We consider polymerization–crystallization waves in a cylindrical reactor, in which monomer is converted to polymer in a planar front. The polymer is subsequently crystallized in a wider zone behind the front. Specifically, we study uniformly propagating polymerization–crystallization waves, and determine profiles of temperature, and concentrations of polymer and crystallized polymer, as well as the propagation velocity. A linear stability analysis of the travelling wave solutions indicates the possibility of Hopf bifurcation, which describes the transition to the experimentally observed spinning mode of propagation, in which a hot spot is observed to propagate along a helical path on the surface of the cylinder. Since conditions at the time of conversion determine the nature of the polymer produced, spiral hollows, which trace out a helical path, appear on the surface of the crystallized polymer product.

The purpose of this paper is to study a mathematical model of lubricating flow between elastic surfaces obeying the linear Hertzian theory when cavitation takes place. Cavitation is a free boundary phenomenon that is described in this paper by the New Elrod–Adams model. This model introduces the concentration of fluid as well as the pressure as unknown functions and is suggested in preference to the classical variational inequality due to its ability to describe inflow and outflow. This leads to a nonlinear variational and nonlocal equation. Herein, an existence theorem is proved by means of two different techniques.

The internal layer behaviour, in one spatial dimension, associated with two classes of Ginzbug–Landau equation with double-well nonlinearities and small diffusivities is investigated. The problems that are examined are the Ginzburg–Landau equation with and without a constant mass constraint. For the constrained problem, steady-state internal layer solutions are constructed using a formal projection method. This method is also used to derive a differential-algebraic system describing the slow dynamics of the constrained internal layer motion. The dynamics of a two-layer evolution is studied in detail. For the unconstrained problem, a nonlinear WKB-type transformation is introduced that magnifies exponentially weak layer interactions and leads to well-conditioned steady problems. A conventional singular perturbation method, without the need for exponential asymptotics, is used on the resulting transformed problem as an alternative method to construct equilibrium solutions and metastable patterns. Exponentially sensitive steady-state internal layer solutions as well as a one-layer evolution are computed accurately using the transformed problem.

A binary liquid that undergoes directional solidification is susceptible to morphological instabilities which cause the solid/liquid interface to change from a planar to a cellular state. This paper presents a numerical study of a class of long-wave equations that describe the evolution of interface morphology. We find new bifurcation points, new solution branches, and the existence of inverted hexagonal nodes and cells.

An existence and uniqueness theorem for travelling waves in a bubbly liquid with a continuous bubble-size distribution is proved.

In this note we develop and analyse a system of equations describing a molten linear polymer being extruded in a capillary rheometer which is operating under the controlled condition that the mean velocity at the capillary inlet is maintained at a constant value. Slipping of the melt at the pipe wall is permitted and an evolution equation for the boundary slip parameter is postulated. The combined system governing the mean cross-sectional velocity and slip parameter is shown to exhibit relaxation oscillations similar to those observed in actual rheometers.

A model for superconductors co-existing with normal materials is presented. The model, which applies to such situations as superconductors containing normal impurities and superconductor/normal material junctions, is based on a generalization of the Ginsburg–Landau model for superconductivity. After presenting the model, it is shown that it reduces to well-known models due to de Gennes for certain superconducting/normal interfaces, and in particular, for Josephson junctions. A provident feature of the modified model is that it can, by itself, account for all of these as well as other physical situations. The results of some preliminary computational experiments using the model are then provided; these include flux pinning by normal impurities and a superconductor/normal/superconductor junction. A side benefit of the modified model is that, through its use, these computational simulations are more easily obtained.

In this paper, we investigate the evolution of N-waves in a medium governed by the modified Burgers' equation. It is shown that the general behaviour when the nonlinearity is of arbitrary odd integer order is the same as for the cubic case. For an N-wave of zero mean displacement, a shock is formed immediately to prevent a multi-valued solution and a second shock is formed at later times. At a finite time, the second shock satisfies a sonic condition and this state persists. The Taylor-type shock structure ceases to be the appropriate description, and instead we have a shock which matches only algebraically to the outer wave on one side. At a larger time still, the other shock is affected but the two shocks remain distinct until the wave dies under linear mechanisms. The behaviour of N-waves of non-zero mean is also examined and it is shown that in some cases, a purely one-signed profile remains.

We discuss the question of uniqueness of planar flames for a simple one-step chemical reaction. We show that when the Lewis number is less than unity (i.e. species diffusion is larger than heat diffusion) uniqueness cannot be generally assumed. An example with three flames, two of them being stable, is exhibited. Other related questions, such as sufficient conditions for uniqueness to hold and high activation energy limits, are discussed.

We develop a technique for obtaining asymptotic properties of the sojourn time distribution in processor-sharing queues. We treat the standard M/M/1-PS queue and its finite capacity version, the M/M/1/K-PS queue. Using perturbation methods, we construct asymptotic expansions for the distribution of a tagged customer's sojourn time, conditioned on that customer's total required service. The asymptotic limit assumes that (i) the traffic intensity is close to one for the infinite capacity model, and (ii) that the system's capacity is large for the finite capacity queue.

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg–Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.

In this paper we present a new formulation of a large class of phase-field models, which describe solidification of a pure material and allow for both surface energy and interface kinetic anisotropy, in terms of the Hoffman–Cahn ξ-vector. The ξ-vector has previously been used in the context of sharp interface models, where it provides an elegant tool for the representation and analysis of interfaces with anisotropic surface energy. We show that the usual gradient-energy formulations of anisotropic phase-field models are expressed in a natural way in terms of the ξ-vector when appropriately interpreted. We use this new formulation of the phase-field equations to provide a concise derivation of the Gibbs–Thomson–Herring equation in the sharp-interface limit in three dimensions.

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.