We study the stretching and bending of line elements transported in random flows with known Eulerian statistics in two and three dimensions. By making use of a cumulant expansion for the log-size of material elements we are able to analyse the exponential stretching they exhibit in random flows and identify conditions under which it will and will not occur. The results are confirmed by our numerical simulation.

We also examine the evolution of curvature in material elements and confirm by numerical simulation that it is governed by an appropriate version of the Pope equation. By modelling this equation as stochastic differential equation we are able to explain the appearance of a power-law tail in the probability distribution for large curvature observed by Pope, Yeung & Girimaji (1989) for surface elements. In two dimensions the appearance of the tail can indeed be attributed to the occurrence of events in which the material element undergoes contraction rather than stretching while subject to bending. In three dimensions the relationship between episodes of contraction and strong bending is less direct. This power-law tail allows us to reconcile the observed asymptotic stability, which we confirm here, of the powers and cumulants of the log-curvature with the unboundedness of powers of the curvature itself.

This paper is concerned with an experimental investigation of the oblique impingement of an unsteady, axisymmetric two-phase jet on heated surfaces. Size and velocity were measured simultaneously with a phase-Doppler velocimeter, and the spatial distributions over the wall jet were found to be correlated with the interfacial activities as inferred from vertical velocity measurements in the vicinity of the wall. These results are discussed together with size measurements by a laser-diffraction technique to quantify the effect of the approach conditions of the inflowing jet droplet field and wall temperature in relation to mechanisms of secondary atomization.

Two mechanisms of secondary atomization were identified; the first did not involve direct wall contact and was due to the strain acting on the droplets by the continuous phase within the impingement region and was enhanced by thermal effects from the wall to cause breakup. The approaching velocity of the inflowing droplets to the plate was important to this process so that higher velocities increased the rate of strain within the impingement region and, consequently, the wall temperature promoting the secondary atomization shifted towards lower values. The second mechanism required direct wall contact and involved atomization of the film deposited on the wall by the impingement of the inflowing two-phase jet. With the penetration of high-speed inflowing droplets into the film, liquid mass was raised into the two-phase medium due to splashes from the film so that a new size class with larger droplets was generated. The resulting large droplets tended to stay close to the wall within the impingement region with small vertical velocities

In between the injections, the suspended droplet field above the film oscillated normal to the plate as a cloud so that the impact of large droplets on the film resulted in coalescence with the film and the ejection of smaller numbers of small droplets. The unsteady wall jet flow, caused by the arrival of the spray at the plate, swept the vertically oscillating droplet cloud radially outwards so that the resulting radial transport caused the dynamics of the unsteady film to be correlated with the size characteristics of the unsteady wall jet. Based on this phenomenological description, a radial droplet transport equation is derived.

The correlation suggests that the secondary atomization with direct wall contact is the dominant process for the generation of a new size class within the wall flow and initiates the mutual interaction between the unsteady film and wall jet droplet field.

The sloshing under gravity is considered for a liquid contained in a horizontal cylinder of uniform cross-section and symmetric about a vertical plane parallel to its generators. Much of the published work on this problem has been concerned with twodimensional, transverse oscillations of the fluid. Here, attention is paid to longitudinal modes with variation of the fluid motion along the cylinder. There are two known exact solutions for all modes; these are for cylinders whose cross-sections are either rectangular or triangular with a vertex semi-angle of ¼π. Numerical solutions are possible for an arbitrary geometry but few calculations are reported in the open literature. In the present work, some general aspects of the solutions for arbitrary geometries are investigated including the behaviour at low and high frequency of longitudinal modes. Further, simple methods are described for obtaining upper and lower bounds to the frequencies of both the lowest symmetric and lowest antisymmetric modes. Comparisons are made with numerical calculations from a boundary element method.

The phenomena of momentum and heat transfer associated with an impulsively started spherical particle in a quiescent argon thermal plasma environment is considered. The changing plasma thermodynamics and transport property effects are studied using a Chebyshev-Legendre spectral method. Steady-state solutions for the case of constant sphere surface temperature are obtained and compared with previously published results. Transient solutions with particle internal heat conduction included are also presented. Results indicate that the magnitude of the drag force increases as the plasma free-stream temperature increases, while the Nusselt number decreases with increasing free-stream temperature. Effects due to different initial particle temperatures on the transient Nusselt number and drag coefficient are demonstrated.

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.

The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.

The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.

We consider model nonlinear wave equations of the form ut + uux = [Hscr ](x,t;u, ux,…) arising in gasdynamics and other fields, [Hscr ] incorporating various linear mechanisms of dissipation and dispersion. If [Hscr ] includes a thermoviscous dissipation term ∈uxx, then it is generally believed that u(x,t) will remain single-valued for all t > 0 and all single-valued u(x, 0), for any ∈ > 0. The question addressed here is whether, if thermoviscous dissipation is excluded from [Hscr ], u(x, t) remains single-valued for all t > 0, or whether certain dissipative-dispersive mechanisms (such as relaxation processes) are in themselves insufficient to prevent wave overturning. To answer this we propose a numerical scheme based on the use of intrinsic coordinates ψ = ψ(s, t) to describe the waveform at each time. In this paper, the method is described and validated by comparisons with the exact solutions for certain [Hscr ] ([Hscr ] = 0, [Hscr ] = -αu, [Hscr ] = ∈uxx). These comparisons show that the scheme is free of numerical viscosity effects which preclude the solution of the problem by finite-difference or spectral methods applied to the signal u(x, t), that it can reliably distinguish between finite-time overturning and merely the formation of steep gradients, and that it can accurately predict the time of overturning when it does occur. Having established the validity of the method, attention can then be turned to those cases where criteria for overturning have not as yet been determined by conventional methods. In Part 2, harmonic wave propagation through a relaxing gas is investigated.

We consider finite-amplitude acoustic disturbances propagating through media in which relaxation mechanisms, such as those associated with the vibration of polyatomic molecules, are significant. While the effect of these relaxation modes is to inhibit the wave steepening associated with nonlinearity, whether a particular mode is sufficient to prevent the occurrence of multi-valued solutions will depend on the form of the disturbance and on the characteristic parameters of the relaxation. Analysis of this condition is necessary in order to reveal which physical mechanisms actually determine the evolution of the wave profile. This then dictates the scaling of any embedded shock regions. Sufficient conditions for the occurrence of multi-valued solutions are obtained analytically for periodic waves, hence proving that in certain circumstances relaxation is in fact insufficient in fully describing the wave propagation. A much more precise criterion is then obtained numerically. This uses the techniques described in Part 1 for analysing the phenomenon of wave overturning using intrinsic coordinates. Illustrations are provided of the development of a harmonic signal for different classes of material parameters.

Two-dimensional time-dependent buoyancy-induced flows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature, are studied numerically. Transitions to unsteady flows are performed by direct simulations for various depths of immersion of the source in the central vertical plane of air-filled vessels. For a square vessel and a line source near the bottom wall, the numerical solutions exhibit a sequence of instabilities, called natural swaying motion of confined plumes, beginning with a periodic regime having a high fundamental frequency followed by a two-frequency locked regime. Then, broadband components appearing in the spectra indicate chaotic behaviour and a weakly turbulent motion arises via an intermittent route to chaos. For rectangular vessels of aspect ratio greater than 2 and depths of immersion greater than the width, the flow undergoes a pitchfork bifurcation. This symmetry breaking is driven by the destabilization of an upper unstable layer of stagnant fluid above the plume. Then a subcritical Hopf bifurcation occurs. On the other hand, if the depth of immersion is lower than the width of the vessel, a stable layer of fluid is at rest below the line source. Then penetrative convection sets the whole air-filled vessel in motion and an oscillatory motion of very low frequency arises through supercritical Hopf bifurcation followed by a two-frequency locked state.

The effect of channel walls on the hydrodynamic characteristics of fixed or oscillating bodies is discussed using classical linear water wave theory. Particular attention is paid to the occurrence of trapped modes persisting local to the fixed body and which are manifested in a non-uniqueness of the corresponding forced problem at the trapped mode frequency. The general ideas are illustrated by consideration of two simple geometries for which semi-analytic solutions are available, namely a circular cylinder either partly immersed or extending throughout the water depth, and a thin vertical plate parallel to the channel walls which extends throughout the water depth. Conclusions are drawn concerning the conditions under which trapped modes may exist and their effect on the hydrodynamic characteristics of more general bodies.

Scaling analyses describe the evolution of an extrusion of viscous or plastic fluid in the presence of surface cooling and solidification, under the assumption that the flows consist of two components: an isothermal interior and a surface crust. The ‘crust’ is a complex thermal and rheological boundary layer which we model using a viscous, plastic or brittle rheology. These models are thought to be relevant to some types of lava flows and address the effects of cooling on their morphology and rate of advance of the flow front. They show that effects of crust strength will dominate over both viscous and yield stresses in the interior when the ratio of crust thickness to flow length exceeds the ratio of effective yield stress of the crust to basal shear stress exerted on the bulk of the flow, a condition that appears likely to be met by many lava flows and small outgrowths on large flows. Similarity solutions are compared with measurements on the spreading of extrusions of wax beneath cold water in the laboratory, where the extruded liquid is viscous but develops a solid crust. Crust strength provides the dominant retarding force for the wax flows in cases where surface solidification occurs rapidly compared with lateral advection. These conditions give flows topped by sheets of solid that buckle or rift apart, or extrusions that enlarge by small bulbous outgrowths (analogous to ‘pillows’ on submarine lavas and ‘toes’ on some sub-aerial basalt flows). A comparison of the models with data for the growth of lava domes in the craters of Mount St Helens and Soufrière of St Vincent volcanoes reveals that spreading of those domes was not controlled by stresses in the flow interior. Instead the data are consistent with a balance between gravity and yield stresses in a thin crustal layer over most of the growth period.

The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2π(T/ρg)1½2 (g = gravity, T = surface tension, ρ = density). The dispersion relation $c^2 = 2(1-\frac{11}{32}\alpha^2_{\max)$ between the speed c and the maximum surface slope αmax is derived from the nonlinear Schrödinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations.

The dynamics of an asymmetric annulus of vorticity in an incompressible, inviscid twodimensional fluid are experimentally studied using a pure electron plasma. A strict fluid analogy requires that the plasma system behave like an ideal fluid in a frictionless cylindrical container. For certain parameters the asymmetric annulus undergoes a complex evolution which is quite different from that of a symmetric annulus. During the first ‘active’, phase the symmetries grow until the annulus collapses, leaving a large vortex at the device centre. In the second, ‘passive’, phase the remainder of the annulus winds around this central vortex into an ever tighter spiral. Finally, slow shear instabilities destroy the structure of the highly evolved spiral.