The classic problem, first treated by Taylor [18], of the dispersion of inert soluble matter in fluid flow continues to attract attention from researchers describing the approach to the asymptotic state [5, 17]. The present article considers some of the complications caused when the solute is chemically active, Dispersing chemically active solutes occur in diverse fields such as chromatography, chemical engineering and environmental fluid mechanics.

The asymptotic large-time analysis of Chatwin [5] is re-worked to handle the case of reactive solutes dispersing in parallel flow. Matching between moderate and large-time solutions requires consideration of the integral moments of the reactive contaminant could, and the Aris method of moments is therefore invoked and modified for reaction effects. The results are applied in detail to the outstanding paractical example—the chemical flow reactor (a device used to measure reaction rates for chemical reactions taking place between fluids). For this case, the paper provides a practical alternative to recent variable diffusion coefficient studies [6, 7, 15], and presents further results for the concentration distribution and for the limiting behaviour under weak and vigorous recactions at the boundary of the flow.

Various problems related to the propagation of small amplitude long waves on the surface of superfluid helium (helium II), usually called third sound, are studied on the basis of the appropriate governing equations. The two-fluid continuum model due to Landau is considered, with the effects of healing and relaxation incorporated, and viscosity, heat conduction and compressibility terms retained. The helium vapour is treated as a classical (Newtonian) compressible gas and the exact jump conditions across the liquid/vapour interface are employed. These liquid, vapour and jump equations constitute the exact problem although, in an effort to reduce the complexity of the equations, a simplified set of ‘model’ surface boundaiy conditions is also introduced. This full set of equations is non-dimensionalised taking care that all physical parameters are defined using only the undisturbed depth of the layer as the appropriate length scale. The ratio depth/wave-lenght (δ) is then a separate parameter as is the wave amplitude/depth ratio (ɛ). The limit which corresponds to the wave under discussion is then ɛ, δ → 0 with all the other parameters fixed.

A number of analyses are presented, four of which describe various aspects of the linearised theory and two examine the nature of the far-field nonlinear problem. Using the simplified surface boundary conditions we discuss in turn: the wave motion in the absence of healing: the rôle of a second wave speed leading to a wave hierarchy; and the effects of healing. The final linearised problem makes use of the full vapour model, but again the healing terms are ignored. This latter analysis suggests that if the upper boundary of the vapour is sufficiently close to the liquid surface then third sound is suppressed.

The complexity of the equations, particularly when the nonlinear terms are to be examined, is such that the incompressible limit is now taken in the absence of both healing and relaxation. Imposing the physically realistic limiting process (ɛ, δ → 0) we show that the only equation valid in the far-field is the Burgers equation. However, we also demonstrate that allowing the other parameters to be functions of e (which is not physically realisable in practice) it is easy to derive, for example, the Korteweg-de Vries equation.

Sufficient conditions are obtained for the existence of a unique linearly asymptotically stable positive periodic solution of an ecosystem model of two species competition in a periodic environment with time lags in interspecific interactions. It is shown that if the self-regulating intraspecific interaction effects are strong enough and act without time delays then time delays of any length in the interspecific interactions cannot destabilise an otherwise stable ecosystem in a periodic environment.

In this paper we develop the Laplace-transform method to solve initial-value problems for the velocity potential describing the generation of infinitesimal capillary-gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles. The two principal problems considered are the forced motions due to a submerged wave source and an immersed vertical plane wave-maker, which begin to operate in a time-dependent manner at a given instant. The transformed potentials are calculated using techniques similar to those which are effective in traditional time-harmonic problems with a free surface. The steady-state development in the time-harmonic example taken demonstrates the existence of outgoing progressive waves under any inertial surface, in contrast to the case of no surface tension when such waves cannot propagate under an inertial surface that is too heavy. The solution is also noted of the Cauchy-Poisson problem for the free motion flowing an intial elevation of the inertial surface, which is obtained by the same method.

The main ideas of Hopf bifurcation theory and its relevance to the development of periodic motions of an autonomous system depending on a parameter are presented, and an algorithm for the computation of the orbits is described. It is then shown that a model system for the motion of a wheelset can be cast in the form amenable to Hopf bifurcation theory. Numerical results for the period and amplitudes of the lateral and yaw motions are obtained in terms of the forward speed of the wheelset, and the wheel-rail profile parameters.

It is found that the period of oscillation decreases while the lateral and yaw motion amplitudes increase as the forward speed increases, for any given rail and wheel profile. While the effect of wheel curvature on the lateral motions seems to be non-existent, its effect on the yaw motion amplitude and the period is to increase them very slightly as the wheel profile changes from a conical to a curved profile. On the other hand, the effect of rail curvature on the lateral amplitude, for instance, is significant; the larger the curvature the smaller the amplitude for a given forward speed.

Finite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.

This paper deals with the question of the existence of a solution to the stationary-point problem corresponding to a given nonlinear nondifferentiable program. An existence theorem for the stationary-point problem is presented under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the nonlinear program.