This volume is dedicated to Professor Stephen H. Davis on the occasion of his seventieth birthday. Steve, as he is known to his many friends and colleagues, has served the fluid mechanics community for a good part of those seventy years. He has been distinguished by election to the National Academy of Engineering and the National Academy of Sciences of the USA. A variety of professional societies have honoured Steve, including the American Physical Society, with the Fluid Dynamics Prize in 1994, and the Society of Engineering Sciences, with the G. I. Taylor Medal in 2001. Among a number of named lectureships, he was selected as the G. K. Batchelor Lecturer in 2003 at the University of Cambridge; this is an honour that befits the third scholar to enjoy a dedicated volume of Journal of Fluid Mechanics (JFM), following those dedicated to George Batchelor himself in 1990 (volume 212) and to Philip Saffman in 2000 (volume 409). Steve's editorial services have benefitted the Annual Review of Fluid Mechanics, the Proceedings of the Royal Society of London, SIAM Monographs on Mathematical Modeling and Computation, and the Cambridge University Press Monographs in Mechanics. However, the community to benefit most from Steve's editorial leadership has surely been the authors and readers of JFM. Steve served as an assistant editor and then associate editor for twenty years, from 1969 to 1989, and then returned as editor of the Journal in 2000.

Celebrating the occasion of his 70th birthday, observations are offered on the life and technical contributions of Stephen H. Davis, Editor of the Journal of Fluid Mechanics.

Extreme events such as intense tornadoes and huge floods, though infrequent, are particularly important because of their disproportionate impact. Our ability to forecast them is poor at present. Large events occur also in intermittent features of turbulent flows. Some dynamical understanding of these features is possible because the governing equations are known and can be solved with good accuracy on a computer. Here, we study large-amplitude events of turbulent vorticity using results from direct numerical simulations of isotropic turbulence in conjunction with the vorticity evolution equation. We show that the advection is the dominant process by which an observer fixed to the laboratory frame perceives vorticity evolution on a short time scale and that the growth of squared vorticity during large excursions is quadratic in time when normalized appropriately. This result is not inconsistent with the multifractal description and is simpler for present purposes. Computational data show that the peak in the viscous term of the vorticity equation can act as a precursor for the upcoming peak of vorticity, forming a reasonable basis for forecasts on short time scales that can be estimated simply. This idea can be applied to other intermittent quantities and, possibly, more broadly to forecasting other extreme quantities, e.g. in seismology.

A series of experiments have been carried out on gravity currents released from locks of various dimensions into a sloping, open channel. Initially all the driving heads of the gravity currents grew by addition of heavier material from a following down-slope flow and by entrainment of ambient fluid, as in Maxworthy & Nokes (J. Fluid Mech., vol. 584, 2007, pp. 433–453). After propagating a distance of the order of 5–10 lock lengths the inflow into the rear stopped, and the head began to lose buoyancy-containing fluid from its rear by the detachment of large, weakly vortical structures. At the same time it was still entraining fluid over the majority of its surface so that its mean density was reduced. Measurements using a semi-direct method, in which dye concentration was used as a surrogate for density, have shown that the buoyancy in the current head increased during the first phase and decreased during the second. At no stage was the buoyancy constant except, of course, at the location and time at which the buoyancy was maximum with a magnitude significantly smaller than the initial value in the lock. Despite this the constant buoyancy theory of Beghin, Hopfinger & Britter (J. Fluid Mech., vol. 107, 1981, pp. 407–422), in which the head location x varies with time t as t2/3 during the later velocity-decay stage of the evolution, was found to be remarkably robust as a description of the evolution over both the latter part of the increasing-buoyancy stage and all of the decreasing-buoyancy phase. Critically, however, the multiplying coefficient had to be smaller than presented by them in order to track the experimental data with precision. This was due principally to the observation that the buoyancy at the beginning of the decay phase was considerably smaller than the initial buoyancy in the lock.

We present an experimental and numerical study of one stratified fluid propagating into another. The two fluids are initially at rest in a horizontal channel and are separated by a vertical gate which is removed to start the flow. We consider the case in which the two fluids have the same mean densities but have different, constant, non-zero buoyancy frequencies. In this case the fluid with the smaller buoyancy frequency flows into the other fluid along the mid-depth of the channel in the form of an intrusion and two counter-flowing gravity currents of the fluid with the larger buoyancy frequency flow along the top and bottom boundaries of the channel. Working from the available potential energy of the system and measurements of the intrusion thickness, we develop an energy model to describe the speed of the intrusion in terms of the ratio of the two buoyancy frequencies. We examine the role of the stratification within the intrusion and the two gravity currents, and show that this stratification plays an important role in the internal structure of the flow, but has only a secondary effect on the speeds of the exchange flows.

Results of experimental and numerical studies are presented for a class of horizontal thermal convection in a long box forced at one horizontal boundary by two regions of destabilizing buoyancy flux separated by a region of stabilizing buoyancy flux. The steady-state circulation with zero net heat input is examined. The circulation generally involves two plumes, one at each end of the box, which drive overturning throughout the domain. The flow is classified into three regimes according to the pattern of interior circulation and depending on the relative heat input applied to the two destabilizing regions. Unequal heat inputs can double the interior stratification above that created by symmetric flow with two identical plumes, and when the heat inputs differ by more than 10%, the interior stratification is set by the stronger plume. The arrangement of boundary forcing broadly parallels the distribution of the zonally averaged surface cooling and heating in the Northern and Southern Hemispheres, and the results suggest that ocean overturning circulations may be sensitive to interhemispheric differences in the buoyancy inputs.

Smith & Davis (J. Fluid Mech., vol. 132, 1983, pp. 119–144) considered the stability of thermocapillary convection in a horizontal fluid layer with an upper free surface generated by a horizontal temperature gradient. They showed that for a return-flow velocity profile, the convection will become unstable in the hydrothermal mode with waves propagating upstream obliquely. Their findings provided a theoretical explanation for the defects often found in crystals grown by the floating-zone technique and in thin-film coating processes. Their predictions were verified experimentally by Riley & Neitzel (J. Fluid Mech., vol. 359, 1998, pp. 143–164) in an experiment with 0.75 mm thick layer of silicone oil. Their results with 1 and 1.25 mm thick layers show that as the thickness of the layer is increased, the angle of propagation, the frequency of oscillation and the phase speed of the hydrothermal wave instability decrease, while the wavelength stays nearly constant. We have extended the linear stability analysis of the problem with the effect of gravity included. It is found that when the Grashof number Gr is increased from zero, the angle of propagation first increases slightly, reaches a maximum and then decreases steadily to zero at Gr = 18. The phase speed, the frequency of oscillation and the wavelength of the instability waves all decrease with increasing Grashof number. For Gr larger than 18, there is the onset of the instability into travelling transverse waves. We have also carried out energy analysis at the time of the instability onset. It is found that the major contribution to the energy of the disturbances is from the surface-tension effect. As the gravitational effect is increased, there is a reduction in the kinetic energy supply to sustain the motion of the disturbances. We also found that it requires more kinetic energy to sustain the hydrothermal mode of instability than that required for the travelling transverse mode of instability. As a result, with increasing Grashof number, the kinetic energy available for the disturbances decreases, causing the angle of propagation to gradually decrease until finally reaching zero at Gr = 18.

We perform linear stability calculations for horizontal bilayers of a two-component fluid that can undergo a phase transformation, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. Critical values for the applied temperature difference across the system that is necessary to produce instability are obtained by a linear stability analysis, using both numerical computations and small wavenumber approximations. Thermophysical properties are taken from the aluminum–indium monotectic system, which includes a liquid–liquid miscibility gap. In addition to buoyant and thermocapillary modes of instability, we find an oscillatory phase-change instability due to the combined effects of solute diffusion and fluid flow that persists at small wavenumbers. This mode is sensitive to the ratio of the layer depths, and for certain layer depths can occur for heating from either above or below.

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniform heated substrate. An equation for the thickness profile of the droplet is derived by employing lubrication approximations. The model includes the effect of a non-zero contact angle introduced through a disjoining–conjoining pressure term. Instead of assuming a fixed shape for the droplet, as in previous works, here we allow the droplet to change its profile with time. We identify and describe three different regimes of behaviour. For small contact angles, the droplet spreads into a long film profile with a capillary ridge near the leading edge, a behaviour that resembles the experiments on Marangoni films reported by Ludviksson & Lightfoot (Am. Inst. Chem. Eng. J., vol. 17, 1971, pp. 1166). For large contact angles, the droplet moves as a single entity, weakly distorted from its static shape. This regime is the usual one reported in experiments on thermocapillary migration of droplets. We also show some intriguing morphologies that appear in the transition between these two regimes. The occurrence of these three regimes and their dependence on various parameters is analysed.

The influence of Marangoni stresses, caused by contaminants adsorbed on the surface of small air bubbles, rising in water, is examined by numerical simulations. A modified level set method is used to represent the deformable bubble interface, extended by a model for the contaminant transport on the bubble surface. We show that surface tension variations of less than 2% are sufficient to generate Marangoni stresses that are strong enough to change the rising characteristics of a bubble to that of a corresponding solid particle. In such situations, we find that the bubble surface is fully covered with contaminant and the shear stress profile resembles the shear stress profile around a solid sphere.

A high-velocity (≈1 ms−1) impact between a liquid droplet (≈1 mm) and a solid surface produces a splash. Classical observations traced the origin of this splash to a thin sheet of fluid ejected near the impact point, though the fluid mechanical mechanism leading to the sheet is not known. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. In this paper, we theoretically and numerically study the events within the time scale of about 1 μs over which the coupled dynamics between the gas and the droplet becomes important. The droplet initially tries to contact the substrate by either draining gas out of a thin layer or compressing it, with the local behaviour described by a self-similar solution of the governing equations. This similarity solution is not asymptotically consistent: forces that were initially negligible become relevant and dramatically change the behaviour. Depending on the radius and impact velocity of the droplet, we show that the solution is overtaken by initially subdominant physical effects such as the surface tension of the liquid–gas interface or viscous forces in the liquid. At low impact velocities surface tension stops the droplet from impacting the surface, whereas at higher velocities viscous forces become important before surface tension. The ultimate dynamics of the interface once droplet viscosity cannot be neglected is not yet known.

We rederive and expand upon a method for finding solutions to the two-dimensional irrotational (inviscid) flow equations in the presence of a free surface, found by Hopkinson. This method allows the flow to be driven by placing singularities, like sources or vortices, in the interior of the flow domain. We then apply the method to find a number of novel solutions: separated flow driven by a source, vortices behind a plate and free-surface flow stirred by a double vortex. Free surfaces generically exhibit cusp singularities with a 2/3 power index, similar to those found in very viscous flow.

Surface tension acting at a cylindrical interface holds an underlying liquid in motionless equilibrium. This static base state is subject to dynamic capillary instability, including Plateau–Rayleigh breakup. If the interface is partially supported by a cylindrical cup-like solid, the extent of the wetting contact can significantly influence the dynamics and the stability of the configuration. The equation for the motion of small disturbances is formulated as an eigenvalue equation on linear operators. A solution is constructed on a constrained function space using a Rayleigh–Ritz procedure. The influence of the extent-of-constraint on the dispersion relation and on modal structures is reported. In the extreme, the support reduces to a wire, aligned axially, and just touching the interface. From prior work, this constraint is known to stabilize the Plateau–Rayleigh limit by some 13%. We report the wavenumber of maximum growth and estimate the time to breakup. The constraint is then bent in-plane to add a weak secondary curvature to the now nearly cylindrical base state. This is referred to as the torus lift of the cylinder. The static stability of these toroidal equilibria, calculated using a perturbation approach, shows that the position of constraint is crucial – constraint can stabilize (outside) or destabilize (inside). The combined influence of secondary curvature and wire constraint on the Plateau–Rayleigh limit is tracked. Finally, attention is restricted to constraints that yield a lens-like cylindrical meniscus. For these lenses, the torus lift is used as apparatus along with a symmetrization procedure to prove a large-amplitude static stability result. Our study is conveniently framed by a classic paper on rivulets by Davis (J. Fluid Mech., vol. 98, 1980, p. 225).

We investigate the dynamics of thin films driven by gravity on the outer surface of a cylinder and sphere. The surface is rigid, stationary and the axis of the cylinder is horizontal. An instantaneous release of a constant volume of fluid at the top of the cylinder or sphere results initially in a two-dimensional or axisymmetric current respectively. The resultant flow of a thin film of fluid is described using lubrication theory when gravity and viscous forces govern the dynamics. We show that the thickness of the flow remains uniform in space and decreases in time like t−1/2 near the top of both the cylinder and the sphere. Analytic solutions for the extent of the flow agree well with our experiments until the advancing front splits into a series of rivulets. We discuss scalings of the flow at the onset of the instability as a function of the Bond number, which characterizes the relative importance of gravity and surface tension. The experiments, conducted within an intermediate range of Bond numbers, suggest that the advancing front becomes unstable after it has propagated a critical distance, which depends primarily and monotonically on the volume of fluid and not on the viscosity of fluid. Releasing a sufficiently large volume of fluid ensures that rivulets do not develop on either a cylinder or sphere.

This paper considers the dynamics of a thin film of viscous liquid of density ρ coating the underside of a horizontal rigid boundary under the action of surface tension σ and gravity g, and in the lubrication limit. Gravitational instability for inverse wavenumbers larger than the capillary length ℓ = (σ/ρg)1/2) leads to the formation of quasi-static pendent drops of radius ≈3.83ℓ. If the boundary conditions are such as to pin the positions of the drops then the drops slowly drain fluid from the regions between them through thin annular trenches around each drop. A similarity solution is derived and verified numerically in which the film thickness in the intervening regions scales like t−1/4 and that in the trenches like t−1/2. A single drop placed far from boundaries on an otherwise uniform film, and given an initial perturbation, undergoes self-induced quasi-steady translation during which it grows slowly in amplitude by leaving a wake where the film thickness is reduced by an average of 90. It is driven by release of gravitational potential energy as fluid is collected from the film into the lower lying drop. Analysis of Landau–Levich regions around the perimeter of the translating drop predicts its speed and the profile of the wake. Two translating drops may coalesce if they collide, in contrast with the non-coalescence of colliding collars in the analogous one-dimensional problem (Lister et al., J. Fluid Mech. vol. 552, 2006b, p. 311). Colliding drops may also bounce off each other, the outcome depending on the angle of incidence through complex interactions between their surrounding capillary wave fields.

Spin-coating is a process used to fabricate thin films for device applications. In this paper, lubrication theory is used to derive an axisymmetric model for the spin-coating of two immiscible vertically stratified Newtonian thin films. The model includes gravitational, van der Waals, capillary and viscous forces, differences in liquid layer properties and evaporation/condensation effects. Thinning calculations focus on the effects of viscosity and condensation/evaporation. In this case, for layers of uniform thickness, the lower layer thins monotonically yet never reaches zero thickness. With evaporation mass loss the upper layer disappears in finite time, whereas with condensation effects the upper layer approaches a steady-state thickness. Fully nonlinear calculations are carried out for films with non-uniform thickness and the deviation of the interfaces from the flat state is monitored. In general, disturbances to the lower layer have a greater effect on the upper layer than those of disturbances of the upper layer on the lower layer. Disturbances along the upper gas–liquid free surface propagate outward more rapidly than those along the lower liquid–liquid interface and disturbances that decrease the film thickness tend to dissipate more slowly.

We present a study of a cylinder of ice melting in warm air in order to quantify the heat-transfer mechanisms controlling the evolution of its shape, which are inherent in a range of phenomena involving phase change and fluid flow. Motivated by the initial melting at the top of a flat-topped cylinder of ice, we analyse laminar, natural convection above a cooled, finite, horizontal plate (or below a heated, finite, horizontal plate) and show that, to a very good approximation, the partial-differential, boundary-layer equations can be separated with self-similar vertical profiles scaled by the boundary-layer thickness. We find that the horizontal evolution of the boundary-layer thickness is governed by equations describing a steady, viscous gravity current fed by diffusive entrainment, and therefore describe such flows as diffusive gravity currents. We first use the predictions of our model to examine previous experimental results in two dimensions. Our experimental results relating to the melting of ice in air are then compared with predictions based on our analysis of the axisymmetric thermal boundary layer. This comparison confirms the vertical thermal structure and shows that melting is governed in roughly equal measure by heat transfer from the air, the latent heat of condensation of water vapour, and the net radiative heat transfer from the surroundings to the ice.

We consider a model for the solidification of an ideal ternary alloy in a mushy layer that incorporates the effects of thermal and solutal diffusion, convection and solidification. Our results reveal that although the temperature and solute fields are constrained to the liquidus surface of the phase diagram, the system still admits double-diffusive modes of instability. Additionally, modes of instability exist even in situations in which the thermal and solute fields are each individually stable from a static point of view. We identify these instabilities for a general model in which the base-state solution and its linear stability are computed numerically. We then highlight these instabilities in a much simpler model that admits an analytical solution.

Uniform suspensions of bottom-heavy, upswimming (gyrotactic) micro-organisms that are denser than water are unstable, through a gravitational mechanism first described by Pedley, Hill & Kessler (J. Fluid Mech., vol. 195, 1988, p. 223). Suspensions of downswimming, head-heavy cells do not experience this instability. In the absence of gravity, a uniform suspension of swimming micro-organisms may be unstable because of the ‘particle stresses’ generated by the swimming cells themselves, each of which acts as a force-dipole or stresslet (Simha & Ramaswamy, Phys. Rev. Lett., vol. 89, 2002, p. 058101). The stresslet strength S is positive for ‘pullers’ such as algae and negative for ‘pushers’ such as bacteria or spermatozoa. In this paper, the combined problem is investigated, with attention being paid also to the effect of rotational diffusivity and to whether the probability density function f(e) for the cells' swimming direction e can be approximated as quasi-steady in calculations of the mean swimming direction, which arises in the cell conservation equation, and the particle stress tensor, which appears in the momentum equation. The existence of both the previous instabilities is confirmed at long wavelength. However, the non-quasi-steadiness of the orientation distribution means that the particle-stress-driven instability no longer arises for arbitrarily small |S|, in the Stokes limit, but requires that the dimensionless stresslet strength (proportional to the product of S and the basic state cell volume fraction no) exceed a critical value involving both viscosity and rotational diffusivity. In addition, a new mode of gravitational instability is found for ‘head-heavy’ cells, even when they exert no particle stresses (S = 0), in the form of weakly growing waves. This is a consequence of unsteadiness in the mean swimming direction, together with non-zero fluid inertia. For realistic parameter values, however, viscosity is expected to suppress this instability.

We model the dynamics of the human tear film during relaxation (after a blink) using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the flux of tear fluid into or out of the domain. The governing nonlinear partial differential equation is solved on an overset grid by a method of lines using finite differences in space and an adaptive second-order backward difference formula solver in time. Our simulations in a two-dimensional domain are computed in the Overture computational framework. The flow around the boundary is sensitive to both our choice of flux boundary condition and the presence of gravity. The simulations recover features seen in one-dimensional simulations and capture some experimental observations of tear film dynamics around the lid margins. In some instances, the influx from the lacrimal gland splits with some fluid going along the upper lid towards the nasal canthus and some travelling around the temporal canthus and then along the lower lid. Tear supply can also push through some parts of the black line near the eyelid margins.