The deformation of a viscous drop, driven by buoyancy towards a solid surface or a deformable interface, is analysed in the asymptotic limit of small Bond number, for which the deformation becomes important only when the drop is close to the solid surface or interface. Lubrication theory is used to describe the flow in the thin gap between the drop and the solid surface or interface, and boundary-integral theory is used in the fluid phases on either side of the gap. The evolution of the drop shape is traced from a relatively undeformed state until a dimple is formed and a long-time quasi-steady-state pattern is established. A wide range of drop to suspending phase viscosity ratios is examined. It is shown that a dimple is always formed, independently of the viscosity ratio, and that the long-time thinning rates take simple forms as inverse fractional powers of time.

The impact of a drop onto a liquid film with a relatively high impact velocity, leading to the formation of a crown-like ejection, is studied theoretically. The motion of a kinematic discontinuity in the liquid film on the wall due to the drop impact, the formation of the upward jet at this kinematic discontinuity and its elevation are analysed. Four main regions of the drop and film are considered: the perturbed liquid film on the wall inside the crown, the unperturbed liquid film on the wall outside the crown, the upward jet forming a crown, and the free rim bounding this jet. The theory of Yarin & Weiss (1995) for the propagation of the kinematic discontinuity is generalized here for the case of arbitrary velocity vectors in the inner and outer liquid films on the wall. Next, the mass, momentum balance and Bernoulli equations at the base of the crown are considered in order to obtain the velocity and the thickness of the jet on the wall. Furthermore, the dynamic equations of motion of the crown are developed in the Lagrangian form. An analytical solution for the crown shape is obtained in the asymptotic case of such high impact velocities that the surface tension and the viscosity effects can be neglected in comparison to inertial effects. The edge of the crown is described by the motion of a rim, formed due to the surface tension.

Three different cases of impact are considered: normal axisymmetric impact of a single drop, oblique impact of a single drop, and impact and interaction of two drops. The theoretical predictions of the height of the crown in the axisymmetric case are compared with experiments. The agreement is quite good in spite of the fact that no adjustable parameters are used.

A small-scale experiment was conducted (in a 3 m long flume) to study interfacial long-waves in a two-immiscible-fluid system (water and petrol were used). Experiments and nonlinear theories are compared in terms of wave profiles, phase velocity and mainly frequency–amplitude relationships. As expected, the KdV solitary waves match the experiments for small-amplitude waves for all layer thickness ratios. The characteristics of ‘large’-amplitude waves (that is when the crest is close to the critical level – approximately located at mid-depth) asymptotically tend to be predicted by a ‘KdV-mKdV’ equation containing both quadratic and cubic nonlinear terms. In addition a numerical solution of the complete Euler equations, based on Fourier series expansions, is devised to describe solitary waves of intermediate amplitude. In all cases, solitary interfacial waves in this numerical theory tally with the experimental data. When the layer thicknesses are almost equal (ratio of lower layer to total depth equal to 0.4 or 0.63) both the KdV-mKdV and the numerical solutions match the experimental points.

High-resolution measurements have been performed of the convective heat current as a function of time when a Rayleigh-Bénard cell is swept through its threshold with a specified time-dependent heat input. The results are interpreted in terms of the amplitude equation which exactly describes the slow variations in space and time of hydrodynamic quantities near the threshold. A phenomenological forcing field is added to this equation, and its form and magnitude are fitted to the onset time of the convective heat current. A deterministic model in which the field is an adjustable constant yields a good fit to the data for both a step and a linear ramp in the heat input. An alternative stochastic model, in which the field is a Gaussian variable with zero mean and a white-noise spectrum, is adequate for the ramp experiments, but cannot fit the step data for any value of the mean-square field. The systematics of the field and onset time versus ramp rate are studied in both the deterministic and stochastic models, and attempts are made to interpret the field in terms of physical mechanisms. When the data for long times are analysed in terms of the amplitude equation, it is found that the state first excited at onset is not the roll pattern which is stable in steady state. Instead, the system goes first to an intermediate state, which we tentatively identify as a hexagonal configuration. The decay of this state is governed by a further adjustable field in the amplitude equation.

Properties of turbulent thermal convection were measured in air between horizontal plates maintained at constant temperatures. Rayleigh numbers of 6·3 × 105, 2·5 × 106 and 1·0 × 107 were studied with a convection chamber designed to allow measurements to be taken along a horizontal path. Vertical profiles are presented of horizontally averaged temperature; r.m.s. fluctuations of temperature, horizontal velocity and vertical velocity; total heat flux; the correlation coefficient between vertical velocity and temperature; all terms in the thermal variance equation; and of terms in the turbulent kinetic energy balance.

One-dimensional spectra of temperature, horizontal velocity and vertical velocity at two different heights all disclose large intensity for waves of length 5 to 15 times the plate separation. Secondary peaks occur for scales of from 0·7 to 1·7 times the separation height.

Many of the observations are consistent with a thermal structure dominated by plumes extending most or all of the distance between plates.

This article is aimed at relating a certain substantial body of established material concerning wave loading on offshore structures to fundamental principles of mechanics of solids and of fluids and to important results by G. I. Taylor (1928a, b). The object is to make some key parts within a rather specialised field accessible to the general fluid-mechanics reader.

The article is concerned primarily to develop the ideas which validate a separation of hydrodynamic loadings into vortex-flow forces and potential-flow forces; and to clarify, as Taylor (1928b) first did, the major role played by components of the potential-flow forces which are of the second order in the amplitude of ambient velocity fluctuations. Recent methods for calculating these forces have proved increasingly important for modes of motion of structures (such as tension-leg platforms) of very low natural frequency.

Experimental studies of the flow topology, leading-edge vortex dynamics and unsteady force produced by pitching and plunging flat-plate aerofoils in forward flight at Reynolds numbers in the range 5000–20 000 are described. We consider the effects of varying frequency and plunge amplitude for the same effective angle-of-attack time history. The effective angle-of-attack history is a sinusoidal oscillation in the range to with mean of and amplitude of . The reduced frequency is varied in the range 0.314–1.0 and the Strouhal number range is 0.10–0.48. Results show that for constant effective angle of attack, the flow evolution is independent of Strouhal number, and as the reduced frequency is increased the leading-edge vortex (LEV) separates later in phase during the downstroke. The LEV trajectory, circulation and area are reported. It is shown that the effective angle of attack and reduced frequency determine the flow evolution, and the Strouhal number is the main parameter determining the aerodynamic force acting on the aerofoil. At low Strouhal numbers, the lift coefficient is proportional to the effective angle of attack, indicating the validity of the quasi-steady approximation. Large values of force coefficients () are measured at high Strouhal number. The measurement results are compared with linear potential flow theory and found to be in reasonable agreement. During the downstroke, when the LEV is present, better agreement is found when the wake effect is ignored for both the lift and drag coefficients.

The morphology and time-dependent integral properties of the multifluid compressible flow resulting from the shock–bubble interaction in a gas environment are investigated using a series of three-dimensional multifluid-Eulerian simulations. The bubble consists of a spherical gas volume of radius 2.54 cm (128 grid points), which is accelerated by a planar shock wave. Fourteen scenarios are considered: four gas pairings, including Atwood numbers −0.8 < A < 0.7, and shock strengths 1.1 < M ≤ 5.0. The data are queried at closely spaced time intervals to obtain the time-dependent volumetric compression, mean bubble fluid velocity, circulation and extent of mixing in the shocked-bubble flow. Scaling arguments based on various properties computed from one-dimensional gasdynamics are found to collapse the trends in these quantities successfully for fixed A. However, complex changes in the shock-wave refraction pattern introduce effects that do not scale across differing gas pairings, and for some scenarios with A > 0.2, three-dimensional (non-axisymmetric) effects become particularly significant in the total enstrophy at late times. A new model for the total velocity circulation is proposed, also based on properties derived from one-dimensional gasdynamics, which compares favourably with circulation data obtained from calculations, relative to existing models. The action of nonlinear-acoustic effects and primary and secondary vorticity production is depicted in sequenced visualizations of the density and vorticity fields, which indicate the significance of both secondary vorticity generation and turbulent effects, particularly for M > 2 and A > 0.2. Movies are available with the online version of the paper.

It is suggested in this paper that axisymmetric holes in thin sheets of fluid in which surface tension forces predominate will open out if they are initially large in relation to the thickness of the sheet; but that small holes will close up. No exact criterion has been found for the critical hole size in a free falling sheet, but the behaviour of the sheet may be closely simulated by the suspension of a soap film between coaxial circular rings. Theoretical results and experimental observations on catenoid films so formed are described.

For a hole in a sheet standing under gravity on a horizontal plane an equilibrium configuration exists, which is shown to be unstable. It is suggested that in this case the equilibrium position serves to distinguish between holes which open and those which close. Experiments on the behaviour of holes in a mercury sheet reveal a well-defined critical size which is in good agreement with that predicted by the unstable equilibrium.

A further series of experiments on holes made in a sheet of water standing on paraffin wax gave no sharp distinction between opening and closing holes, and holes of a wide range of sizes could remain stationary. This behaviour is associated with changes in the angle of contact with the plane. Independent meniscus observations similar to those of Ablett for a steadily moving meniscus show that the angle of contact θa, for a meniscus about to advance is greater than the value θr for a meniscus on the point of receding. It is seen that this difference will produce a range of hole diameters within which a hole will be trapped and remain stationary. Observations on the minimum size of hole on a water sheet which will remain open are reported. But it was found that the largest holes which would remain stationary were too large in relation to the size of the sheet for reliable results to be obtained.

The stability of three axisymmetric jet profiles is reviewed. These profiles represent the development of an incompressible jet from a nearly top-hat profile to a fully developed jet profile. The disturbance equations for arbitrary mode number in a region of zero shear, which provide the boundary conditions for the numerical solution, are solved analytically through use of the disturbance vorticity equations. Numerical solutions for the spatial stability for the axisymmetric (n = 0) disturbance and the asymmetric n = l disturbance are presented. Previously published calculations of least stable modes are shown to be incorrectly interpreted and their actual mode types are given. The critical Reynolds number is found to increase as the profile varies from a top-hat to a fully developed jet form. Closed contours of constant amplification, which are unusual in free shear flows, are shown to exist for the n = 1 disturbance in the fully developed jet region. A fluctuation energy balance is used to justify the occurrence of this destabilizing effect of decreasing Reynolds number.

Turbulent mixing and combustion are investigated in a gaseous shear layer formed between two streams: one containing a low concentration of hydrogen in nitrogen and the other containing a low concentration of fluorine in nitrogen. The resulting temperature field is measured simultaneously at eight points across the width of the layer using fast-response cold-wire thermometry. The results show the presence of large, hot structures separated by tongues of cool fluid that enter the layer from either side. The usual bell-shaped mean-temperature profiles therefore result from a duty cycle whereby a fixed probe sees alternating hot and cool fluid, which results in the local mean. The adiabatic flame temperature is not achieved in the mean, at any location across the layer. For fixed velocities, it is found that, in general, two different mean-temperature profiles result from a given pair of reactant compositions if the sides of the layer on which they are carried are exchanged (‘flipped’). This finding is a direct consequence of the asymmetric entrainment of fluid into the layer. Results are compared with the predictions of Konrad and discussed in the context of the Broadwell–Breidenthal model. By comparison with the liquid result of Breidenthal, the amount of product formed in the layer at high Reynolds number is found to be dependent upon the Schmidt number. Results for a helium–nitrogen layer are discussed briefly.

A technique for the quantitative measurement of fluid velocities in the range 0–5 cm/sec is described. The technique uses a pH indicator, is applicable in aqueous solutions and permits visualization and measurement of three-dimensional flow fields.

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as \[ \Gamma^\prime\propto (1-\eta^{-\frac{1}{2}})(\sin\alpha)(1+M^{-1}+2M^{-2})(M-1)(\gamma^{\frac{1}{2}}/\gamma + 1)\] where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.

We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.

The paper describes an experimental study of the structure of air-water mixtures flowing vertically. Resistivity probe techniques were applied to measurements of local void properties, including void fraction, gas-phase convection velocity, bubble size distributions and space-time correlation functions. The axial development of flows for six different air-water mixing conditions were examined. Measurements up to 108 diameters from inlet indicated that, while flow patterns for the different mixers may be significantly different initially, the flows tend to develop towards a common structure determined only by the flux rates of the two phases. This was evidenced by the convergence of the void and velocity profiles, and particularly the bubble size distributions, as the flow developed. Estimates of bubble sizes for these more developed conditions, based on a balance of energy between the interfacial structure and the turbulent structure, gave values of diameters which were on average 13% above experimental values. The void distributions obtained for bubbly flow conditions, after an adequate settling length, appear to be characterized by a local minimum at the centre of the pipe.

The three-dimensional structure of the flow behind a heaving and pitching finite-span wing is investigated using dye flow visualization at a Reynolds number of 164. Phase-locked image sequences, which are obtained from two orthogonal views, are combined to create a set of composite images that give an overall sense of the three-dimensional structure of the flow. A model of the vortex system behind the wing is constructed from the image sequences. Variations of the Strouhal number, pitch amplitude and heave/pitch phase angle are qualitatively shown to affect the structure of the wake.

The ability of small drops or bubbles to stick to non-horizontal solid surfaces is analysed. The principal results consist of identifying the critical value of the volume of a drop (or bubble) beyond which it will dislodge and move down (or up) the surface of the solid, and determining the speed at which it will move. In addition, the area of the solid wetted by the drop (or dried by the bubble) is calculated when its volume is at its critical value. All of the results are expressed in terms of experimentally measurable material properties. The most limiting restriction on the validity of the results is the assumption that the value of the contact angle hysteresis is small.

The instability of steady natural convection of a stably stratified fluid between vertical surfaces maintained at different temperatures is analysed. The linear stability theory is employed to obtain the critical Grashof and Rayleigh numbers, for widely varying levels of the stable background stratification, for Prandtl numbers ranging from 0·73 to 1000 and for the limiting case of infinite Prandtl number. The energetics of the critical disturbance modes also are investigated. The numerical results show that, if the value of the Prandtl number is in the low to moderate range, there is a transition from stationary to travelling-wave instability if the stratification exceeds a certain magnitude. However, if the Prandtl number is large, the transition, with increasing stratification, is from travelling-wave to stationary instability. The theoretical predictions are in excellent agreement with the experimental observations of Elder (1965) and of Vest & Arpaci (1969), for stationary instability, and in fair to good agreement with the experimental results of Hart (1971), for travelling-wave instability.

A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re < 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modelled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 < Re < 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed.

Marine ice sheets are continental ice masses resting on bedrock below sea level. Their dynamics are similar to those of land-based ice sheets except that they must couple with the surrounding floating ice shelves at the grounding line, where the ice reaches a critical flotation thickness. In order to predict the evolution of the grounding line as a free boundary, two boundary conditions are required for the diffusion equation describing the evolution of the grounded-ice thickness. By analogy with Stefan problems, one of these conditions imposes a prescribed ice thickness at the grounding line and arises from the fact that the ice becomes afloat. The other condition must be determined by coupling the ice sheet to the surrounding ice shelves. Here we employ matched asymptotic expansions to study the transition from ice-sheet to ice-shelf flow for the case of rapidly sliding ice sheets. Our principal results are that the ice flux at the grounding line in a two-dimensional ice sheet is an increasing function of the depth of the sea floor there, and that ice thicknesses at the grounding line must be small compared with ice thicknesses inland. These results indicate that marine ice sheets have a discrete set of steady surface profiles (if they have any at all) and that the stability of these steady profiles depends on the slope of the sea floor at the grounding line.

Previous studies investigating the effect of aspect ratio ($\mathit{AR}$) for insect-like regimes have reported seemingly different trends in aerodynamic forces, however no detailed flow observations have been made. In this study, the effect of $\mathit{AR}$ and Reynolds number on the flow structures over insect-like wings is explored using a numerical model of an altered fruit fly wing revolving at a constant angular velocity. Increasing the Reynolds number for an $\mathit{AR}$ of 2.91 resulted in the development of a dual leading-edge vortex (LEV) structure, however increasing $\mathit{AR}$ at a fixed Reynolds number generated the same flow structures. This result shows that the effects of Reynolds number and $\mathit{AR}$ are linked. We present an alternative scaling using wing span as the characteristic length to decouple the effects of Reynolds number from those of $\mathit{AR}$. This results in a span-based Reynolds number, which can be used to independently describe the development of the LEV. Indeed, universal behaviour was found for various parameters using this scaling. The effect of $\mathit{AR}$ on the vortex structures and aerodynamic forces was then assessed at different span-based Reynolds numbers. Scaling the flow using the wing span was found to apply when a strong spanwise velocity is present on the leeward side of the wing and therefore may prove to be useful for similar studies involving flapping or rotating wings at high angles of attack.