The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

A blob of viscous Newtonian fluid is surrounded by inviscid fluid and sandwiched in the narrow gap between two plane parallel surfaces, so that initially its plan view occupies a simply connected domain. Recently Entov, Etingoff & Kleinbock (1993) produced some steady-state solutions for the blob placed in a quadrupole driven flow, and including the effects of surface tension. Here a numerical solution of the time-dependent problem using a Boundary Integral algorithm finds that for low values of the flow rate there exist two solutions. We find that one, which is close in shape to a circle, is stable, while the other, more deformed equilibrium, is unstable. The analysis also reveals that for certain flow strengths stable non-convex shapes also exist. If the flow strength is too large no stable equilibrium is possible.

The probability density for the solution yn of a stochastic difference equation is considered. Following Knessl et al. [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problem

ut =J*u−u−f (u),

the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.

The Pólya–Szegö dipole tensors are employed for the analysis of plane elasticity problems in non-homogeneous media. Classes of equivalence for defects (cavities and rigid inclusions) are specified for the Laplacian operator and elasticity equations: composite materials with defects of the same class have the same effective elastic moduli. Explicit asymptotic formulae for effective compliance matrices of dilute composites are presented.

The motivation for this work arises from the study of the processes involved in the manufacturing of a class of composite materials, in particular, those that are obtained by injecting a resin through a porous preform. A one-dimensional model that describes the non-isothermal filtration of an incompressible fluid is presented, and it also includes the possibility of curing, i.e. the polymerization of the penetrating resin. It comes out a fully coupled system consisting of the heat diffusion equation, Darcy's law and an equation related to the kinetics of the chemical reaction. The system is regarded as a free boundary problem for the heat equation with non-constant discontinuous coefficients. Its weak formulation is studied and the local existence of solutions is proved.

In this paper a generalized model for the growth of avascular tumours is presented. The formulation leads naturally to the incorporation of free boundaries which define the outer tumour surface explicitly and various inner surfaces implicitly. A combination of numerical simulations, asymptotic analysis and perturbation techniques is used to study the model and yields results, which agree well with experimentally-observed phenomena.