A fiftieth birthday encourages one to look both backwards and forwards. Could the behaviour of the infant, in the first year or two of life, give a glimpse of the adult to come? And can the present state of that adult be used to make predictions for the future? Here we refer not to the existence or eminence of the Journal of Fluid Mechanics, both of which seem to us to be perfectly secure, even with the onset of electronic and open-access publishing, but to its content and the nature of fluid mechanics itself. To give substance to our reflections we have looked through volumes 1 and 2 (May 1956 to August 1957) and compared them with the three most recent volumes to hand at the time of writing, numbers 532 to 534 (June to July 2005). The total number of pages in each case was almost the same, though there were 80 papers in the first two volumes and only 49 in the recent three.

We consider a viscous drop, loaded with an insoluble surfactant, spreading over a flat plane that is covered initially with a thin liquid film. Lubrication theory allows the flow to be modelled using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the multi-region asymptotic structure of the spatially one-dimensional spreading flow and derive a simplified ODE model that captures its dominant features at large times. The model includes a version of Tanner's law accounting for a Marangoni flux through the drop's effective contact line, the magnitude of which is influenced by a rarefaction wave in the film ahead of the contact line. Focusing on the neighbourhood of the contact line, we then examine the stability of small-amplitude disturbances with spanwise variation, using long-wavelength asymptotics and numerical simulations to describe the growth-rate/wavenumber relationship. In addition to revealing physical mechanisms and new scaling properties, our analysis shows how initial conditions and transient dynamics have a long-lived influence on late-time flow structures, spreading rates and contact-line stability.

The influence of surface roughness on the slip behaviour of a Newtonian liquid in steady planar shear is investigated using three different approaches, namely Stokes flow calculations, molecular dynamics (MD) simulations and a statistical mechanical model for the friction coefficient between a corrugated wall and the first liquid layer. These approaches are used to probe the behaviour of the slip length as a function of the slope parameter $ka\,{=}\,2 \pi a/\lambda$, where $a$ and $\lambda$ represent the amplitude and wavelength characterizing the periodic corrugation of the bounding surface. The molecular and continuum approaches both confirm a monotonic decay in the slip length with increasing $ka$ but the rate of decay as well as the magnitude of the slip length obtained from the Stokes flow solutions exceed the MD predictions as the wall feature sizes approach the liquid molecular dimensions. In the limit of molecular-scale wall corrugation, a Green–Kubo analysis based on the fluctuation–dissipation theorem accurately reproduces the MD results for the behaviour of the slip length as a function of $a$. In combination, these three approaches provide a detailed picture of the influence of periodic roughness on the slip length which spans multiple length scales ranging from molecular to macroscopic dimensions.

A drop of low viscosity hitting a solid may bounce, provided that the material is highly hydrophobic. As a model of such a situation, we consider here the case of a very hot solid. Then, as discovered by Leidenfrost, a thin layer of vapour sustains the drop, preventing any contact with the substrate. On hitting such a solid, a drop rebounds, and we discuss here the elasticity of the shock. Two very different cases are described: at a large velocity, the weaker the impact velocity, the weaker the elasticity; at a small velocity, a quasi-elastic regime is found. The boundary between the two domains is set by a Weber number, which compares the kinetic and surface energies of the drop, of order unity.

A state of permanent, isothermal non-wetting of a solid surface by a normally wetting liquid may be achieved if the surface moves tangentially to a liquid drop that is pressed against it. Surrounding gas is swept into the space between the liquid and solid creating a lubricating film that prevents wetting. The length scales of the drop and the film are typically three or more orders of magnitude different, making numerical simulation difficult from a resolution standpoint. The present paper focuses on a hybrid approach employing lubrication theory for the thinnest portions of the gas film and numerical simulation for the liquid and outer gas phases.

A small liquid drop can be kept bouncing on the surface of a bath of the same fluid for an unlimited time when this substrate oscillates vertically. With fluids of low viscosity the repeated collisions generate a surface wave at the bouncing frequency. The various dynamical regimes of the association of the drop with its wave are investigated first. The drop, usually a simple ‘bouncer’, undergoes a drift bifurcation when the forcing amplitude is increased. It thus becomes a ‘walker’ propagating at a constant velocity on the interface. This transition occurs just below the Faraday instability threshold, when the drop becomes a local emitter of a parametrically forced wave. A model of the particle–wave interaction accounts for this drift bifurcation. The self-organization of several identical bouncers is also investigated. At low forcing, bouncers form bound states or crystal-like aggregates. At larger forcing, the collisions between walkers reveal that their interaction can be either repulsive or attractive, depending on their distance apart. The attraction leads to the spontaneous formation of orbiting pairs, the possible orbit diameters forming a discrete set. A theoretical model of the non-local interaction resulting from the interference of the waves is given. The nature of the interaction is thus clarified and the various types of self-organization recovered.

The bulk viscosity of a suspension relates the deviation of the trace of the macroscopic or averaged stress from its equilibrium value to the average rate of expansion. For a suspension the equilibrium macroscopic stress is the sum of the fluid pressure and the osmotic pressure of the suspended particles. An average rate of expansion drives the suspension microstructure out of equilibrium and is resisted by the thermal motion of the particles. Expressions are given to compute the bulk viscosity for all concentrations and all expansion rates and shown to be completely analogous to the well-known formulae for the deviatoric macroscopic stress, which are used, for example, to compute the shear viscosity. The effect of rigid spherical particles on the bulk viscosity is determined to second order in volume fraction and to leading order in the Péclet number, which is defined as the expansion rate made dimensionless with the Brownian time scale. A repulsive hard-sphere-like interparticle force reduces the hydrodynamic interactions between particles and decreases the bulk viscosity.

The nature of the stress in a suspension of equal homogeneous spheres all subject to the same force, such as weight, is considered; inertial effects are neglected. This study builds upon some of the well-known work devoted to this problem by the founder of the Journal of Fluid Mechanics, Professor George K. Batchelor. After developing a general theory, the antisymmetric part of the stress tensor is considered in detail. It is shown that, in addition to a term already found by Batchelor and characterized by an axial vector, the antisymmetric stress contains another term characterized by the curl of a polar vector. As a consequence, a suspension will possess, in addition to an axial vortex viscosity, a polar vortex viscosity. Appendix C presents a calculation of the hindrance function for rotation correct to the first order in the particle volume fraction.

We present a mathematical model of the unidirectional solidification of a suspension of hard-sphere colloids. Similarity solutions are obtained for the volume fraction and temperature profiles ahead of a planar solidification front. The highly nonlinear functional dependence of the diffusion coefficient on the volume fraction gives rise to a range of behaviours. For small particles, Brownian diffusion dominates and the system behaviour is reminiscent of binary-alloy solidification. Constitutional supercooling occurs at the interface under certain conditions, leading potentially to an instability in the shape of the interface. For larger particles, Brownian diffusion is weak and the particles form a porous layer above the interface. In this case constitutional supercooling reaches a maximum near the surface of the layer, and the porous medium itself is potentially unstable. In stable systems there exists the possibility of secondary nucleation of ice.

Following a novel realization of low-Reynolds-number swimming (Dreyfus et al., Nature, vol. 436, 2005, p. 862), in which self-assembled filaments of paramagnetic micron-sized beads are tethered to red blood cells and then induced to swim under crossed uniform and oscillating magnetic fields, the dynamics of magnetoelastic filaments is studied. The filament is modelled as a slender elastica driven by a magnetic body torque. The model is applied to experiments of Goubault et al. (Phys. Rev. Lett., vol. 91, 2003, art. 260802) to predict the lifetimes of metastable static filament conformations that are known to form under uniform fields. A second experimental swimming scenario, complementary to that of Dreyfus et al. (2005), is described: filaments are capable of swimming even if not tethered to red blood cells. Yet, if both ends of the filament are left free and the material and magnetic parameters are uniform along its length then application of an oscillating transverse field can only generate homogeneous torques, and net translation is prohibited by symmetry. It is shown that fore–aft symmetry is broken when variation of the bending stiffness along the filament is accounted for by including elastic defects, which produces results consistent with the swimming phenomenology.

Multiple scenarios have been discovered by which laminar flow undergoes a transition to turbulence in Newtonian fluids. Here we show in non-Newtonian fluids a transition sequence to ‘elastic turbulence’ due to elasticity from polymers, with negligible inertia. Multiple dynamic states are found linking the base flow to ‘elastic turbulence’ in the flow between a rotating and stationary disk, including circular and spiral rolls. Also, a surprising progression from apparently ‘chaotic’ flow to periodic flow and then to ‘elastic turbulence’ is found. These transitions are found in experiments where either shear stress or shear rate is incrementally increased and then held at fixed values; the modes found following stable base flow are ‘stationary ring’, ‘competing spirals’, ‘multi-spiral chaotic’ and ‘spiral bursting’ modes, followed then by ‘elastic turbulence’. Each mode has a distinct rheological signature, and accompanying imaging of the secondary-flow field (simultaneous with rheological measurement) reveals kinematic structures including stationary and time-dependent rolls. The time-dependent changes in the secondary-flow structure can be related to the time-dependent viscosity in the case of several of the modes. Finally, the effect of polymer concentration on the transitional pathway modes is studied systematically.

There is wide interest in the role of the endothelial surface layer (ESL) in transmitting blood shear stress to the intracellular cytoskeleton of the endothelial cell. However, very little is known about the mechanical properties of the glycocalyx or the flexural rigidity of the core proteins that comprise it. Vink, Duling & Spaan (FASEB J., vol. 13, 1999, p. A 11) measured the time-dependent restoration of the ESL after it had been nearly completely compressed by the passage of a white blood cell (WBC) in a tightly fitting capillary. Using this initial experiment, Weinbaum et al. (Proc. Natl. Acad. Sci. USA, vol. 100, 2003, p. 7988) predicted that the core proteins have a flexural rigidity EI of 700 pN nm$^{2}$, which is $\sim$1/20 the measured value for an actin filament. However, their analysis assumes small deflections and only the fibre motion is considered. In the present paper we report additional experiments and apply large-deformation theory for ‘elastica’ to describe the restoration of the fibres in a Brinkman medium which absorbs fluid as the ESL expands. We find that there are two phases in the fibre recoil: an initial phase for large compressions where the ESL thickness is $<0.36$ its undisturbed thickness, and the ends of the fibres overlap and are parallel to the capillary wall; and a second phase where the fibres assume a shape that is close to the solutions for an elastic bar with linearly distributed vertical loading. The predicted time-dependent change in thickness of the ESL provides remarkably good agreement with experiment and yields an estimate of 490 pN nm$^{2}$ for the flexural rigidity EI of the core protein fibres, which is unexpectedly close to that predicted by the linear theory in Weinbaum et al. (2003).

It is shown that a vertically vibrated binary granular gas mixture of light and heavy particles can segregate (in the presence of gravity) in such a way that the bottom and top layers are composed mostly of light particles, even if all other parameters (including size) are the same for both species. The corresponding concentration profile possesses certain universal properties. It is also shown that such mixtures can segregate when the only difference between the species is the value of the coefficient of restitution. These findings follow from a set of hydrodynamic equations for granular gas mixtures which we derived from the pertinent Boltzmann equation. The above results comprise the second part of this article, the first part of which is devoted to a brief and somewhat biased review of the main physical properties of granular gases. This includes their (generic) tendency to coagulate into clusters and other micro- and macrostructures. A fundamental property of granular materials in general, and granular gases in particular, is the lack of scale separation; an explanation and some consequences are presented. The answer to the basic question of whether the dynamics of granular gases lends itself to description by (appropriate) hydrodynamic equations seems to be positive, though some restrictions apply.

A modified shallow-water model is presented for the collapse of tall columns of grains. The flow is divided in two parts. Depth-averaged shallow-water equations are applied to a thin horizontally spreading layer which is subjected to Coulombic friction. The falling mass of grains is gradually added to the zone of the initial column during the free-fall time of the column. This ‘rain’ is assumed to have no horizontal momentum. The results obtained here are in agreement with both planar and axisymmetric experiments over a range of aspect ratio $a$. In particular, the runout distance is found to vary as $a^{0.65}$ (planar) and $a^{0.52}$ (axisymmetric). The flow dynamics compares well with discrete simulations which have been successfully compared with experiments.

In the last few decades cooperation among fluid dynamicists and geomorphologists has allowed the construction of a rational framework for the quantitative understanding of several geomorphologic processes involved in the shaping of the Earth's surface. Particular emphasis has been given to the dynamics of sedimentary patterns, features arising from the continuous dynamic interaction between the motion of a sediment-carrying fluid and an erodible boundary. It is this interaction which ultimately gives rise to the variety of natural forms, often displaying a high degree of regularity, observed in rivers, estuaries, coasts, as well as in the deep submarine environment. Theoretical analyses and laboratory experiments have shown that the nature of most of the observed patterns is related to fundamental instability mechanisms whose particular character lies in the fact that it is the mobile interface between the fluid and the erodible boundary, rather than the flow itself, that is unstable. Developments have been general enough to reach the status of a distinct branch of fluid mechanics, geomorphological fluid mechanics. This paper concentrates on the mechanics of fluvial meandering. Our aim is to provide the reader with a systematic overview of the fundamental aspects of the subject, assessing, with the help of recent and novel results, settled as well as unsettled issues.

Gravity currents, driven by horizontal differences in buoyancy, play a central role in fluid mechanics, with numerous important natural and industrial applications. The first quantitative, fluid-mechanical study of gravity currents, by von Kármán in 1940, was carried out before the birth of this Journal; the next important theoretical contribution was in 1968 by Brooke Benjamin, and appeared in this Journal more than a decade after its birth. The present paper reviews some of the material that has built on this auspicious start. Part of the fun and satisfaction of being involved in this field is that its development has been based on both theoretical and experimental contributions, which have at times been motivated and supported by field observations and measurements.

This paper discusses the fluid dynamics of an underfloor air distribution (UFAD) ventilation system in which, in contrast to conventional air-conditioning systems, cool air is delivered from diffusers in the floor rather than from overhead vents. In order to produce more realistic models of UFAD systems, we extend previous work on a simplified system consisting of a single heat source at floor level and a single cooling diffuser, to the case of multiple cooling diffusers and a single heat source located at different heights above the floor. We carry out experiments in which the heat source is represented as a buoyant plume, and the cooling diffusers are modelled using negatively buoyant vertical jets. The experiments show that the properties of the system are determined by the entrainment into the plumes and the negatively buoyant jets. We characterize these entrainment processes by entrainment coefficients, and develop a theoretical model based on layered models of ventilation flows. The model predictions are compared with the laboratory experiments, and used to determine the dependence of these parameters on the cooling load, the ventilation rate and the properties of the cooling diffusers.

Rapidly rotating convection in spherical geometry outside the tangent cylinder is investigated using the quasi-geostrophic approximation. The validity of the approximation is discussed, and numerical simulations using these equations are performed, reaching Ekman numbers, $E$, down to $10^{-6}$. The results are compared with experiments and fully three-dimensional numerical simulations. We find that the inertial scaling developed to study rapidly rotating convection does not represent the Prandtl-number dependence of our results adequately. Instead, we find that even in strongly supercritical situations the dominant wavenumbers at the onset of convection still have a strong influence on the behaviour. We find that the local Péclet number, the product of the typical convective velocity and local convective length scale divided by the thermal diffusivity, is helpful for understanding the dynamics of rapidly rotating convection. We explore the zonal flows driven by Reynolds stresses with no-slip boundaries and explore their Prandtl-number dependence. We also study the convective heat transport at low $E$, and consider the boundary layer structures that can form at large Rayleigh number, slowing down the rate of growth of the Nusselt number with Rayleigh number.

The nonlinear evolution of hydrodynamically unstable flames is studied numerically within the context of a hydrodynamic model, where the flame is confined to a surface separating the fresh mixture from the hot combustion products. The numerical scheme uses a variable-density Navier–Stokes solver in conjunction with a level-set front-capturing technique for the numerical treatment of the propagating front. Unlike most previous studies that were limited to the weakly nonlinear Michelson–Sivashinsky equation valid for small density changes, the present work places no restriction on the density contrast and thus elucidates the effect of thermal expansion on flame dynamics. It is shown that the nonlinear development leads to corrugated flames with a transverse dimension that is significantly larger than the wavelength corresponding to the most amplified disturbance predicted by the linear theory, and which is determined by the overall size of the system. The flame structure consists of wide troughs and relatively narrow cusp-like crests, and propagates ‘steadily’ at a constant speed, larger than the speed of a planar flame. The propagation speed increases as the cells widen, but eventually reaches a constant value that remains independent of the mixture's composition and of the transverse length. The dependence of the incremental increase in speed on thermal expansion is found to be nearly linear; for realistic values of thermal expansion it may be as large as 15% to 20%. In sufficiently large domains the dynamics is found to be extremely sensitive to background noise that may result, for example, from weak turbulence. Small-scale wrinkles appear sporadically on the flame surface and travel along its surface, causing a significant increase in the overall speed of propagation, up to twice the laminar flame speed.

Since the experiments of Monkewitz et al. (J. Fluid Mech. vol. 213, 1990, p. 611), sufficiently hot circular jets have been known to give rise to self-sustained synchronized oscillations induced by a locally absolutely unstable region. In the present investigation, numerical simulations are carried out in order to determine if such synchronized states correspond to a nonlinear global mode of the underlying base flow, as predicted in the framework of Ginzburg–Landau model equations. Two configurations of slowly developing base flows are considered. In the presence of a pocket of absolute instability embedded within a convectively unstable jet, global oscillations are shown to be generated by a steep nonlinear front located at the upstream station of marginal absolute instability. The global frequency is given, within 10% accuracy, by the absolute frequency at the front location and, as expected on theoretical grounds, the front displays the same slope as a $k^-$-wave. For jet flows displaying absolutely unstable inlet conditions, global instability is observed to arise if the streamwise extent of the absolutely unstable region is sufficiently large: while local absolute instability sets in for ambient-to-jet temperature ratios $S \le 0.453$, global modes only appear for $S \le 0.3125$. In agreement with theoretical predictions, the selected frequency near the onset of global instability coincides with the absolute frequency at the inlet. For lower $S$, it gradually departs from this value.