Numerical solutions have been obtained for steady viscous flow past a finite flat plate attached normally to an infinite wall at Reynolds numbers Re up to 2800. Separately, Levi-Civita's method has been used to obtain the flow solution based on the free-streamline theory. Apart from the case of uniform flow past an isolated obstacle, the present flow problem appeared to exhibit the validity of the free-streamline model for the global structure of the flow field. The position of the separation point could be predicted by two terms of the asymptotic solution based on the Sychev–Smith model within 0.6% error for the range 100 [les ] Re [les ] 2800.

In the vicinity of a caustic of a dispersive wave system, where the group velocity is stationary and hence dispersive effects are relatively weak, the nonlinear Schrödinger equation (NLS) breaks down, and the propagation of the envelope of a finiteamplitude wavepacket is governed by a modified nonlinear Schrödinger equation (MNLS). On the basis of the MNLS, a search for wave envelopes of permanent form is made near a caustic. It is shown that possible solitary wave envelopes satisfy a nonlinear eigenvalue problem. Numerical evidence is presented of symmetric, double-hump solitary-wave solutions. Also, a variety of periodic envelopes are computed. These findings are discussed in connection with previous analytical and numerical work.

A boundary-element method is applied to solve the equations describing the deformation of a two-dimensional liquid region under the influence of gradients of the curvature of its outer boundary. This research is motivated by a desire to obtain a better understanding of viscous sintering processes in which a granular compact is heated to a temperature at which the viscosity of the constituent material becomes low enough for surface tension to cause adjacent particles to deform and coalesce. The boundary-element method is capable of showing how a moderately curved initial shape transforms itself into a circle. Initial shapes showing more extreme curvature gradients, which are relevant in the initial stages of a sintering process, cannot be dealt with by the boundary-element method in its present form. The numerical solution of the continuous model shows a tendency to create oscillations in the outer boundary of the liquid region. On the other hand, an analytical small-amplitude analysis shows that rapid oscillations vanish exponentially fast.

This paper treats the kinematics of particles advected passively by flow of an incompressible fluid. It is shown that for steady irrotational flow without circulation, and for many monochromatic waves in a fluid the particle paths are not chaotic, i.e. do not depend sensitively on initial conditions. However, if the flow is a time-periodic potential flow or a superposition of waves then the particle paths may be chaotic. This is shown by the application of the theory of Melnikov to the breakup of a heteroclinic orbit (which connects two stagnation points and may bound a region of closed streamlines) and the onset of chaos in two examples of two-dimensional flow. The first example is a simple unbounded irrotational flow comprising a steady flow with two stagnation points which has a time-periodic perturbation. The second example is of two Rossby waves with a mean zonal flow; the particle paths are examined geometrically and numerically, and consequences for pollutant dispersion are discussed in physical terms. Also the combination of the effects of chaotic advection and molecular diffusion on the transport of a solute are examined.

In this paper we study the statistical laws of relative dispersion in two-dimensional turbulence by deriving an exact equation governing its evolution in time, then evaluating the magnitude of its various terms in numerical experiments, which allows us to check the validity of the classical dispersion laws: the equivalent to the Richardson-Obukhov t3 law in the energy cascade range, and the Kraichnan-Lin exponential law in the enstrophy cascade range. We examine theoretically and experimentally the conditions of validity of both laws. It is found that the t3 law is obtained in the energy inertial range provided the separation scale of the particles is smaller by an order of magnitude than the injection scale. When the t3 law is reached, the relative acceleration correlations are observed to have reached a statistical quasistationary stage: this would tend to justify in the energy inertial range of two-dimensional turbulence a working hypothesis formulated by Lin & Reid (1963); also, the necessity of starting from very small initial separations to get the t3 law may be explained by the time necessary for relative acceleration correlations to reach the statistical quasi-stationary regime. On the other hand, the Kraichnan-Lin exponential law is, strictly speaking, never observed; it is in fact reduced to a very short transient stage when the relative dispersion characteristic time reaches its minimum value, as predicted by Batchelor.

Steady and oscillatory convection in rigid vertical cylinders heated from below studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. Both adiabatic and ideal conducting sidewalls are considered. The effect of the geometry of the container on the onset of convective instability and the structure and symmetry of the flow are analysed and compared with the results of linear stability theories. The nonlinear evolution and stability of convective flows at Rayleigh numbers beyond the critical number for the onset of convective motion are investigated for Prandtl numbers of 0.02 to 6.7. The limits of stable axisymmetric solutions are an important finding of this study. The onset and the frequency of oscillatory instability are calculated for the small Prandtl number 0.02 and compared with experimental data. Calculated stream patterns and velocity profiles illustrate the three-dimensional structure of steady convection and the time-dependent behaviour of oscillatory flows.

The aim of the paper is to examine the nonlinear dynamics of a truncated system modelling low-Prandtl-number thermal convection. The model describes situations where the primary flow is not a straight roll and the dynamics is dominated by the production of axial flow along the axis of bent rolls or of swirl along ring vortices. The physical mechanism for these processes is a spontaneous growth (i.e. bifurcation) of a vertical vorticity mode, breaking the two-dimensional symmetry of the system. A description of the model can be found in Massaguer & Mercader (1988) where the physics and the numerical results have been checked against laboratory experiments. The nonlinear dynamics of that model will be discussed in the more academic case of free boundaries, as it has been shown that for sufficiently small Prandtl numbers straight rolls cannot be expected to be the primary flow near the onset of convection (Busse & Bolton 1984). Two clearly differentiated time-dependent regimes have been found and they correspond to small and intermediate Péclet numbers. In the former regime there exists a transition to chaos with the whole scenario being dependent on a symmetry invariance common to a large number of confined flows. The route to chaos is made up of a sequence of homoclinic explosions giving rise to a cascade of period doublings, with the whole scenario being different from that of a Feigenbaum's cascade.

The coagulation rate of a dilute polydisperse suspension of particles is considered for small Péclet number, which provides a measure of the ratio of the relative gravity-induced motion to Brownian motion between two rigid spheres. In particular, a fourterm expansion for the dimensionless coagulation rate (Nusselt number) as function of the Péclet number is developed by making use of a singular perturbation method. In the limit of the radius of one of the two spheres becoming small, the result agrees with Acrivos & Taylor's (1962) work on mass transfer to spheres at small Péclet number.

The supersonic flow past a thin straight circular cylinder is investigated. The associated boundary-layer flow (i.e. the velocity and temperature field) is computed; the asymptotic, far downstream solution is obtained, and compared with the full numerical results.

The inviscid, linear, axisymmetric (temporal) stability of this boundary layer is also studied. A so-called ‘doubly generalized’ inflexion condition is derived, which is a condition for the existence of so-called ‘subsonic’ neutral modes. The eigenvalue problem (for the complex wavespeed) is computed for two free-stream Mach numbers (2.8 and 3.8), and this reveals that curvature has a profound effect on the stability of the flow. The first unstable inviscid mode is seen to disappear rapidly as curvature is introduced, whilst the second (and generally the most important) mode suffers a substantially reduced amplification rate.

An investigation is carried out on the effects of Brownian agitation in the motion of small particles in a carrier gas in situations far from equilibrium. Although the standard near-equilibrium closure of the hydrodynamic equations is not valid for the heavy particles, the smallness of their speed of thermal agitation allows an alternative systematic hypersonic closure. The hypersonic equations are solved for two known instances where the kinetic Fokker–Planck equation describing the non-equilibrium particle distribution function admits exact solutions. These problems are characterized by a null or a spatially constant value for the gradient of the velocity field in the carrier gas, both being free from boundary surfaces. In the first case, where the background velocity is uniform, a fundamental solution (expressed as an integral) is obtained for the steady flow of particles from a point source; this result has obvious applications for the description of the Brownian broadening of particle streamlines. An asymptotic integration of the fundamental solution yields analytical expressions for the particle hydrodynamic properties valid everywhere except near the source, where a direct integration of the Vlasov equation completes the description. The exact solution for the second example, where the background velocity field gradient is uniform, is taken from the literature. Once these reference solutions have been established, the hypersonic equations are attacked by a variety of methods. In particular, for the uniform steady flow, a boundary-layer analysis yields analytical results identical to those obtained from the asymptotic evaluation of the kinetic fundamental solution. In both problems, the agreement found between kinetic and hydrodynamic solutions is excellent even for values of order one of the inverse particle Mach number, the expansion parameter of the hypersonic theory.