The incompressible slow viscous flow of a power-law shear-thinning fluid in a wedge-shaped region is considered in the specific instance where the stress is a very small power of the strain rate. Asymptotic analysis is used to determine the structure of similarity solutions. The flow is shown to possess an outer region with boundary layers at the walls. The boundary layers have an intricate structure consisting of a transition layer 0(ɛ) adjoining an inner layer O(ɛlnɛ), which further adjoins an inner-inner layer 0(ɛ) next to the wall. Explicit solutions are found in all the regions and the existence of ‘dead zones’ in the flow is discussed.

The diffraction of time-harmonic waves in a nonlinear medium with periodic structure is studied in this paper. In particular second harmonic generation – an important phenomenon in nonlinear optics-is modelled. The model, derived from a general nonlinear system of Maxwell's equations, is shown to have a unique solution for all but a discrete number of frequencies. The problem is solved numerically by combining a method of finite elements and a fixed-point iteration scheme. Numerical experiments for some simple grating structures are presented and discussed.

We present a method of solution of a class of fracture problems in the theory of elasticity. The method can be applied to any problem reducible to Poisson's equation, e.g. heat conduction and mass diffusion in solids, theory of consolidation and the like. The novelty of the paper is that we address regions of layered composites with notches, or, in a particular case, with a crack. Within the framework of classical analysis, we apply Fourier and Mellin transforms, 'fit' them together, and reduce the problem to solving a singular integral equation with fixed singularities on a semi-axis. We show the existence and uniqueness of solutions of the equations under consideration, and justify the asymptotics necessary for applications. We show the practical usefulness of the method on the examples of an antiplane problem of fracture mechanics. From our solution, we are able to find the stress intensity factor in the case when a crack tip penetrates a layered composite consisting of 60 layers, and show the limits of applicability of the anisotropic model of such composites.

A general method of placing variational bounds on the effective scalar conductivity of random heterogeneous solids is proposed. The method utilizes the Hashin-Shtrikman variational principle and trial fields which are obtained upon truncating the functional series expansion for the polarization field in the medium. It is shown how the earlier variational procedures due to Hashin and Shtrikman, Milton and Phan-Thien and Willis find natural places in the proposed general scheme. The new results reported include, for example, the fact of nonexistence of certain Gaussian-like random media and, for dispersions of spheres with volume fraction c, an exact O(c2) relation for the effective conductivity which contains absolutely convergent integrals only.

Ovsiannikov's method of partially invariant solutions of differential equations can be considered to be a special case of the method of differential constraints introduced by Yanenko and by Olver and Rosenau. Differential constraints are used to construct non-reducible partially invariant solutions of the boundary layer or Prandtl equations.

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.