This paper presents analytical results for the temperature and pressure fluctuations in a droplet or bubble pulsating in a sound wave, the related damping coefficients, as well as the corresponding sound attenuation coefficients for dilute suspensions. The study is limited to small-amplitude motions but includes the effects of compressibility and heat conduction in the fluid outside the particle. Results are obtained for both average and surface values of the particle's temperature and pressure fluctuations that are applicable to droplets in gases and liquids, and to gas bubbles in liquids. In the latter instance, it is found that the bubble's response exhibits a clear resonant peak at the isothermal natural frequency, that acoustic radiation is the dominant dissipation mechanism near resonance, and that the disturbances produced by the bubble in the liquid significantly reduce the thermal damping at most frequencies. Similar conclusions apply for droplets in liquids, except that the effects of resonance are significantly diminished.

An analytical theory is developed for a class of stable quasi-geostrophic vortices propagating either in the westward direction with supercritical velocities (c<−βR2d) or eastward, where Rd is the radius of deformation and β is the gradient of the Coriolis parameter. The numerical spectral calculations, initiated by the theoretical solutions, indicate that the supercritical vortices move initially with the predicted velocity, but later slow down to the speed of the long planetary waves. The period of time during which an eddy is propagating with its initial velocity is analysed as a function of its strength and (initial) speed. The theoretical solution has a distinctive streamline signature (‘alpha gyre’), which also appears in the final state of the numerical calculations, including those for an initially completely symmetric vortex placed on the β-plane.

Turbulent stratified flow over topography is studied using layered quasi-geostrophic models. Mean flows develop under random forcing, with lower-layer mean stream-function positively correlated with topography. When friction is sufficiently small, upper-layer mean flow is weaker than, but otherwise resembles, lower-layer mean flow. When lower-layer friction is larger, upper-layer mean flow reverses and can exceed lower-layer mean flow in strength. The mean interface between layers is domed over topographic elevations. Eddy fluxes of potential vorticity and layer thickness act in the sense of driving the flow toward higher entropy. Such behaviour contradicts usual eddy parameterizations, to which modifications are suggested.

Two-dimensional, unsteady, compressible flow fields produced by the interactions between a single vortex or a pair of vortices and a shock wave are simulated numerically. The Navier–Stokes equations are solved by a finite difference method. The sixth-order-accurate compact Padé scheme is used for spatial derivatives, together with the fourth-order-accurate Runge–Kutta scheme for time integration. The detailed mechanics of the flow fields at an early stage of the interactions and the basic nature of the near-field sound generated by the interactions are studied. The results for both a single vortex and a pair of vortices suggest that the generation and the nature of sounds are closely related to the generation of reflected shock waves. The flow field differs significantly when the pair of vortices moves in the same direction as the shock wave than when opposite to it.

This paper presents the results of a series of laboratory experiments aimed at understanding the processes associated with surface freezing of a two-layer fluid. The flow configuration consists of a layer of cold, salty water overlying a relatively deep bottom layer of warm, saltier water. This situation is common in high-latitude oceans during periods of rapid ice formation. The experiments were conducted in a tank with well-insulated side and bottom walls, placed in a walk-in freezer with air temperatures from −12 to −20°C. A system of thermocouples was used to measure the temperatures at fixed levels in water, ice and air. Microscale conductivity and temperature probes were used to obtain vertical profiles of temperature and salinity in the water. In general, when external uxes of heat and salt are absent, such a system enhances static stability, in the sense that the net density difference between the layers increases with time. When external uxes of heat (because of surface cooling) and salt (rejected during ice formation) are applied, however, this fluid system may become unstable and overturning of fluid layers is possible. In addition, heat transport from the warmer bottom layer to the colder upper layer may be important, possibly leading to a reduction in the rate of ice formation compared to that of a homogeneous fluid with temperature and salinity identical to the upper layer. Descriptions of such physical processes are given using laboratory experiments, and quantitative measurements of salient parameters are compared with the predictions of a theoretical model developed to explicate the flow evolution.

The formation of stationary cross flow vortices in a three-dimensional boundary layer due to surface roughness located near the leading edge of a swept wing is investigated using numerical solutions of the compressible Navier–Stokes equations. The numerical solutions are used to evaluate the accuracy of theoretical receptivity predictions which are based on the parallel-flow approximation. By reformulating the receptivity theory to include the effect of surface curvature, it is shown that convex surface curvature enhances receptivity. Comparisons of the parallel-flow predictions with Navier–Stokes solutions demonstrate that non-parallel effects strongly reduce the initial amplitude of stationary cross flow vortices. The curvature and non-parallel effects tend to counteract one another; but, for the cases considered here, the non-parallel effect dominates leading to significant over-prediction of receptivity by parallel-flow receptivity theory. We conclude from these results that receptivity theories must account for non-parallel effects in order to accurately predict the amplitude of stationary crossflow instability waves near the leading edge of a swept wing.

This paper is concerned with the effect of free-stream turbulence on the pretransitional flat-plate boundary layer. It is assumed that either the turbulent Reynolds number or the downstream distance (or both) is small enough that the flow can be linearized. The dominant disturbances in the boundary layer, which are of the Klebanoff type, are governed by the linearized unsteady boundary-region equations, i.e. the linearized Navier–Stokes equations with the streamwise derivatives neglected in the viscous and pressure-gradient terms. The turbulence is represented as a superposition of vortical free-stream Fourier modes and the corresponding Fourier component solutions to the boundary-region equations are obtained numerically. The results are then superposed to compute the root mean square of the fluctuating streamwise velocity in the boundary layer produced by the actual free-stream turbulence. It is found that the disturbances computed with isotropic free-stream turbulence do not reach the levels measured in experiments. However, good quantitative agreement is obtained with the relatively low turbulent Reynolds number data of Kendall when the measured strong anisotropy of the low-frequency portion of his spectrum is accounted for. Data at higher turbulent Reynolds numbers are affected by nonlinearity, which manifests itself through the generation of small spanwise length scales. We attempt to model this within the context of the linear theory by choosing a free-stream spectrum whose energy is concentrated at larger transverse wavenumbers and achieve very good agreement with the data. The results suggest that even small deviations from pure isotropy can be an important factor in explaining the large amplitudes of the Klebanoff modes in the pre-transitional boundary layer, and also point to the importance of nonlinear effects. We discuss some additional effects that may need to be accounted for in order to obtain a complete description of the Klebanoff modes.

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

Rayleigh instability on an axisymmetric viscous film around a vertical fibre produces localized wave structures (pulses) on a flat substrate film that can grow by an order of magnitude to form large capillary drops. We show that this drop formation process is driven by a unique mechanism in the form of an ever-growing pulse which leaves behind a trailing film thinner than the one it advances into. In addition to accumulating liquid from the film, the growing pulse also captures smaller and slower pulses in coalescence cascades. We construct this supercritical growing pulse by matched asymptotics and show that it will eventually evolve into a capillary drop if the local film thickness h is larger than hc=1.68R3H−2 where R is the fibre radius and H the capillary length. We also show that subcritical pulses with h<hc will equilibrate into stationary pulses that do not grow or coalesce readily to form drops.

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

Sound transmission through straight circular ducts with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion. A natural extension for ducts with axially slowly varying properties (diameter and mean flow, wall impedance) is a multiple-scales solution. It is shown in the present paper that a consistent approximation of boundary condition and isentropic mean flow allows the multiple-scales problem to have an exact solution. Since the calculational complexities are no greater than for the classical straight duct model, the present solution provides an attractive alternative to a full numerical solution if diameter variation is relevant. A unique feature of the present solution is that it provides a systematic approximation to the hollow-to-annular cylinder transition problem in the turbofan engine inlet duct.

A vertical jet of water impinging on a horizontal surface produces a radial film flow followed by a circular hydraulic jump. We report a phenomenon where fairly large (1 ml) drops of liquid levitate just upstream of the jump on a thin air layer between the drop and the film flow. We explain the phenomenon using lubrication theory. Bearing action both in the air film and the water film seems to be necessary to support large drops. Horizontal support is given to the drop by the hydraulic jump. A variety of drop shapes is observed depending on the volume of the drop and liquid properties. We show that interaction of the forces due to gravity, surface tension, viscosity and inertia produces these various shapes.

This study investigates the existence and stability of static liquid bridges in non-axisymmetrically buckled elastic tubes. The liquid bridge which occludes the tube is formed by two menisci which meet the tube wall at a given contact angle along a contact line whose position is initially unknown. Geometrically nonlinear shell theory is used to describe the deformation of the linearly elastic tube wall in response to an external pressure and to the loads due to the surface tension of the liquid bridge. This highly nonlinear problem is solved numerically by finite element methods.

It is found that for a large range of parameters (surface tension, contact angle and external pressure), the compressive forces generated by the liquid bridge are strong enough to hold the tube in a buckled configuration. Typical meniscus shapes in strongly collapsed tubes are shown and the stability of these configurations to quasi-steady perturbations is examined. The minimum volume of fluid required to form an occluding liquid bridge in an elastic tube is found to be substantially smaller than predicted by estimates based on previous axisymmetric models. Finally, the implications of the results for the physiological problem of airway closure are discussed.

Vapour cavities in liquid flows have long been associated with cavitation damage to nearby solid surfaces and it is thought that the final stage of collapse, when a high- speed liquid jet threads the cavity, plays a vital role in this process. The present study investigates this aspect of the motion of laser-generated cavities in a quiescent liquid when the distance (or stand-off) of the point of inception from a rigid boundary is between 0.8 and 1.2 times the maximum radius of the cavity. Numerical simulations using a boundary integral method with an incompressible liquid impact model provide a framework for the interpretation of the experimental results. It is observed that, within the given interval of the stand-off parameter, the peak pressures measured on the boundary at the first collapse of a cavity attain a local minimum, while at the same time there is an increase in the duration of the pressure pulse. This contrasts with a monotonic increase in the peak pressures as the stand-off is reduced, when the cavity inception point is outside the stated interval. This phenomenon is shown to be due to a splash effect which follows the impact of the liquid jet. Three cases are chosen to typify the splash interaction with the free surface of the collapsing cavity: (i) surface reconnection around the liquid jet; (ii) splash impact at the base of the liquid jet; (iii) thin film splash. Hydrodynamic pressures generated following splash impact are found to be much greater than those produced by the jet impact. The combination of splash impact and the emission of shock waves, together with the subsequent re-expansion, drives the flow around the toroidal cavity producing a distinctive double pressure peak.

When a fluid is pushed by a less viscous one the well-known Saffman–Taylor instability phenomenon arises, which takes the form of fingering. Since this phenomenon is important in a wide variety of applications involving strongly non-Newtonian fluids – in other words, fluids that exhibit yield stress – we undertake a full theoretical examination of Saffman–Taylor instability in this type of fluid, in both longitudinal and radial flows in Hele-Shaw cells. In particular, we establish the detailed form of Darcy's law for yield-stress fluids. Basically the dispersion equation for both flows is similar to equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability contain the yield stress. As a consequence the wavelength of maximum growth can be extremely small even at vanishing velocities. Additionally an approximate analysis shows that the fingers which are left behind at the beginning of destabilization should tend to stop completely. Fingering of yield-stress fluids therefore has some peculiar characteristics which nevertheless are not sufficient to explain the fractal pattern observed with colloidal systems.