Exact analytical representations are obtained describing self-similar unsteady flows of multi-phase immiscible fluids in the vicinity of non-circular, but constant strength, fronts. It is assumed that Darcy's law holds for each phase and that the mobilities are known functions of the saturations. Equivalent representations are obtained for Hele-Shaw cell flows that are produced when a viscous fluid is injected into a region containing some other viscous fluid. The fluids may be Newtonian fluids or non-Newtonian fluids for which the coefficients of viscosity depend on the shear stress. Even though the flows are unsteady and two dimensional, the representations are obtained by using hodograph techniques.

An analytical version of the discrete-ordinates method is used here to solve the classical temperature-jump problem based on the BGK model in rarefied-gas dynamics. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield very accurate results for the temperature jump and the complete temperature and density distributions in the gas. The algorithm is easy to use, and the developed code runs typically in less than a second on a 400 MHz Pentium-based PC.

The linearized stability problem for fibre spinning of Newtonian and power law fluids, in a one-dimensional, isothermal setting, is reduced to a single Volterra integral equation. A rigorous linear stability criterion is presented, and we compute the corresponding ‘critical stretch ratios’, using methods of complex analysis. The results of this paper prove the correctness of earlier results by Matovich, Pearson and Shah.

In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use.

We present a model of sedimentation in a subsiding fluvio-deltaic basin with steady sediment supply and unsteady base level. We demonstrate that mass transfer in a fluvio-deltaic basin is analogous to heat transfer in a generalized Stefan problem, where the basin's shoreline represents the phase front. We obtain a numerical solution to the governing equations for sediment transport and deposition in this system via an extension of a deforming-grid technique from the phase-change literature. Through modification of the heat-balance integral method, we also develop a semi-analytical solution, which agrees well with the numerical solution. We construct a space of dimensionless groups for the basin and perform a systematic exploration of this space to illustrate the influence of each group on the shoreline trajectory. Our model results suggest that all subsiding fluvio-deltaic basins exhibit a standard autoretreat shoreline trajectory in which a brief period of shoreline advance is followed by an extended period of shoreline retreat. Base-level cycling produces a shoreline response that varies relative to the autoretreat signal. Contrary to previous studies, we fail to observe either a strong phase shift between shoreline and base level or a pronounced attenuation of the amplitude of shoreline response as the frequency of base-level cycling decreases. However, the amplitude of shoreline response to base-level cycling is a function of the basin's age.

A problem for a backward–forward parabolic partial differential equation with a small diffusion coefficient ε is analyzed. The problem arises in describing the stochastic behaviour of a data-handling system with a large number of sources which turn on or off at random. We solve the problem asymptotically for ε small by using the ray method and the boundary layer method.

Coarsening of solutions of the integro-differential equation

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where Ω ⊂ ℝn, J(·) [ges ] 0, ε > 0 and f(u) = u3 − u (or similar bistable nonlinear term), is examined, and compared with results for the Allen–Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when ε is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen–Cahn equation. The special case J(·) ≡ 1 is explored in detail, giving insight into the behaviour in the more general case J(·) [ges ] 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.

The aim of this paper is to describe a technique based on matched asymptotic expansions that allows us to derive the variation of the stress intensity factors in a homogeneous isotropic elastic medium under plane strain deformation. The case of antiplane shearing is also considered.

We conjecture that the mean-field model of superconducting vortices given in [10] is ill-posed wherever the electric current j has some component in the same direction as the vorticity vector ω (which gives the average density and direction of the superconducting vortices). The conjecture is illustrated with a linear stability analysis of a certain solution to the model. A regularised model is then proposed, and this is used to demonstrate the instability of force-free steady states in a certain geometry.

In this paper we investigate the uniqueness of solutions of the Ginzburg–Landau system for superconductivity in the regime where the thickness 2a of the film is small. We analyse the equations of first variation with respect to two shooting parameters and obtain estimates on all relevant quantities at the end of the film where the boundary conditions are prescribed. Using these estimates, we prove that the bifurcation curve of symmetric solutions is given by a decreasing function of the order parameter β when the film is thin enough. In addition, we prove that there is no curve of asymmetric solutions branching from the symmetric curve.

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling-wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.

In this paper we describe solutions of a one-dimensional model of inviscid fluid jets that develop finite time singularities in a fractal-like manner. We also discuss the extreme sensitivity of the solutions of this problem with respect to small perturbations of the initial data.

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. For simplicity, the fine structure is here taken to be periodic, first in one dimension, and then a lattice of squares in two dimensions. A method of multiple scales is employed, with a classical free-boundary problem being used to model the evolution of the two-phase microstructure. Then a macroscopic model for the mush is obtained by an averaging procedure. The free-boundary temperature is taken to vary according to Gibbs–Thomson and/or kinetic-undercooling effects.

Some new global results are given about solutions to the boundary value problem for the Euler–Lagrange equations for the Ginzburg–Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for solution of the linear equations. Finally, an error in Odeh's work on this problem is discussed.

We study an inverse boundary problem for the diffusion equation in ℝ2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (cμa + iω0) u = 0; ω0 ≠ 0, where D and μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μa inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω0, determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.

This work presents an asymptotic algorithm for the derivation of equations of thin elastic shells. The algorithm is based on the analysis of a boundary value problem for the Navier system in a thin region. The analysis covers both the membrane theory and the moment theory of elastic shells, including the eigenvalue problems.

Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins–Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow us to treat such free boundary problems in both IR2 and IR3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method, we compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

We consider the eigenvalue problem associated with the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size of the microstructure ε is shorter than the wavelength of the eigenfunctions 1/√λε. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions (ε ∼ 1/√λε) singular phenomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/√λε is larger than the critical size ε. We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to ε. Our analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic expansion requires one more derivative of the density. Thus, a full description requires the density to be C∞ smooth.