Experiments are reported on Faraday waves in a circular cylinder, which are internally resonant with either the subharmonic mode (with frequency ¼ that of the forcing) or the superharmonic mode (with frequency equal to that of the forcing). For subharmonic resonance both modes achieved comparable amplitudes that were steady, or were modulated with one or two periods, or exhibited quasi-periodic or chaotic motions. A stability map of these responses is presented. Theoretical predictions of linear stability and growth rates are tested. Measurements of steady amplitudes, limit-cycle frequencies and wave slopes at breaking are presented. Some of the measured phase-plane trajectories are shown to have theoretical counterparts. For superharmonic resonance the amplitude of the superharmonic was never comparable with that of the Faraday wave. For low modes an energy exchange occurred during the initial period of growth, and a precession instability sometimes developed. For high modes for which both frequencies and wavenumbers are in a 2:1 ratio, superharmonic resonance occurred irreproducibly; it appeared to be overwhelmed by 1:1 interactions among the possible Faraday-wave modes.

Standing radial cross-waves in an annular wave tank are investigated using Whitham's average-Lagrangian method. For the simplest case, in which a single radial cross-wave is excited, energy is transferred from the wavemaker to the cross-wave through the spatial mean motion of the free surface, as described by Garrett (1970) for a purely transverse cross-wave in a rectangular tank. In addition, energy is transferred through spatial coupling since, in contrast to the purely transverse cross-wave in the rectangular tank, the (non-axisymmetric) radial cross-wave is three-dimensional. It is shown in an Appendix that this spatial coupling does occur for a three-dimensional cross-wave in a rectangular tank. The equations that govern this single-mode resonance are isomorphic to those that govern the Faraday resonance of surface waves in a basin of fluid subjected to vertical excitation (Miles 1984a).

It is found that the second-order Stokes-wave expansion for deep-water, standing gravity waves, which is regular for rectangular containers, may become singular for circular containers (Mack (1962) noted these resonances for finite-depth, standing gravity waves in circular containers). The evolution equations that govern two distinct types of resonant behaviour are derived: (i) 2:1 resonance between a radial cross-wave and a resonantly forces axisymmetric wave, corresponding to approximate equality among the driving frequency, a natural frequency of the directly forced wave, and twice the natural frequency of a cross-wave; (ii) 2:1 internal resonance between a radial cross-wave and a non-axisymmetric second harmonic, corresponding to approximate equality among the driving frequency, the natural frequency of a non-axisymmetric wave of even azimuthal wavenumber, and twice the natural frc.quency of the cross-wave. The axisymmetric, directly forced wave in (i) is resonantly excited and exchanges energy with the subharmonic cross-wave through spatial coupling, whereas the cross-wave in (ii) is parametrically excited and exchanges energy with the non-axisymmetric second harmonic through spatial coupling. The equations governing case (i) are shown to exhibit chaotic motions; those governing (ii) are shown to be isomorphic to the equations governing 2:1 internal resonance in the Faraday problem (Miles 1984a, §6), which have been shown to exhibit chaotic motions (Gu & Sethna 1987).

Preliminary experiments on standing radial cross-waves are reported in an Appendix, and theoretical predictions of mode stability are in qualitative agreement with these experiments. For the single-mode theory, the interaction coefficient that is a measure of the energy exchange between the wavemaker and the cross-wave is evaluated numerically for a particular wavemaker. The maximum interaction coefficient for a fixed azimuthal wavenumber of the cross-wave typically occurs for that radial mode number for which the turning point of the cross-wave radial profile is nearest the wavemaker. The present experiments for standing radial cross-waves are compared with those of Tatsuno, Inoue & Okabe (1969) for progressive radial cross-waves.

This paper extends recent theoretical work on sub-inertial trapped modes in bays to consider trapping of energy in the neighbourhood of estuary mouths on coastal shelves. The qualitative form of the theoretical predictions accords well with recent observations on the Scotian Shelf that show energy trapped near the Laurentian Channel at a frequency higher than that of the propagating waves on the shelf.

The trapping and scattering of shelf waves is modelled for a shelf–estuary or shelf–headland system by considering barotropic waves in a straight, infinite channel with an attached rectangular estuary or interrupted by a rectangular headland. Taking the depth to increase exponentially with distance from the coast and expanding in cross-shelf modes reduces the problem to a system of real linear algebraic equations.

Trapped modes with frequencies above the cutoff frequency of propagating waves are found near the mouth of the estuary. Waves propagating towards the estuary are strongly scattered and, for particular frequencies, incident energy can be either perfectly transmitted or totally reflected. An incident wave can be in resonance with the estuary causing energy to penetrate the estuary. Bounds on the frequencies of trapped and resonant solutions are given and allow an easy modal interpretation.

If the frequency of an incident wave is sufficiently high, waves cannot propagate past a headland. Energy at these frequencies can however tunnel through the region and appear as an attenuated wave on the far side. For particular frequencies all energy passes the headland and none is reflected. For headlands long compared with the incident wave, transmission coefficients for single-mode scattering follow from spatially one-dimensional wave mechanics.

The nonlinear, three-dimensional Euler equations can be reduced to a simple linear equation when the flow has helical symmetry and when the flow consists of a rigidly rotating basic part plus a Beltrami disturbance part (with vorticity proportional to velocity or a slight generalization of this). Solutions to this linear equation represent steadily rotating, non-axisymmetric waves of arbitrary amplitude. Exact solutions can be constructed in the case of flow in a straight pipe of circular cross-section. Analogous results are obtained for the incompressible, non-dissipative equations of magnetohydrodynamics. In addition to a rigidly rotating basic flow, there may exist a toroidal magnetic field varying linearly with radius.

Film evaporation of a spherical droplet over a hot surface is investigated in this paper. In view of the radial evaporation-induced velocity at the liquid-gas interface, an improvement over the classical flow field solution of Stimson & Jeffery (1926) needs to be employed. In addition to the flow, the energy equation internally and externally to the droplet and the species equation in the gas phase are also solved. The boundary conditions at the droplet surface couple the temperature, species and flow field. Analytical expressions for the hydrodynamic force and its components (viscous and pressure) that the droplet experiences are obtained. It is shown that, depending on the droplet separation distance from the hot surface and the type of liquid, there may be a substantial temperature variation along the droplet surface. Furthermore, considering a quasi-steady approximation for the droplet regression rate and balancing at each time step the weight of the droplet with the hydrodynamic force it experiences, time histories are obtained numerically for various quantities of interest. Thus, it is predicted that the droplet moves away from the hot surface while evaporating and that the initially substantial temperature variation along the droplet surface decreases with time and diminishes towards the end of the droplet lifetime. It is also shown that the droplet surface temperature is more uniform at higher hot-surface temperature.

The response of a salinity-stratified fluid to sidewall heating is examined. Experiments were conducted using a linear salt gradient alone and with a combination of salt and thermal gradient in which the thermal stratification destabilized the density. The dominant feature of the flow was intrusions which propagated into the interior of the fluid. These were studied using shadowgraph, vertical temperature and salinity profiles, and dye streaks. Using the theory for convection in a long shallow cavity, an expression is derived for the velocity of propagation of the intrusions. The mechanism by which intrusions merge is also discussed to explain the observed formation of wavy interfaces during merging.

Several aspects of the oscillations of a gas bubble in a slightly compressible liquid are discussed by means of a simplified model based on the assumption of a spatially uniform internal pressure. The first topic considered is the linear initial-value problem for which memory effects and the approach to steady state are analysed. Large-amplitude oscillations are studied next in the limit of large and small thermal diffusion lengths obtaining, in the first case, an explicit expression for the internal pressure, and, in the second one, an integral equation of the Volterra type. The validity of the assumption of uniform pressure is then studied analytically and numerically. Finally, the single-bubble model is combined with a simple averaged-equation model of a bubbly liquid and the propagation of linear and weakly nonlinear pressure waves in such a medium is considered.

The flow of water in straight open channels with prismatic complex cross-sections is considered. Lateral distributions of depth-mean velocity and boundary shear stress are derived theoretically for channels of any shape, provided that the boundary geometry can be discretized into linear elements. The analytical model includes the effects of bed-generated turbulence, lateral shear turbulence and secondary flows. Experimental data from the Science and Engineering Research Council (SERC) Flood Channel Facility are used to illustrate the relative importance of these three effects on internal shear stresses. New experimental evidence concerning the spatial distribution of Reynolds stresses τyx and τzx is presented for the particular case of compound or two-stage channels. In such channels the vertical distributions of τzx are shown to be highly nonlinear in the regions of strongest lateral shear and the depth-averaged values of τyx are shown to be significantly different from the depth mean apparent shear stresses. The importance of secondary flows in the lateral shear layer region is therefore established. The influence of both Reynolds stresses and secondary flows on eddy viscosity values is quantified. A numerical study is undertaken of the lateral distributions of local friction factor and dimensionless eddy viscosity. The results of this study are then used in the analytical model to reproduce lateral distributions of depth-mean velocity and boundary shear stress in a two stage channel. The work will be of interest to engineers engaged in flood channel hydraulics and overbank flow in particular.

Using the theory of homogenization we examine the correction to Darcy's law due to weak convective inertia of the pore fluid. General formulae are derived for all constitutive coefficients that can be calculated by numerical solution of certain canonical cell problems. For isotropic and homogeneous media the correction term is found to be cubic in the seepage velocity, hence remains small even for Reynolds numbers which are not very small. This implies that inertia, if it is weak, is of greater importance locally than globally. Existing empirical knowledge is qualitatively consistent with our conclusion since the linear law of Darcy is often accurate for moderate flow rates.

At large Reynolds number (Re > 10), waves on inclined films grow rapidly downstream in both amplitude and wavelength to the extent that linear stability theory cannot adequately describe their velocity–wavenumber relationship. The wavelength increases indefinitely until solitary waves are formed very far downstream. In a recent experiment of Brauner & Maron (1982), this evolution to long waves is observed to occur by successive wavelength doubling. In this analysis, we develop a second-order integral boundary-layer approximation for long waves at intermediate Re of O(ε−1), where ε is the dimensionless wavenumber scaled with respect to the film thickness. (A second-order theory is needed because it introduces important dissipation terms which allow periodic and solitary waveforms to exist when surface tension is negligible.) After showing that this model can adequately describe infinitesimal waves at inception, we verify the existence of solitary waves and long-wavelength periodic waves near the critical Reynolds number with a weakly nonlinear analysis. These finite-amplitude waves are then numerically continued into the more important high-Re and strongly nonlinear regions. It is shown that the solitary wave speed approaches 1.67 times the Nusselt velocity, and the thickness of the substrate film approaches 0.47 times the Nusselt film thickness at large Re. These results are favourably compared to experimental data of Chu & Dukler (1974, 1975). We also confirm the period-doubling scenario of Brauner & Maron by showing that short finite-amplitude monochromatic waves are unstable to subharmonic instability.