We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter ε appears in this problem. The formal limit as ε → 0 is a free boundary problem. We show that the solution for ε > 0 tends, in a suitable norm, to the solution of the formal limit.

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.

Subsonic potential flow around the leading edge of a thin two-dimensional general aerofoil with a parabolic nose at small angles of attack is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed Mach number in terms of the thickness ratio of the aerofoil in an outer region around the aerofoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of linearized aerofoil theory and its second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary-value problem is formulated in the inner region for the solution of a compressible uniform flow with the same Mach number, past a semi-infinite two-dimensional parabola. The analysis shows that the drag of any general aerofoil with a round nose is zero for every subsonic Mach number below the critical Mach number, where the drag component that is calculated by the integration of the outer pressure distribution over the aerofoil is cancelled exactly by the drag of the parabolic nose. The numerical solution of the flow field in the inner region results in the pressure distribution on the parabolic nose. The stagnation point tends to shift toward the leading edge due to the compressibility effects as the Mach number is increased at a fixed angle of attack and camber of the aerofoil. Good agreement is found in the leading edge region between the present asymptotic solution and numerical solutions of the Euler equations. Using the outer linearized solution and the nose solution a uniformly valid pressure distribution on the entire aerofoil surface in a subsonic flow is derived.

The basic constitutive relations for elastoplasticity and elastoviscoplasticity are shown to form a typical boundary layer-type stiff system of ordinary differential equations. Three numerical algorithms are discussed: (i) The singular perturbation method (O'Malley, 1971a, b; Hoppensteadt, 1971; Miranker, 1981; Smith, 1985), which yields accurate results for both the rate-independent and rate-dependent cases, where in the former case, the algorithm is explicit, whereas in the latter case, it is implicit and requires the solution of a nonlinear equation; therefore it is impractical as a constitutive algorithm for large-scale finite-element applications, where the constitutive algorithm is used a great number of times at each finite-element node. (ii) The new constitutive algorithm (Nemat-Nasser, 1991; Nemat-Nasser & Chung, 1989, 1992) which is explicit and accurate for both the rate-independent and rate-dependent cases; the underlying mathematical feature of this new method is investigated, and it is shown that it can be classified as a simplified perturbation method; computable error bounds for this algorithm are obtained, and when the flow rule is given by the commonly used power law, it is shown that the errors are very small, (iii) A modified outer-solution method, which combines the above two techniques, and is simple, explicit, and accurate.

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.

Asymptotic methods are applied to the outdiffusion of a diffusant whose behaviour is governed by a power-law diffusivity. Both semi-infinite and finite domain problems are considered. The ‘fast’ diffusion (negative power-law) case is of particular interest, and the results are significantly different from those for the corresponding indiffusion problem.

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)

They have the form

where the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

Nonstationary two-dimensional filtration in a porous medium is considered, whereby part of the medium is saturated, another part is unsaturated but wet, and the remaining part is dry. The saturated/unsaturated and unsaturated/dry interfaces are free boundaries. It is shown that there exists a unique solution, and that the saturation function is continuous in the wet portion of the medium; this implies that the two interfaces are separated. Under some monotonicity-type conditions on the initial and boundary data it is shown that the free boundaries are continuous.