The interfacial behaviour of surfactant-laden drops squeezing through tight constrictions in a uniform far-field flow is modelled with respect to capillary number, drop-to-medium viscosity ratio and surfactant contamination. The surfactant is treated as insoluble and non-diffusive, and drop surface tension is related to surfactant concentration by a linear equation of state. The constriction is formed by three solid spheres held rigidly in space. A characteristic aspect of this confined and contaminated multiphase system is the rapid development of steep surfactant-concentration gradients during the onset of drop squeezing. The interplay between two physical effects of surfactant, namely the greater interface deformability due to decreased surface tension and interface immobilization due to Marangoni stresses, results in particularly rich drop-squeezing dynamics. A three-dimensional boundary-integral algorithm is used to describe drop hydrodynamics, and accurate treatment of close squeezing and trapped states is enabled by advanced singularity subtraction techniques. Surfactant transport and hydrodynamics are coupled via the surface convection equation (or convection–diffusion equation, if artificial diffusion is included), the interfacial stress balance and a solid-particle contribution based on the Hebeker representation. For extreme conditions, such as drop-to-medium viscosity ratios significantly less than unity, it is found that upwind-biased methods are the only stable approaches for modelling surfactant transport. Two distinct schemes, upwind finite volume and flow-biased least squares, are found to provide results in close agreement, indicating negligible numerical diffusion. Surfactant transport is enhanced by low drop-to-medium viscosity ratios, at which extremely sharp concentration gradients form during various stages of the squeezing process. The presence of surfactant, even at low degrees of contamination, significantly decreases the critical capillary number for droplet trapping, due to the accumulation of surfactant at the downwind pole of the drop and its subsequent elongation. Increasing the degree of contamination significantly affects surface mobility and further decreases the critical capillary number as well as drop squeezing times, up to a threshold above which the addition of surfactant negligibly affects squeezing dynamics.

The detailed structure of the relaxation region in shock waves in oxygen was investigated using Blackman's experimental results. Oxygen was found to display a behaviour similar in many ways to that found previously for carbon dioxide with the relaxation frequency, as defined by the simple relaxation equation, depending on the departure from equilibrium as well as on temperature. The previous results for carbon dioxide were further analysed by means of a separate relaxation equation for each mode.

An analytical approximation is derived for the flow field in the vicinity of a transition zone between electrolytes of different mobility in isotachophoretic transport through a channel. Due to the difference in electroosmotic mobility and electric field on both sides of the transition zone, the flow field consists of a superposition of electroosmotic and pressure-driven flow. The corresponding convective ion transport inherently reduces the resolution of isotachophoretic separation processes. The derived analytical result is adequate for both wide and narrow transition zones and valid in the limit of thin electric double layers, relevant for most situations where isotachophoresis is employed. In this way, it complements and generalizes the results obtained for wide transition zones in the lubrication approximation. The analysis is extended to multiple sample zones with ions of different electrophoretic mobility, a scenario characteristic for applications in the field of analytical chemistry. The results are validated by comparison to finite-element calculations accounting for the transport of different ionic species governed by the coupled Nernst–Planck and Stokes equations, both for situations with only a single transition zone as well as for several transition zones. Excellent agreement is obtained between the analytical and the numerical results for realistic parameter values encountered in ITP experiments. This suggests using the analytical expression for the flow field in the framework of numerical studies of species transport in ITP experiments, since the time-consuming computation of the velocity field is essentially eliminated. The latter is successfully demonstrated using an iterative procedure, numerically solving the Nernst–Planck equation for a given flow field, and using the resulting concentration fields as an input for the derived analytical expression.

We study the effects of having an opening in a vertical cylinder of an arbitrary cross-sectional shape when subjected to incident waves from the outside field. Both the diffraction and radiation problems of linear potential-flow theory are addressed. The cylinders considered are bottom-mounted with vanishing thickness and the problem is formulated as a hypersingular boundary-integral method. A simple higher-order procedure is presented to handle the strong singularity. Comparisons are made between the hydrodynamic properties of open and closed cylinders, and the effects of increasing the opening size are discussed and explained. Results for square, circular and elliptical open cylindrical shells, presented as examples, indicate that wave loads on the structures could be dramatically decreased to zero (effectively) at certain frequencies and opening sizes. This leads to the surprising conclusion that directing an open structure into the incident-wave field results in lower loads on the structure. A model of an open harbour with a frontal breakwater in a wave field, inspired by the idea of the Portunus Project, is also analysed.

The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney–Hasegawa–Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $\lambda$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate ${\rm d}\ell/{\rm d} t$ of the characteristic length scale $\ell$ representing the energy centroid. It is found that ${\rm d}\ell/{\rm d} t\le2\|q\|_\infty/(\ell_s\lambda^2)$, where $\|q\|_\infty$ is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime ($\ell\ge\ell_s\,{\gg}\,\lambda^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.

The turbulent separated flow through an asymmetric diffuser with and without manipulation of incoming turbulence-driven mean secondary vortices (MSVs) from a rectangular duct is investigated by large-eddy simulations. The simulations carried out for two diffuser geometries reveal that by introducing a small amount of mean-flow kinetic energy via the MSVs into the flow, the complex three-dimensional separation behaviour and pressure recovery can be effectively controlled. Manipulated MSVs were found to enhance cross-sectional transport of high-momentum fluid, which determined the location, shape, and size of the separation bubble. The integral effect was a delay or expedition in the onset of separation. This change strongly affected the conversion of mean-flow kinetic energy to pressure, in particular for the front part of the diffuser. In addition, a substantial reduction in total pressure loss could be achieved. The manipulation of the MSVs is an efficient mechanism for performance enhancement in the cases investigated. The results have important implications for both control and statistical modelling of turbulent separated flow in rectangular diffusers.

We present the first solution to a boundary motion planning problem for the Navier–Stokes equations, linearized around the parabolic equilibrium in a three-dimensional channel flow. The pressure and skin friction at one wall are chosen as the reference outputs as they are the most readily measurable ‘wall-restricted’ quantities in experimental fluid dynamics and also because they play a special role as performance metrics in aerodynamics. The reference velocity input is applied at the opposite wall. We find the exact (method independent) solution to the motion planning problem using the PDE (partial differential equation) backstepping theory. The motion planning solution results in open-loop controls, which produce the reference output trajectories only under special initial conditions for the flow velocity field. To achieve convergence to the reference trajectory from other (nearby) initial conditions, we design a feedback controller. We also present a detailed examination of the closed-form solutions for gains and the behaviour of the motion planning solution as the wavenumbers grow or the Reynolds number grows. Numerical results are shown for the motion planning problem.

We develop asymptotic solutions for passive suspended sediment transport under a flood surge on a uniform slope. Our solutions provide predictions of the net scour under a surge, and simple estimates of the conditions under which it may ‘bulk up’ into a mud or debris flow, as well as illustrating their sensitivity to sediment entrainment rates.

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

We perform a detailed experimental study of large-scale vortices propagating in the rotating shallow-water layer in a paraboloidal vessel. A specific data acquisition technique is used in order to ensure precise measurements of the free-surface elevation. We find two qualitatively different types of vortex behaviour controlled by the relative elevation value. For small elevations we observe a standard quasi-geostrophic pattern with an asymmetric secondary circulation around an initially symmetric vortex which leads to a meridional drift and Rossby wave radiation. This type of behaviour is exhibited by both cyclonic and anticyclonic vortices. For relative elevations larger than 1 (nonlinear regime) the necessarily anticyclonic vortices are drifting strictly zonally maintaining their circular symmetry during the viscous decay. By varying the initial latitude of the vortex we were able to check that in the nonlinear regime the vortex lifetime is not sensitive to the beta-effect, while it is the case in the quasi-geostrophic regime. In the same way we show that the observed difference in cyclone–anticyclone lifetimes is not influenced by the beta-effect.

We address two aspects of the influence of double diffusion on convection from point buoyancy sources (plumes). First we consider double-diffusive convection acting in conjunction with buoyancy from an isolated source in an unconfined environment, and we study the problem of when fluxes of properties occur in the opposite direction to that of the initial buoyancy. A simple dimensional analysis is in good agreement with the experimental data for these ‘counterbuoyant’ fluxes of properties. Secondly, we consider a double-diffusive plume discharging into a confined environment. The plume now does not always reach the bottom (or top) of the confined environment, but can spread out into the environment at intermediate depths. Double-diffusive interfaces form, and we describe their development and the evolution of the density profile in the confined environment. A quantitative measure of the importance of double-diffusive convection in relation to the normal filling-box mechanism is obtained in the experiments by using the measured density profile, and this correlates well with a theoretically derived parameter Bρ.

An experimental investigation of resonant standing water waves in a rectangular tank with a corrugated bottom is reported. The study was stimulated by the theory of Howard & Yu (J. Fluid Mech., vol. 593, 2007, pp. 209–234) predicting the existence of normal modes that can be significantly affected by Bragg reflection/scattering. As a result, the amplitude of the standing waves (normal modes) varies exponentially along the entire length of the tank, or from the centre out in each direction, depending on the phase of the corrugations at the tank endwalls. Experiments were conducted in a 5 m tank fitted with a sinusoidal bottom with one adjustable endwall. Waves were excited by small-amplitude sinusoidal horizontal movement of the tank using an electrical motor drive system. Simultaneous time-series data of standing oscillations were recorded at well-separated positions along the tank to measure the growth in amplitude. Waveforms over a section of the tank were filmed through the transparent acrylic walls. Except for very shallow depths and near the tank endwalls, the experimental measurements of resonant frequencies, mean wavelengths, free-surface waveforms and amplitude growth are found in essential agreement with the Bragg resonant normal mode theory.

The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23) is re-cast in terms of the non-dimensional plume radius, the plume ‘laziness’ defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339–358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248). We are able to explain the unstable behaviour previously found when the power lies in the range (−4, −8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

Consider an infinite two-dimensional acoustic waveguide containing a long rectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequencies which decay down the waveguide. We provide analytic estimates for trapped mode frequencies and prove that the number of trapped modes is asymptotically proportional to the length of the obstacle.

Experiments are described in which the populations of metastable levels in ionizing argon are measured through spatially resolved hook interferometry. The results are compared with the present model for shock-induced ionization and a recently proposed mechanism to explain observed flow instabilities. It is found that the experimental measurements support the presently accepted model, which states that electron–atom collisions play the dominant role in the excitation process, but contradicts recent proposals which predict a rapid build-up of anomalously high metastable populations through atom–atom collisions.

The stability of an infinite layer of fluid of uniform thickness at rest with two horizontal boundaries is investigated when the difference between the temperatures at the top and bottom boundaries has a component which is a fluctuating periodic function of time in addition to a constant part. When both boundaries are free small fluctuations have a stabilizing effect on the layer, while large fluctuations tend to make it less stable, consistently with the numerical results of Yih & Li (1972); the effect is attributable entirely to the variation of the temperature gradient with time. An approximate relation between the mean Rayleigh number and the amplitude of the fluctuations is found which separates stable situations from unstable ones. This is compared with the criteria deduced by Homsy (1974) using energy arguments for disturbances of any amplitude.

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.

A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.