Boussinesq-type nonlinear equations for waves propagating over a sloping bottom are shown to accurately model the evolving bispectra of a spectrum of non-breaking shoaling ocean-surface gravity waves. The model response to a variation of the gentle, constant beach slope and the amount of nonlinear (i.e. non-random) phase coupling in the initial conditions is also examined. Variation of these quantities results in relatively little change in the overall structural evolution of the bicoherence and biphase (related to the nonlinear modification of the wave shape). The apparent unimportance of bottom slope motivates consideration of constant-depth KdV equations. Simple analytic solutions are found for harmonic growth in the special case of a monochromatic primary wavetrain. The associated bispectral evolution is qualitatively similar to field observations and to predictions based on the full Boussinesq model for a sloping bottom.

A new method is presented for deriving an integral expression for calculating the large-time longitudinal shear dispersivity in laminar or turbulent two-dimensional channel flow or tube flow.

The problem of quasi-geostrophic two-layer flow past a vertical cylinder on a β-plane is investigated analytically and numerically. Two parameter regimes are considered: (i) 0 ≤ E½/ε ≤ ∞ and β = O(1); (ii) E½/ε [Gt ] 1 and βε/E½ = O(1). ε is the Rossby number, E is the Ekman number and β is the beta parameter. In the first parameter regime the nonlinear interior and boundary-layer equations are integrated to determine if and when the wall shear stress vanishes so that an estimate of the condition for separation in the classical sense can be obtained. The results seem to explain the enhancement/suppression of separation in retrograde/prograde flows and the east-west asymmetry observed in the experiments of Boyer & Davies (1982). In the second parameter regime the analysis is linear and the vorticity balance is dominated by the β-effect and Ekman suction. When the flow at infinity is vertically sheared, two large standing interior eddies can be generated next to the cylinder. Only the interior solutions are given in (ii) since the boundary-layer flow is irrelevant to the large-scale behaviour.

An experimental investigation of the onset of convection in shallow fluid layers heated uniformly from below and cooled from above by an air layer has been made. If the depth of the silicone layer is smaller than 2 mm the onset of convection takes place in two stages. There is first a weak pattern, which is characterized by its appearance at ever smaller temperature gradients as the depth of the fluid is decreased. When the temperature difference across the fluid is increased a second strong pattern forms near the predicted critical Marangoni number. The cells in this pattern are hexagonal and seem to be what one has always referred to as Bénard cells. The temperature gradient at which this pattern appears increases with decreased depth. The heat transfer through the fluid has been measured. The critical temperature gradient for the formation of the hexagonal pattern has been determined from the break of the heat transfer curve.

The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead's point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point-vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.

We consider the evolution of finite uniform-vorticity regions in an unbounded in viscid fluid. We perform a perturbation analysis based on the assumption that the regions are remote from each other and nearly circular. Thereby we obtain a self-consistent infinite system of ordinary differential equations governing the physical-space moments of the individual regions. Truncation yields an Nth-order moment model. Special attention is given to the second-order model where each region is assumed elliptical. The equations of motion conserve local area, global centroid, total angular impulse and global excess energy, and the system can be written in canonical Hamiltonian form. Computational comparison with solutions to the contour-dynamical representation of the Euler equations shows that the model is useful and accurate. Because of the internal degrees of freedom, namely aspect ratio and orientation, two like-signed vorticity regions collapse if they are near each other. Although the model becomes invalid during a collapse, we find a striking similarity with the merger process of the Euler equations.

When a horizontal magnetic field B(z) is sheared vertically on a lengthscale L in a diffusionless fluid, critical layers occur at 2c where the local Alfvén speed V(zc) matches the phase speed c of the wave. However, when a vertical field Bz is introduced, all the critical layers disappear. The present study investigates the solution in the neighbourhood of zc when Bz/B is very small, in order to clarify the manner in which the vertical field annihilates the critical layers. It is found that the solution across the critical layer is adjusted in a thin magnetic layer whose thickness is determined by the parameter ε2 (= U/αV, where U, V are measures of the vertical and horizontal components of the Alfvén velocity and α/L is the horizontal wavenumber). The vertical field increases the order of the equation governing the vertical variations of the amplitude of the perturbations from two to four. Within the magnetic layer the two extra Alfvén waves, one upgoing and the other downgoing, interact with those due to the horizontal field to make the solution regular everywhere. The mean vertical wave energy flux varies continuously from one constant value far on one side of the layer to another constant value far on the other side of the layer.

The influence of the vertical field on the resistive instabilities present in its absence is also examined. It is found that the relative importance of resistivity and vertical field is measured by the ratio of the thicknesses of the resistive and magnetic layers. In general, the influence of the vertical field is to suppress resistive instabilities. The slow exchange resistive instabilities are suppressed by the presence of the vertical field if $\epsilon \ges a(S\alpha)^{-\frac{1}{3}}$ while the localized gravitational modes are inhibited for ε ≥ b(α2S)−¼ where a, b are constants whose values depend on the profile of the horizontal field and on the gravitational parameter G; and S is the Lundquist number.

The calculation of dynamic laser-light scattering by dilute suspensions of Brownian particles is reviewed. It is shown that present theories of diffusion can provide approximations for the autocorrelation of the intensity of the scattered light that are only uniformly accurate for correlation times up to order (D0k2)−l where D0 is the diffusivity of a single particle and k is the scattering wave vector. The meanings of, and connections between, down-gradient, self- and tracer diffusion for both short and long times are established and it is shown how these may be inferred from light-scattering experiments for optically monodisperse and polydisperse systems.

For dilute systems, equations giving the time evolution of the intermediate and self-intermediate scattering functions, F(k, t) and Fs(k, t), accurate to first order in the volume concentration of particles are constructed, and are solved for suspensions of hard spheres with and without hydrodynamic interaction. For short and long times (semi-) analytic solutions are given; for intermediate times numerical results are presented. The formal correspondence of the limiting values of the time-dependent solutions with the results of Batchelor (1976, 1983) and others for steady sedimentation in polydisperse systems is established.

The trapping of gravity waves over a circular sill, originally studied by Longuet-Higgins (1967), is re-examined in the limit δ ≡ dl/d ↓ 0 with λ ≡ σ2d/g = O(1), where d1 is the depth over the sill, d the outer depth, and a the angular frequency of the incident wave. Explicit results are obtained for the resonance curves. These results are in qualitative agreement with the corresponding shallow-water approximations (Λ [Lt ] 1) of Longuet-Higgins, which have been questioned by Renardy (1983). A remarkably simple result is obtained for the mean-square response to a broadband, randomly phased incident wave.

Spin-stabilized projectiles with liquid payloads can experience a severe flight instability characterized by a rapid yaw-angle growth and a simultaneous loss in spin rate. Laboratory experiments and field tests have shown that this instability originates from the internal fluid motion in the range of high viscosity. After evaluation of the experimental data and analysis of the equations for the fluid motion in a spinning and nutating cylinder, we have developed a simple model of this flow. Disregarding the finite length of the cylinder, this model provides the flow field and the viscous contribution to the liquid moments in analytical form. At low Reynolds number, the flow field agrees well with computational results for the centre section of a cylinder of aspect ratio 4.3. The roll moment caused by this flow largely agrees with experimental data for a wide range of Reynolds numbers. Estimates of the temperature variation indicate that discrepancies at very low Reynolds numbers may originate from associated changes of the viscosity during the experiments.

This paper treats the liquid-metal flow in a straight circular pipe with a thin metal wall. A strong magnetic field is applied by a magnet with parallel poles that end abruptly. In the plane midway between the magnet poles: (a) far upstream, the flow is uniform, fully developed in a uniform magnetic field; (b) as the flow enters the non-uniform magnetic field near the end of the magnet, the flow moves away from the central part of the pipe and becomes concentrated as two jets near the points where the magnetic field is tangential to the pipe wall; (c) further downstream where the magnetic field strength is O(c⅙) compared with its value upstream, the flow migrates from these jets back towards a uniform flow distributed over the entire pipe cross-section. Here, c is the wall conductance ratio, which is assumed to be small. The analysis also applies to flow into the magnetic field, because inertial effects and induced magnetic fields are neglected. There are circulations of electric current in planes parallel to the magnet poles. These currents produce a pressure drop in addition to that for two fully developed flows in a uniform magnetic field and in no magnetic field, joined at the end of the magnet. This pressure drop is given by 0.9336 $\beta^{\frac{1}{3}}c^{\frac{2}{3}}\sigma$V0B20L, where β is a measure of the magnetic field gradient with a minimum value of 2/π, σ is the liquid metal's electrical conductivity, V0 is the average velocity, B0 is the strength of the uniform magnetic field, and L is the inside radius of the pipe. This three-dimensional pressure drop is $O(c^{-\frac{1}{3}}L) $ times the pressure gradient for the fully developed flow in the uniform magnetic field.

In this paper we describe the design and operating characteristics of a computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear flows. The control system is based upon the use of: a video camera to visualize the instantaneous position of the drop or particle; a PDP 11/23 computer, with a pipeline processor acting as an interface between the camera and computer, to calculate the position and implement a control strategy, and d.c. stepping motors to convert an electronic signal to angular velocities of the four rollers. The control objective is to keep the centre of mass of the drop/particle at the centre of the region between the rollers where there is a stagnation point in the undisturbed flow, while maintaining the shear-rate and the ratio of vorticity to strain rate in the flow at fixed values. The resulting system is suitable for studies of: the rotational motions of single solid particles; the deformation and burst of single droplets; or the hydrodynamic interactions of two particles or drops, one of which is held with its centre-of-mass fixed at the stagnation point of the undisturbed flow. In all cases, the flow can be varied from pure rotation to pure strain, and the shear rate can be either steady or changing as a prescribed function of time.

We consider the deformation and burst of small fluid droplets in steady linear, two-dimensional motions of a second immiscible fluid. Experiments using a computer-controlled, four-roll mill to investigate the effect of flow type are described, and the results compared with predictions of several available asymptotic deformation and burst theories, as well as numerical calculations. The comparison clarifies the range of validity of the theories, and demonstrates that they provide quite adequate predictions over a wide range of viscosity ratio, capillary number, and flow type.

An experimental and theoretical study of the propagation and stability of long nonlinear internal waves over slope–shelf topography is presented. A generalized Korteweg–de Vries (KdV) equation, including the effects of nonlinearity, dispersion, dissipation and varying bottom topography, is formulated and solved numerically for single and rank-ordered pairs of solitary waves incident on the slope. The results of corresponding laboratory experiments in a salt-stratified system are reported. Very good agreement between theory and experiment is obtained for a range of stratifications, topography and incident-wave amplitudes. Significant disagreement is found in some cases if the effects of dissipation and higher-order (cubic) nonlinearity are not included in the theoretical model. Weak shearing and strong breaking (overturning) instabilities are observed and found to depend strongly on the incident-wave amplitude and the stratification on the shelf. In some cases the instability of the lowest-mode wave leads to the generation of a second-mode solitary wave. The application of these findings to the prediction and interpretation of field data is discussed.

The investigation of Gill (1977) on the effects of a finite upstream depth upon frictionless flow through a rotating box-like channel has been extended to take into account a parabolic geometry. In addition to being more geophysically realistic, this type of topography with continuously sloping lateral boundaries has the advantage that it yields a unified solution. In contrast to the case of a rectangular channel, no separation of the geostrophically balanced downchannel flow from the sidewalk can take place. The resulting algebraic problem can be resolved either using an iterative technique or by the construction of perturbation series solutions. One of the most important results to emerge from the analysis is that the classical concept of hydraulic control is only applicable for a limited range of the parameters governing the problem. It is finally argued that this behaviour of the solutions is not due to the specific choice of geometry, but rather represents a common feature for all topographies characterized by a continuously sloping cross-stream bottom profile.

A theory is presented for the development of the nonlinear critical layer below an unsteady free surface wave, of amplitude ε, described by the Korteweg-de Vries (KdV) equation. The problem is formulated (via the Euler equations) for wave propagation over an arbitrary shear in a two-dimensional channel which contains a critical level. The equations are scaled so as to be valid in the far-field regime of the surface wave, appropriate to the existence of the KdV equation, i.e. long waves. The regions above and below the critical layer are solved (to O(ε), as ε → 0) and thence expanded in the neighbourhood of the critical layer itself. The symmetry of the critical layer solution, assuming that it exists, is then sufficient to determine the Burns integral for the linearized wave speed, and the relevant KdV equation. These turn out to be the classical results evaluated in terms of finite parts.

The critical layer, of thickness O(ε½), is analysed to O(ε) and matched to the outer regions of the flow field. The initial configuration is taken to contain no closed streamlines, and so the vorticity can, presumably, be assigned from the undisturbed conditions at infinity. The initial surface profile must therefore contain a single peak, but by virtue of the KdV equation this can evolve into any number of solitons. Between consecutive pairs of peaks there will now appear regions of closed streamlines (cat's-eyes) with known vorticity. No recourse to a viscous argument is necessary to uniquely determine this vorticity. However, it is shown that the vorticity cannot be prescribed arbitrarily at all orders, initially: the long-wave assumption imposes a certain structure on the problem, and then the continuity of stream function and particle velocity fixes the vorticity. This agrees with the work of Varley & Blyth (1983) on the hydraulic equations. The vorticity inside the separating streamlines is obtained to O(ε), but it is shown that for unsteady motion this asymptotic expansion is not uniformly valid as the bounding streamlines are approached. An alternative method, which exploits Varley & Blythe's approach, is used to confirm the correctness of our results away from these boundaries, and to indicate that a non-uniformity is present near the separating streamlines. Thus the model requires the inclusion of a vortex sheet; for steady flow a jump in vorticity is sufficient. The removal of the discontinuity by allowing a distortion of the main flow outside the critical layer is briefly discussed.

Some results are presented for the formation of a single cat's-eye by using the exact 2-soliton solution of the KdV equation.

The particle paths of the Arnold-Beltrami-Childress (ABC) flows \[ u = (A \sin z+ C \cos y, B \sin x + A \cos z, C \sin y + B \cos x). \] are investigated both analytically and numerically. This three-parameter family of spatially periodic flows provides a simple steady-state solution of Euler's equations. Nevertheless, the streamlines have a complicated Lagrangian structure which is studied here with dynamical systems tools. In general, there is a set of closed (on the torus, T3) helical streamlines, each of which is surrounded by a finite region of KAM invariant surfaces. For certain values of the parameters strong resonances occur which disrupt the surfaces. The remaining space is occupied by chaotic particle paths: here stagnation points may occur and, when they do, they are connected by a web of heteroclinic streamlines.

When one of the parameters A, B or C vanishes the flow is integrable. In the neighbourhood, perturbation techniques can be used to predict strong resonances. A systematic search for integrable cases is done using Painlevé tests, i.e. studying complex-time singularities of fluid-particle trajectories. When ABC ≠ 0 recursive clustering of complex time singularities occurs that seems characteristic of non-integrable behaviour.

Consider the unsteady motion of a fluid–fluid interface over and attached to a solid surface with one-dimensional periodic roughness in the limit of small capillary number [Copf ]. We show that the macroscopic behaviour of the interface can be described, to leading order in [Copf ], in terms of well-defined continuum quantities, even though the complicated fluid motion in the neighbourhood of the contact line cannot. The key is that contact-angle hysteresis makes it possible to isolate the viscous stress singularity at the contact line, since for a sufficiently slowly moving fluid—fluid interface all the movement of the contact line occurs in a time much shorter than the macroscopic timescale. The effective ‘slip length’ for the macroscopic description is shown to be velocity dependent and equal to a[Copf ]−1, where a is the wavelength of the roughness on the solid surface. Finally, we consider surfaces with two-dimensional random roughness, and argue that they too would exhibit velocity dependent ‘slip lengths’, though the velocity dependence would be stronger in this case. These results combined with earlier work (Jansons 1985) explain why observed ‘slip lengths’ can be much larger than roughness dimensions, and why the degree of ‘stick-slip’ decreases with increasing speed.

The 203rd Euromech Colloquium, on developments in the theoretical modelling of homogeneous and heterogeneous combustion was held at Cranfield Institute of Technology from 2 to 4 December 1985. The emphasis of the meeting was on the ability of analytic, numerical and approximate methods to predict, or interpret, events which occur in the laboratory or in the field. There were forty-five participants in the Colloquium from six different countries.

A theory is developed to predict the motion of N neutrally buoyant spheres suspended in laminar flow between parallel plates. The spheres are at large separation yet nearer each other than the duct walls, and the Reynolds number is small. In this parameter range, viscous interactions are larger than inertial effects, and can be represented in terms of a superposition of ‘strainlets’. Several examples are given to show this viscous interaction effect. Near the leading edge of a front of spheres or near the trailing edge significant lateral migration velocities can occur, being at least one order of magnitude larger than inertially induced migration velocities. This phenomenon may have a negative effect on ‘chromatographic’ separation schemes, affecting particle concentration, recovery and resolution.