The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.

The paper de als with the plane unsteady problem of the collision of two rigid undeformable and shallow surfaces, one of which is covered by a thin layer of an ideal incompressible liquid. At the initial instant of time, a dry surface touches the liquid free boundary at a single point and then starts to penetrate the liquid layer. The flow region is divided into four parts: the region beneath the entering surface, the jet root, the spray jet and outer region. Inside each of those subdomains the flow patterns have their own peculiarities and are analysed separately. The matching conditions allow us to obtain the uniformly valid asymptotic solution of the original problem. The relative body motion and the characteristics of the spray jets generated under the impact are determined. The condition on the shapes of the bodies, under which the velocity of the impact of the rigid surfaces is non-zero, is derived.

A modified slip boundary condition is obtained to represent the effects of small roughness-like perturbations to an otherwise-plane fixed wall which is acting as a boundary to steady laminar flow of a viscous fluid. In its simplest form, for low local Reynolds number and small roughness slope, this boundary condition involves a constant apparent backflow at the mean surface or, equivalently, represents a shift of the apparent plane boundary toward the flow domain. Extensions of the theory are also made to include finite local Reynolds number and finite roughness slope.

The dynamics of a free-surface slender two-dimensional stream (liquid sheet) issuing from a nozzle in the gravitational field in still air, under the effect of surface-active agents, are analysed experimentally. The particular test section geometry (the liquid is forced to assume a bidimensional form between two vertical guides and a horizontal plate placed at a certain variable distance from the nozzle exit section) employed in this study gives rise to various flow regimes depending on the governing parameters: liquid flow rate, sheet height, surface pressure, gravity. Two basic phenomena are observed: thinning of the sheet (with recirculating motion inside it) and sheet-threadlines transition. For a certain surfactant (bulk) concentration, there exists a minimum critical flow rate value for which the sheet is seen to thin starting at both of the sheet bottom corners. A ridge, usually referred to as a Reynolds ridge in the literature, separates the sheet from the thin-film regions. The thin films exhibit recirculating flows (caused by the onset of surfactant-induced surface-pressure-driven convection in the gravitational field) and extend to the entire rectangular interface as the flow rate is reduced. At zero flow rate the thinned sheet resembles a plane vertical soap film showing a recirculating cellular structure. These phenomena are linked to the presence of surface-active material adsorbed at the liquid-air interface and occur when the sheet height exceeds a critical value. Otherwise, at a critical flow rate value the liquid sheet breaks up into an array of (more or less regularly distributed) discrete threadlines (vertical jets), whose spacing depends on the surface tension of the test liquid.

The level-set approach of Osher & Sethian to tracking interfaces is successfully adapted to the simulation of a premixed turbulent open V-flame including the effects of exothermicity and baroclinicity. In accord with experimental observations this algorithm, along with a flame anchoring scheme, predicts flame cusping for a case in which a strong vortex pair interacts with the flame front. The computed velocity and scalar statistics obtained for the turbulent V-flame compare reasonably well with experimental results by Cheng & Shepherd, and demonstrate the importance of flame-generated vorticity in the determination of flame dynamics and product velocity characteristics.

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of order R−1 in the boundary-layer Reynolds number R. In Blasius flow, the present approach is consistent with that of Bertolotti et al. (1992) to O(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis to O(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.

However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal order R−1 except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

Asymptotic results for the free decay of shape oscillations of viscous liquid spheres have been extended to include higher-order terms in the ratios of the inner and outer viscous penetration lengths to the radius. The new expressions are shown to be important for studies in which the two fluids have dissimilar densities and viscosities such as air/liquid systems. The analysis also includes an expansion for the frequency of maximum response of driven oscillations. The calculations are supported by measurements of the small-amplitude quadrupole mode free decay of nearly spherical bubbles acoustically levitated in clean water. The bubble radii ranged from 400 μm to 1400 μm. The results are interpreted in light of the initial-value problem. The lack of excess damping suggests that the interface behaves ideally for times up to two hours after bubble injection. Measurements were also carried out on bubbles in 0.5 m NaCl solution and in sea water. Larger bubbles (radius > 800 μm) in clean water exhibit damping two to four times larger than predicted by theory. The transition from this anomalous damping to theoretical damping is a very abrupt function of radius. All observations were carried out with similar acoustic fields for counteracting buoyancy. The excess damping appears to be associated with some nonlinear response of the bubble.

We show that rotation appears owing to bifurcation in primarily pure meridional steady motion of viscous incompressible fluid. This manifestation of the laminar axisymmetric 'swirl dynamo’ occurs in flows inside liquid conical menisci with the cone half-angle θc < 90°. The liquid flows towards the cone apex near the surface and moves away along the axis driven by (i) surface shear stresses (typical for electrosprays) or (ii) by body Lorentz forces (e.g. in the process of cathode eruption). When the motion intensity increases and passes a critical value, new swirling regimes appear resulting from the supercritical pitchfork bifurcation. This agrees with recent observations of swirl in Taylor cones. We find that when the swirl Reynolds number Γc reaches a threshold value, flow separation occurs and the meridional motion becomes two-cellular with inflows near both the surface and the axis, and an outflow near the cone θ = θs, 0 < θs < θc. In the limit of high Γc, the angular thickness of the near-surface cell tends to zero. In case (i) the swirl is concentrated near the surface while the motion inside the inner cell becomes purely meridional with the radial velocity being uniform. We also study the two-phase flow of a liquid inside and a gas outside the meniscus. Flow separation occurs in both media and then swirl is concentrated near the interface. In case (ii) we reveal another interesting effect: a cascade of flow separations near the axis. As the driving forces increase, meridional motion becomes multi-cellular although very slow in comparison with swirl. To cover all ranges of parameters we combine numerical calculations and asymptotic analyses.

The flow of a uniform stream of pure saturated vapour past a cold, semi-infinite vertical plate is examined. The formulation incorporates the limits of both pure forcedconvection and pure body-force-convection laminar film condensation. Detailed asymptotic and exact numerical solutions are obtained and comparisons drawn with approximate methods and experimental results reported in the literature.

Deformation due to hydrodynamic interactions between two deformable buoyant drops may result in the alignment and coalescence of horizontally offset drops. Three-dimensional boundary integral calculations are presented for systems containing two, three or four drops and it is argued that the interactions which occur between three drops or four drops may be characterized qualitatively by the two-drop interactions. In a dilute monodisperse suspension, the rate of coalescence of deformable drops is calculated using far-field analytical results and is found to be proportional to the Bond number. The rate of coalescence in a dilute polydisperse suspension of bubbles in corn syrup is determined by performing a large number of laboratory experiments for Bond numbers based on the larger bubble radius 15 < [Bscr ] < 120. The rate of coalescence is enhanced (by a factor of 10 for [Bscr ] = 10), owing to the effects of deformation, compared to the predictions of models which include hydrodynamic interactions and van der Waals forces among spherical bubbles. The rate of coalescence is greater than the rate predicted by the Smoluchowski model which ignores all hydrodynamic interactions. The experimental results are used to calculate the evolution of the bubble size distribution in suspensions using a standard one-dimensional population dynamics model; deformation affects the size distribution in suspensions, resulting in a wider range of bubble sizes.

Separation of a supersonic boundary layer (or equivalently a hypersonic boundary layer in a region of weak global interaction) near a compression ramp is considered for moderate wall temperatures. For small ramp angles, the flow in the vicinity of the ramp is described by the classical supersonic triple-deck structure governing a local viscous-inviscid interaction. The boundary layer is known to exhibit recirculating flow near the corner once the ramp angle exceeds a certain critical value. Here it is shown that above a second and larger critical ramp angle, the boundary-layer flow develops an instability. The instability appears to be associated with the occurrence of inflection points in the streamwise velocity profiles within the recirculation region and develops as a wave packet which remains stationary near the corner and grows in amplitude with time.

We present results of numerical experiments on two-dimensional compressible convection in a polytropic layer with an imposed horizontal magnetic field. Our aim is to determine how far this geometry favours the occurrence of travelling waves. We therefore delineate the region of parameter space where travelling waves are stable, explore the ways in which they lose stability and investigate the physical mechanisms that are involved. In the magnetically dominated regime (with the plasma beta, $\hat{\beta}$ = 8), convection sets in at an oscillatory bifurcation and travelling waves are preferred to standing waves. Standing waves are stable in the strong-field regime ($\hat{\beta}$ = 32) but travelling waves are again preferred in the intermediate region ($\hat{\beta}$ = 128), as suggested by weakly nonlinear Boussinesq results. In the weak-field regime ($\hat{\beta}$ ≥ 512) the steady nonlinear solution undergoes symmetry-breaking bifurcations that lead to travelling waves and to pulsating waves as the Rayleigh number, $\circ{R}$, is increased. The numerical experiments are interpreted by reference to the bifurcation structure in the ($\hat{\beta}$, $\circ{R}$)-plane, which is dominated by the presence of two multiple (Takens-Bogdanov) bifurcations. Physically, the travelling waves correspond to slow magnetoacoustic modes, which travel along the magnetic field and are convectively excited. We conclude that they are indeed more prevalent when the field is horizontal than when it is vertical.

A high-resolution, quasi-geostrophic numerical model is utilized to examine two-layer baroclinic flow in a cylinder. Particular attention is given to the role of horizontal shear of the basic state induced by viscosity near the cylinder wall, and to the desymmetrization brought about by the cylindrical geometry, in the transition to baroclinic chaos. Solutions are computed for both f-plane and β-plane situations, and the results are compared to previous laboratory experiments. Agreement in the former case is found to be good, although the onset of chaos occurs at slightly lower forcing in the laboratory when its basic flow is prograde, and at higher forcing amplitude when the experimental basic azimuthal currents are retrograde. This suggests that the modest discrepancies may be attributable to computationally neglected ageostrophic effects in the interior fluid and Ekman boundary layers. When β ≠ 0, the numerical and laboratory results are in excellent agreement. The computational simulations indicate that the viscous sidewall boundary layer plays a pivotal role in the dynamics. Moreover, in contrast to previous studies performed in a periodic, rectilinear channel, the route to chaos is largely temporal and involves relatively few spatial modes. The reduction in symmetries upon going from f-plane channel to either f-plane or β-plane cylinder models leads to fewer secondary instabilities and fewer spatial modes that are active in the dynamics.

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.

Moore's (1954) inviscid linear analysis of the interaction of a shock wave with a plane acoustic wave is evaluated by comparison to computation. The analysis is then extended to study the interaction of an isotropic field of acoustic waves with a normal shock wave. The evolution of fluctuating kinetic energy, sound level and thermodynamic fluctuations across the shock wave are examined in detail.

The interaction of acoustic fluctuations with the shock is notably different from that of vortical fluctuations. The kinetic energy of the acoustic fluctuations decreases across the shock wave for Mach numbers between 1.25 and 1.8. For Mach numbers exceeding 3, the kinetic energy amplifies by levels that significantly exceed those found in the interaction of vortical fluctuations with the shock. Upon interacting with the shock wave, the acoustic waves generate vortical fluctuations whose contribution to the far-field kinetic energy increases with increasing Mach number. The level of sound increases across the shock wave. The rise in the sound pressure level across the shock varies from 5 to 20 dB for Mach number varying from 1.5 to 5. The fluctuations behind the shock wave are nearly isentropic for Mach number less than 1.5, beyond which the generation of entropy fluctuations becomes significant.