We study the motion of voids in conductors subject to intense electrical current densities. We use a free-boundary model in which the flow of current around the insulating void is coupled to a law of motion (kinematic condition) for the void boundary. In the first part of the paper, we apply a new complex variable formulation of the model to an infinite domain and use this to (i) consider the stability of circular and flat front travelling waves, (ii) show that, in the unbounded metal domain, the only travelling waves of finite void area are circular, and (iii) consider possible static solutions. In the second part of the paper, we look at a conducting strip (which can be used to model interconnects in electronic devices) and use asymptotic methods to investigate the motion of long wavelength voids on the conductor boundary. In this case we derive a nonlinear parabolic PDE describing the evolution of the free boundary and, using this simpler model, are able to make some predictions about the evolution of the void over long times.

The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.

Solutions to a boundary-value problem involving a second-order linear functional differential equation with an advanced argument are investigated in this paper. The boundary conditions imposed on the differential equation are analogous to conditions defining various singular Sturm-Liouville problems, and if an eigenvalue parameter is introduced certain properties of the spectrum can be deduced having analogues with the classical problem. Dirichlet series solutions are constructed for the problem and it is established that the spectrum contains an infinite number of real positive eigenvalues. A Laplace transform analysis of the problem then reveals that the spectrum does not generically consist of isolated points and that there may be an infinite number of eigenfunctions corresponding to a given eigenvalue. In contrast, it is also shown that there is a subset of eigenvalues that correspond to the zeros of an entire function for which the corresponding eigenfunctions are unique.

In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

Not only are thin fluid films of enormous importance in numerous practical applications, including painting, the manufacture of foodstuffs, and coating processes for products ranging from semi-conductors and magnetic tape to television screens, but they are also of great fundamental interest to mathematicians, physicists and engineers. Thin fluid films can exhibit a wealth of fascinating behaviour, including wave propagation, rupture, and transition to quasi-periodic or chaotic structures. More details of various aspects of thin-film flow can be found in the recent review articles by Oron, Davis & Bankoff (1997) and Myers (1998), and in the volumes edited by Kistler & Schweizer (1997) and Batchelor, Moffatt & Worster (2000).

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

When a solid plate is removed from a pool of fluid, a film of fluid is attached to the plate. There are two possible outcomes. The edge of the fluid may be raised through a finite distance, with the edge slipping on the plate. Alternatively, a continuous film of a certain thickness may be drawn up. For plates which have a small slope, it is shown that the first alternative holds when the speed of withdrawal is sufficiently small, and that, when a critical speed is exceeded, the height of the edge above the fluid level in the pool increases with time. A related problem concerns the shape of a receding meniscus in a channel. If the static contact angle is small, lubrication theory can be applied to the film of fluid adjacent to the wall of the channel, and the results of this part of the solution can be matched to the central part of the meniscus, which is controlled by capillarity and gravity, but for which lubrication theory does not apply. As in the draw-up problem, for small speeds the shape of the meniscus is time-independent, but above a critical speed, a tail of fluid of increasing length remains in the tube.

The purpose of this paper is to analyse the free boundary problem for the Black–Scholes equation for pricing the American call option on stocks paying a continuous dividend. Using the Fourier integral transformation method, we derive and analyse a nonlinear singular integral equation determining the shape of the free boundary. Numerical experiments based on this integral equation are also presented.

The shape stability of the reaction interface for reactive flow in a porous medium is investigated. Previous work showed that the Reaction-Infiltration Instability could cause the reaction zone to lose stability when the Peclet number exceeded a critical value. The new feature of this study is to include a velocity-dependent hydrodynamic dispersion. A mathematical model for this phenomenon is given in the form of a moving free-boundary problem. The spectrum of the linearized problem is obtained, and the related analysis and numerical calculations show that the onset of the instability is not eliminated by the new dispersive terms. The details of analysis show that the instability is reduced especially by the transverse dispersion.

A leader is to be elected from n people using the following algorithm. Each person flips a coin. Those people who wind up with tails (which occurs with probability p, 0 < p < 1) move on to the next stage. Those with heads are eliminated. Let Hn denote the number of stages needed until there is a single winner. We analyze the moments and the probability distribution of Hn. In the symmetric model we have an unbiased coin with p = 1/2; in the asymmetric model p ≠ 1/2. We analyze these models asymptotically, for n → ∞, using a variety of analytical and numerical approaches. This leads to simple derivations of some existing results, as well as some new results for the asymmetric case. Our analysis makes some assumptions about the forms of various asymptotic expansions as well as their asymptotic matching.

In a previous paper the author and Demay advanced a model to explain the melt fracture instability observed when molten linear polymer melts are extruded in a capillary rheometer operating under the controlled condition that the inlet flow rate was held constant. The model postulated that the melts were a slightly compressible viscous fluid and allowed for slipping of the melt at the wall. The novel feature of that model was the use of an empirical switch law which governed the amount of wall slip. The model successfully accounted for the oscillatory behavior of the exit flow rate, typically referred to as the melt fracture instability, but did not simultaneously yield the fine scale spatial oscillations in the melt typically referred to as shark skin. In this note, a new model is advanced which simultaneously explains the melt fracture instability and shark skin phenomena. The model postulates that the polymer is a slightly compressible linearly viscous fluid but assumes no-slip boundary conditions at the capillary wall. In simple shear the shear stress τ and strain rate d are assumed to be related by d = Fτ, where F ranges between F2 and F1 > F2. A strain-rate dependent yield function is introduced and this function governs whether F evolves towards F2 or F1. This model accounts for the empirical observation that at high shears polymers align and slide more easily than at low shears, and explains both the melt fracture and shark skin phenomena.

Rapid solidification fronts are studied using a phase field model. Unlike slow moving solutions which approximate the Mullins–Sekerka free boundary problem, different limiting behaviour is obtained for rapidly moving fronts. A time-dependent analysis is carried out for various cases and the leading order behaviour of solidification front solutions is derived to be one of several travelling wave problems. An analysis of these problems is conducted, leading to expressions for front speeds in certain limits. The dynamics leading to these travelling wave solutions is derived, and conclusions about stability are drawn. Finally, a discussion is made of the relationship to other solidification models.

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment emanating from the vertex, so we have liquid at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundary, the region occupied by the fluid is the image of a semidisc (a domain bounded by a semicircle and its diameter) in the ζ-plane under a conformal map given by a function of the form ζα times a rational function of ζ. The form of this rational function can be written down, and the parameters appearing in it then determined as the solution to a set of algebraic equations. Examples of such flows are given (including one which shows that, in a certain sense, injection can produce a cusp), and the limiting situation in the wedge configuration as one injection point is moved to infinity is also considered.

Fibre drawing is an important industrial process for synthetic polymers and optical communications. In the manufacture of optical fibres, precise diameter control is critical to waveguide performance, with tolerances in the submicron range that are met through feedback controls on processing conditions. Fluctuations arise from material non-uniformity plague synthetic polymers but not optical silicate fibres which are drawn from a pristine source. The steady drawing process for glass fibres is well-understood (e.g. [11, 12, 20]). The linearized stability of steady solutions, which characterize limits on draw speed versus processing and material properties, is well-understood (e.g. [9, 10, 11]). Feedback is inherently transient, whereby one adjusts processing conditions in real time based on observations of diameter variations. Our goal in this paper is to delineate the degree of sensitivity to transient fluctuations in processing boundary conditions, for thermal glass fibre steady states that are linearly stable. This is the relevant information for identifying potential sources of observed diameter fluctuation, and for designing the boundary controls necessary to alter existing diameter variations. To evaluate the time-dependent final diameter response to boundary fluctuations, we numerically solve the model nonlinear partial differential equations of thermal glass fibre processing. Our model simulations indicate a relative insensitivity to mechanical effects (such as take-up rates, feed-in rates), but strong sensitivity to thermal fluctuations, which typically form a basis for feedback control.

In this paper we prove the existence of solutions of the Keller–Segel model in chemotaxis, which blow up in finite or infinite time. This is done without assuming any symmetry properties of the solution.

Long-wavelength models for van der Waals driven rupture of a free thin viscous sheet and for capillary pinchoff of a viscous fluid thread both give rise to families of first-type similarity solutions. The scaling exponents in these solutions are independent of the dimensionality of problem. However, the structure of the similarity solutions exhibits an intriguing geometric dependence on the dimensionality of the system: van der Waals driven sheet rupture proceeds symmetrically, whereas thread rupture is inherently asymmetric. To study the bifurcation of rupture from symmetric to asymmetric forms, we generalize the governing equations with the dimension serving as a control parameter. The bifurcation is governed by leading-order inviscid dynamics in which viscous effects are asymptotically small but nevertheless provide the selection mechanism.

In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

The often-studied problem known as Kramers' problem, in the general area of rarefied-gas dynamics, is investigated in terms of a linearized, variable collision frequency model of the Boltzmann equation. A convenient change of variables is used to reduce the general case considered to a canonical form that is well suited for analysis by analytical and/or numerical methods. While the general formulation developed is valid for an unspecified collision frequency, a recently developed version of the discrete-ordinates method is used to compute the viscous-slip coefficient and the velocity defect in the Knudsen layer for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity) and the rigid-sphere model.