Σ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ε; λ is the ratio of the rotary diffusion time D-1r to the torque relaxation time B (balancing gravitational and viscous torques), while ε is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker–Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s−1). The continuum model for small ε is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Σ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions.

Elongated-body theory has been fruitfully applied over twenty years to the biofluiddynamic analysis of modes of locomotion of elongated fishes by means of body flexure, with special emphasis on the anguilliform mode using undulatory body movements, and on the carangiform mode where oscillatory movements of only a fish's posterior end(including the caudal fin) exhibit phase lag of posterior movements behind anterior movements just as in an undulation yet not nearly as much as a whole wavelength is apparent at any one time. The extension of elongated-body theory to analyse the locomotion of elongated fishes with elongated median fins (dorsal and/or anal) in modes where the body (together with any caudal fin) remains rigid, being propelled forwards by undulations or oscillations of those median fins, has long been recognized as desirable but is here presented for the first time.

In many large groups of fishes, evolutionary adaptation to limited environments (such as coral reefs) favoured a development of defensive ‘armour’ at the expense of speed, to such an extent that bodies became essentially inflexible, with locomotion achieved by fin movements alone. In one principal group of such fishes, however (the sub-order Balistoidei including the trigger-fishes), a later evolutionary development restored a capacity for relatively high-speed movement even though the body remained essentially rigid. The balistiform mode of locomotion, with propulsion achieved by synchronized movements of the dorsal and anal fins, exists in two alternative forms, with either undulatory or oscillatory movements of these median fins, that are analogous to the anguilliform and carangiform modes of body flexure, respectively.

Analysis in this paper throws light on the puzzling question of why trigger-fishes are able to move so fast notwithstanding the modest extent of their fin movements. A form of the large-amplitude elongated-body theory, specially adapted to balistiform locomotion, allows a direct comparison of thrust and efficiency for different modes of propulsion. The conclusions in brief are that thrust is dominated by the mean rate of shedding of backward momentum at the posterior end of the fish's propulsive apparatus and that, for movements of median fins attached to a deep, essentially rigid body, this momentum is increased (above the momentum expected for the same movements of the fins ‘on their own’) by a momentum enhancement factor β of around 3 or a little more. Yet there is no such enhancement of the rate of shedding of ‘unproductive’ energy into the wake; accordingly, overall efficiency is improved. Also, especially for the undulatory mode of balistiform locomotion, sideforces are minimized so that the fish body avoids sideslip and yaw; accordingly, the body drag which fm thrust must overcome is reduced by another large factor.

Alongside discussion of the Balistoidei, this paper reviews and analyses balistiform locomotion as observed in several other groups, including groups of flexible-bodied fishes that regularly use this mode as a low-energy-cost alternative to locomotion by means of body flexure. Finally, we similarly analyse gymnotiform locomotion, in which the body is again held rigid, being propelled by undulations in just a single (ventral) fin, and compare and contrast different interpretations of its advantages.

Intense fluid motions can be generated by the solidification of a binary liquid. This review paper describes systematically some of the concepts involved in the fluid mechanics of solidification. It also presents quantitative calculations for the fluid motion, the rate of growth of solid and the evolution of both the thermal and the compositional fields in various geometries. The results of many of the calculations are favourably compared with data from laboratory experiments using aqueous solutions.

A binary liquid undergoes unidirectional solidification. The one-dimensional steady state is susceptible to morphological instability that causes the solid/liquid interface to change from a planar state to a cellular pattern. This paper examines the effects on this transition of volume-change convection, buoyancy-driven convection or forced flows. It emphasizes how flows alter stability limits, create scale and pattern changes in morphology, and create, through coupling, new instabilities. Emphasis is placed on the physical mechanisms of the interactions.

The percolation of fluids through rock fabrics or through fracture networks, continued over millions of years, is associated with selective dissolution, cementation, fabric alteration in metamorphosis and the formation of certain massive ore deposits in specific locations. The degree of mineral alteration and its spatial distribution are both controlled by the patterns of interstitial flow and three distinct types of flow-controlled reactions are reviewed and analyzed. Isothermal reaction fronts propagate from mineralogical boundaries in the direction of flow at a speed proportional to but less than the fluid transport velocity; their occurrence can be recognized in banded or bimodal mineralogical patterns. Gradient reactions occurs pervasively throughout a fabric at rates proportional to the temperature and pressure gradients and the fluid velocity; they produce gradually changing mineral assemblages throughout, though their rates of reaction are greatest in high permeability lenses and in thermal boundary layers. Mixing zone reactions occur when two fluid masses intermingle and are usually highly localized. In each case, simple but general analytical expressions are given that express the rates of reaction in terms of the flow and geochemical variables.

As an acid flows through porous rock, it etches the rock and so increases the permeability. This propagating reaction front suffers an instability, rather like the viscous fingering instability, in which the acid prefers to follow high-permeability channels which it has already etched. We have examined the linear stability, obtaining analytic results for small and large wavenumbers and for small variations of the permeability, and obtaining numerical results in other cases.

Morton's series for fully developed laminar flow through a uniformly heated horizontal pipe is simplified by assuming high Prandtl number, and then extended by computer to 31 terms in powers of a parameter that is the product of the Prandtl, Rayleigh, and Reynolds numbers. As in the analogous problems, treated previously, of flow through a loosely coiled pipe and a slowly rotating pipe, convergence is limited by a conjugate pair of square-root singularities on the imaginary axis. For the global heat flux, an Euler transformation and extraction of the nearest singularity at infinity yield an approximation in good agreement with existing experiment and numerical solution. The Nusselt number is found to grow asymptotically as the $\frac{2}{15}$-power of the parameter, whereas boundary-layer analyses have suggested a 1/5-power.

This paper describes the fluid mechanics of the natural ventilation of a space connected to a large body of stationary ambient fluid. The flows are driven by buoyancy differences between the interior and exterior fluids. Connections with the ambient fluid are high level and low level openings. Two main forms of ventilation are identified: mixing ventilation and displacement ventilation. Mixing ventilation occurs when the incoming ambient fluid mixes with the fluid within the space, as is, the case if dense fluid enters through a high level inlet. In this case vertical stratification is weak. Displacement ventilation occurs when dense fluid enters at low levels and displaces the lighter fluid within the space out through high level openings. A strong stable stratification develops in this case, and there is little mixing between the incoming fluid and that in the interior. Both of these modes of ventilation are studied theoretically and the results are compared with laboratory experiments. Transient draining flows which occur when a space initially contains fluid of a density different from the ambient are examined.

The presence of internal sources of buoyancy allows steady states to be established, and the effects of point, line and vertically distributed sources are studied. These steady states are extensions of filling box models, with the addition of continuous exchange of fluid with the environment outside the space. A major result of this work is that the form of the stratification within the space depends on the entrainment caused by the convective elements (plumes) produced by the buoyancy sources, but is independent of the strength of the sources. The strength of the stratification and the magnitudes of the velocities do, however, depend on the source strength. The effects of opening size(s) and configurations are determined, and criteria for producing a particular stratification within the space are established. Applications of this work to the ventilation of buildings are presented.

The general incompressible flow uQ(x), quadratic in the space coordinates, and satisfying the condition uQ = · n = 0 on a sphere r = 1, is considered. It is shown that this flow may be decomposed into the sum of three ingredients – a poloidal flow of Hill's vortex structure, a quasi-rigid rotation, and a twist ingredient involving two parameters, the complete flow uQ(x) then involving essentially seven independent parameters. The flow, being quadratic, is a Stokes flow in the sphere.

The streamline structure of the general flow is investigated, and the results illustrated with reference to a particular sub-family of ‘stretch–twist–fold’ (STF) flows that arise naturally in dynamo theory. When the flow is a small perturbation of a flow u1(x) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant’ associated with the perturbation field. When the flow u1 is dominated by its twist ingredient, the particles can migrate from one such surface to another, a phenomenon that is clearly evident in the computation of Poincaré sections for the STF flow, and that we describe as ‘trans-adiabatic drift’. The migration occurs when the particles pass a neighbourhood of saddle points of the flow on r = 1, and leads to chaos in the streamline pattern in much the same way as the chaos that occurs near heteroclinic orbits of low-order dynamical systems.

The flow is believed to be the first example ofa steady Stokes flow in a bounded region exhibiting chaotic streamlines.