Many studies of weakly nonlinear surface waves are based on so-called reduced integrodifferential equations. One of these is the widely used Zakharov four-wave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four- and five-wave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called five-wave, the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravity-wave Hamiltonian in integral-power series in normal variables. For capillary–gravity waves on a fluid of finite depth, expansion of the Hamiltonian in integral-power series in a normal variable with accuracy up to the fifth-order terms is also given.

The solidification of hot fluid flowing in a thin buoyancy-driven layer between cold solid boundaries is analysed in a series of two papers. As an approximation to flow in a crack in a weakly elastic solid or to free-surface flow beneath a thin solidified crust, the boundaries are considered to be flexible and to exert negligible resistance to lateral deformation. The resultant equations of continuity and motion reduce to a kinematic-wave equation with a loss term corresponding to the accumulation of solidified material at the boundaries. The Stefan problem for the solidification is coupled back to the flow through the advection of heat by the fluid, which competes with lateral heat loss by conduction to the solid. Heat and mass conservation are used to derive boundary conditions at the propagating nose of the flow. In this paper the two-dimensional flow produced by a line release of a given volume of fluid is investigated. It is shown that at short times the flow solidifies completely only near the point of release where the flow is thinnest, at later times complete solidification also occurs near the nose of the flow where the cooling rates are greatest and, eventually, the flow is completely solidified along its depth. Some transient melting of the boundaries can also occur if the fluid is initially above its solidification temperature. The dimensionless equations are parameterized only in terms of a Stefan number S and a dimensionless solidification temperature Θ. Asymptotic solutions for the flow at short times and near the source are derived by perturbation series and similarity arguments. The general evolution of the flow is calculated numerically, and the scaled time to final solidification, the length and the thickness of the solidified product are determined as functions of S and Θ. The theoretical solutions provide simple models of the release of a pulse of magma into a fissure in the Earth's lithosphere or of lava flow on the flanks of a volcano after a brief eruption. Other geological events are better modelled as flows fed by a continual supply of hot fluid. The solidification of such flows will be investigated in Part 2.

The model developed in Part 1 (Lister 1994) for the solidification of hot fluid flowing in a thin buoyancy-driven layer between cold solid but freely deformable boundaries is extended to study the case of continual release of fluid. In this model lubrication theory was applied to reduce the equations of mass and heat conservation to a kinematic-wave equation and an advection-diffusion equation, which were coupled by the rate of solidification. The equations allow the source flux to be specified, and the cases of constant input and of flux proportional to a power of time are considered here. The structure of the flow differs significantly from the case of constant-volume release considered in Part 1. The advective resupply of heat prevents the flow from solidifying completely at the source and, if the initial fluid temperature is greater than the melting temperature of the solid, will in fact lead to rapid melting near the source. A perturbation expansion is used to describe the development of thermal boundary layers at the flow margins and the initial self-similar extension of the zone of melting. As the flow propagates beyond its thermal entry length, the fluid temperature falls to the liquidus value and melting gives way to solidification. At large times nearly all of the fluid supplied solidifies against the margins of the flow but, provided the source flux decreases less rapidly than t−½, sufficient reaches the nose of the flow that the flow continues to increase in length indefinitely. Analytic solutions are given for this long-time regime showing, for example, that the length increases asymptotically like t1/2 for constant-flux input. The theoretical solutions, which are calculated by a combination of analytic and numerical methods, may be used to describe the propagation of a dyke fed by a large body of magma through the Earth's lithosphere or the flow of lava down the flanks of a volcano during an extensive period of eruption.

The classical boundary-layer scaling laws proposed by Howard for Rayleigh–Bénard convection at high Rayleigh number extend to the analogous case of convection in saturated porous media. We computationally study two-dimensional porous-media convection near the onset of this scaling behaviour. The main result of the paper is the observation and study of instabilities that lead to deviations from the scaling relations.

At Rayleigh numbers below the scaling regime, boundary-layer fluctuations born at a Hopf bifurcation strengthen and eventually develop into thermal plumes. The appearance of plumes corresponds to the onset of the boundary-layer scaling behaviour of the oscillation frequency and mean Nusselt number, in agreement with the classical theory. As the Rayleigh number increases further, the flow undergoes instabilities that lead to ‘bubbles’ in parameter space of quasi-periodic flow, and eventually to weakly chaotic flow. The instabilities disturb the plume formation process, effectively leading to a phase modulation of the process and to deviations from the scaling laws. We argue that these instabilities correspond to parametric resonances between the timescale for plume formation and the characteristic convection timescale of the flow.

The commonly observed phenomenon of steady, viscous, free-surface flow on the outer surface of a rotating cylinder is investigated by means of an iterative, integral-equation formulation applied to the Stokes approximation of the Navier-Stokes equations. The method of solution places no restriction on the thickness of the fluid layer residing on the cylinder surface; indeed, results are presented for cases where the layer thickness is of the same order of magnitude as the cylinder radius.

Free-surface profiles and free-surface velocity distributions are presented for a range of flow parameters. Where appropriate, comparisons are made with the results of thinfilm theory; excellent agreement is observed.

For all film thicknesses and surface tensions, results show a high degree of symmetry about a horizontal axis even though the gravity field is vertical. A proof is presented that, for vanishing surface tension, this is a consequence of the Stokes approximation.

The long-time self-diffusivity in concentrated colloidal dispersions is determined from a consideration of the temporal decay of density fluctuations. For hydrodynamically interacting Brownian particles the long-time self-diffusivity, Ds∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, Ds(ϕ), and a contribution that depends on the distortion of the equilibrium structure caused by a diffusing particle. An argument is advanced to show that as maximum packing is approached the long-time self-diffusivity scales as Ds∞(ϕ) ∼ Ds0(ϕ)/g(2; ϕ), where g(2; ϕ) is the value of the equilibrium radial-distribution function at contact and ϕ is the volume fraction of interest. This result predicts that the longtime self-diffusivity vanishes quadratically at random close packing, ϕm ≈ 0.63, i.e. Ds∞D0(1-ϕ/ϕm)2 as ϕ→ϕm, where D0 = kT/6πνa is the diffusivity of a single isolated particle of radius a in a fluid of viscosity ν. This scaling occurs because Ds0(ϕ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1 -ϕ/ϕm)−1. A model is developed to determine the structural deformation for the entire range of volume fractions, and for hard spheres the longtime self-diffusivity can be represented by: Ds∞(ϕ) = Ds∞(ϕ)/[1 + 2ϕg(2;ϕ)]. This formula is in good agreement with experiment. For particles that interact through hard-spherelike repulsive interparticle forces characterized by a length b(> a), the same formula applies with the short-time self-diffusivity still determined by hydrodynamic interactions at the true or ‘hydrodynamic’ volume fraction ϕ, but the structural deformation and equilibrium radial-distribution function are now determined by the ‘thermodynamic’ volume fraction ϕb based on the length b. When b ≫ a, the long-time self-diffusivity vanishes linearly at random close packing based on the ‘thermodynamic’ volume fraction ϕbm. This change in behaviour occurs because the true or ‘hydrodynamic’ volume fraction is so low that the short-time self-diffusivity is given by its infinite-dilution value D0. It is also shown that the temporal transition from short- to long-time diffusive behaviour is inversely proportional to the dynamic viscosity and is a universal function for all volume fractions when time is nondimensionalized by a2/Ds∞(ϕ).

Multiple forms for standing waves in deep water periodic in both space and time are obtained analytically as solutions of Zakharov's equation and its modification, and investigated computationally as irrotational two-dimensional solutions of the full nonlinear boundary value problem. The different forms are based on weak nonlinear interactions between the fundamental harmonic and the resonating harmonics of 2, 3,…times the frequency and 4, 9,…respectively times the wavenumber. The new forms of standing waves have amplitudes with local maxima at the resonating harmonics, unlike the classical (Stokes) standing wave which is dominated by the fundamental harmonic. The stability of the new standing waves is investigated for small to moderate wave energies by numerical computation of their evolution, starting from the standing wave solution whose only initial disturbance is the numerical error. The instability of the Stokes standing wave to sideband disturbances is demonstrated first, by showing the evolution into cyclic recurrence that occurs when a set of nine equal Stokes standing waves is perturbed by a standing wave of a length equal to the total length of the nine waves. The cyclic recurrence is similar to that observed in the well-known linear instability and sideband modulation of Stokes progressive waves, and is also similar to that resulting from the evolution of the new standing waves in which the first and ninth harmonics are dominant. The new standing waves are only marginally unstable at small to moderate wave energies, with harmonics which remain near their initial amplitudes and phases for typically 100–1000 wave periods before evolving into slowly modulated oscillations or diverging.

This paper describes exact solutions of two-dimensional vortex structures that were published by Chaplygin (1899, 1903) at the turn of the last century, which seem to have escaped the attention of later investigators in this field. Chaplygin's solutions include that of an elliptical patch of uniform vorticity in an exterior field of pure shear and that of a (symmetric or non-symmetric) dipolar vortex with a continuous distribution of vorticity translating steadily along a straight path. In addition, a solution is presented for a non-symmetric vortex dipole moving along a circular trajectory. A concise account of Chaplygin's solutions is given, complemented by a more detailed analysis of some of their relevant properties.

A three-dimensional turbulent boundary layer (3DTBL) was generated on the floor of a low-speed wind tunnel by the imposition of a cross-stream pressure gradient using a 30° bend in the horizontal plane. The surface streamlines were deflected by as much as 22° relative to the local tunnel centreline. Downstream of the bend, the 3DTBL gradually relaxed towards a 2DTBL; this was an impulse-and-recovery experiment which focused on the outer layer. Mean velocities were measured with a three-hole yawmeter and turbulence quantities, which included the Reynolds-stress tensor and the triple products, were measured with a cross-wire hot-wire anemometer. The experiment isolated the effects of crossflow from those of adverse streamwise pressure gradients, which may have clouded interpretations of previous 3DTBL experiments. In particular, the detailed developments of the cross-stream shear stress and of the stress/energy ratio become clearer. The shear-stress vector lagged behind the velocity-gradient vector as crossflow developed; however, the two vectors became more closely aligned downstream of the bend. Reductions in the stress/energy ratio implied that crossflow made shear-stress production less efficient. Another effect of three-dimensionality was a change of sign in the vertical transport of turbulent kinetic energy by turbulence, in the inner part of the boundary layer.

A canonical Hamiltonian structure for the semi-geostrophic equations is presented and from this a reduced non-canonical Hamiltonian structure is derived, providing a fully nonlinear version of the approximate linearized vorticity advection representation. The structure of this model is described naturally within the framework of contact geometry. A Hamiltonian approach leading to a symplectic algorithm for calculating solutions to the equations of motion is formulated. Basic necessary functional methods are introduced and the Lagrangian and Eulerian kinematic structures are discussed, together with their relevance to symplectic integrating algorithms.

The inviscid instability of O(ϵ) two-dimensional periodic flows to spanwise-periodic longitudinal vortex modes in parallel O(1) shear flows is considered. In such cases, not only is the effect of fluctuations upon the mean state important but also the influence of the developing mean flow on the fluctuating part of the motion. The former is described by a generalized Lagrangian-mean formulation; the latter by a modified Rayleigh equation. Of specific interest is whether the spanwise distortion of the wave field feeds back to enhance or inhibit instability to longitudinal vortex form. Two cases are considered in detail: uniform shear between wavy walls and non-uniform shear beneath free-surface waves. In both cases wave distortion acts to inhibit, and in some circumstances curtail, instability for all but the shortest waves.

The behaviour of internal gravity wave packets approaching a critical level is investigated through numerical simulation. Initial-value problems are formulated for both small- and large-amplitude wave packets. Wave propagation and the early stages of interaction with the mean shear are two-dimensional and result in the trapping of wave energy near a critical level. The subsequent dynamics of wave instability, however, are fundamentally different for two- and three-dimensional calculations. Three-dimensionality develops by transverse convective instability of the two-dimensional wave. The initialy two-dimensional flow eventually collapses into quasi-horizontal vortical structures. A detailed energy balance is presented. Of the initial wave energy, roughly one third reflects, one third results in mean flow acceleration and the remainder cascades to small scales where it is dissipated. The detailed budget depends on the wave amplitude, the amount of wave reflection being particularly sensitive.

The inertial migration of a small rigid spherical particle, suspended in a fluid flowing between two plane boundaries, is investigated theoretically to find the effect on the lateral motion. The channel Reynolds number is of order unity and thus both boundary-induced and Oseen-like inertial migration effects are important. The particle Reynolds number is small but non-zero, and singular perturbation techniques are used to calculate the component of the migration velocity which is directed perpendicular to the boundaries of the channel. The particle is non-neutrally buoyant and thus its buoyancy-induced motion may be either parallel or perpendicular to the channel boundaries, depending on the channel alignment. When the buoyancy results in motion perpendicular to the channel boundaries, the inertial migration is a first-order correction to the magnitude of this lateral motion, which significantly increases near to the boundaries. When the buoyancy produces motion parallel with the channel boundaries, the inertial migration gives the zeroth-order lateral motion either towards or away from the boundaries. It is found that those particles which have a velocity exceeding the undisturbed shear flow will migrate towards the boundaries, whereas those with velocities less than the undisturbed flow migrate towards the channel centreline. This calculation is of practical importance for various chemical engineering devices in which particles must be filtered or separated. It is useful to calculate the forces on a particle moving near to a boundary, through a shear flow. This study may also explain certain migration effects of bubbles and crystals suspended in molten rock flow flowing through volcanic conduits.

The experimental results for an equilibrium boundary layer in a strong adverse pressure gradient flow are reported. The measurements show that similarity in the mean flow and the turbulent stresses has been achieved over a substantial streamwise distance where the skin friction coefficient is kept at a low, constant level. Although the Reynolds stress distribution across the layer is entirely different from the flow at zero pressure gradient, the ratios between the different turbulent stress components were found to be similar, showing that the mechanism for distributing the turbulent energy between the different components remains unaffected by the mean flow pressure gradient. Close to the surface the gradient of the mixing length was found to increase from Kl ≈ 0.41 to Kl ≈ 0.78, almost twice as high as for the zero pressure gradient case. Similarity in the triple correlations was also found to be good. The correlations show that there is a considerable diffusion of turbulent energy from the central part of the boundary layer towards the wall. The diffusion mechanism is caused by a second peak in the turbulence production, located at y/δ ≈ 0.45. This production was for the present case almost as strong as the production found near the wall. The energy budget for the turbulent kinetic energy also shows that strong dissipation is not restricted to the wall region, but is significant for most of the layer.

Convex transverse curvature effects in wall-bounded turbulent flows are significant if the boundary-layer thickness is large compared to the radius of curvature (large γ = δ/a). The curvature affects the inner part of the flow if a+, the cylinder radius in wall units, is small.

Two direct numerical simulations of a model problem approximating axial flow boundary layers on long cylinders were performed for γ = 5 (a+ ≈ 43) and γ = 11 (a+ ≈ 21). Statistical and structural data were extracted from the computed flow fields. The effects of the transverse curvature were identified by comparing the present results with those of the plane channel simulation of Kim, Moin & Moser (1987), performed at a similar Reynolds number. As the curvature increases, the skin friction increases, the slope of the logarithmic region decreases and turbulence intensities are reduced. Several turbulence statistics are found to scale with a curvature dependent velocity scale derived from the mean momentum equation. Near the wall, the flow is more anisotropic than in the plane channel with a larger percentage of the turbulent kinetic energy resulting from the streamwise velocity fluctuations. As the curvature increases, regions of strong normal vorticity develop near the wall.

The effects of convex transverse curvature on the wall pressure fluctuations were studied through direct numerical simulations. The flow regime of interest is characterized by large ratio of the shear-layer thickness to the radius of curvature (γ = δ/a) and by small a+, the radius of curvature in wall units. Two direct numerical simulations of a model problem approximating axial flow boundary layers on long cylinders were performed for γ = 5 (a+ ≈ 43) and γ = 11 (a+ ≈ 21). The space-time characteristics of the wall pressure fluctuations of the plane channel flow simulation of Kim, Moin & Moser (1987), which were studied by Choi & Moin (1990) are used to assess the effects of curvature.

As the curvature increases the root-mean-square (r.m.s.) pressure fluctuations decrease and the ratio of the streamwise to spanwise lengthscales of the wall pressure fluctuations increases. Fractional contributions from various layers in the flow to the wall r.m.s. pressure fluctuations are marginally affected by the curvature. Curvature-dependent timescales and lengthscales are identified that collapse the high-frequency range of the wall pressure temporal spectra and the high wave-number range of the wall pressure streamwise spectra of flows with different curvatures. Taylor's hypothesis holds for the wall pressure fluctuations with a lower convection velocity than in the planar case.