We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

Consider the classical Hele–Shaw situation with two parallel planes separated by a narrow gap, and suppose the plan-view of the region occupied by fluid to be confined to an infinite strip by barriers in the form of two infinite parallel lines. With the fluid initially occupying a bounded, simply-connected region that touches both barriers along a single line segment, we seek to predict the evolution of the plan-view as the blob of fluid is driven along the strip by a pressure difference between its two free boundaries. Supposing the relevant free boundary condition to be one of constant pressure (but a different constant pressure on each free boundary), we show that the motion is characterized by (a) the existence of two functions, analytic in disjoint half-planes, that are invariants of the motion and (b) the centre of area of the plan-view of the blob has a component of velocity down the infinite strip that is simply related to the imposed pressure difference. These features allow explicit analytic solutions to be found; generically, the mathematical solution breaks down when cusps appear in the retreating free boundary. A rectangular blob, of course, moves down the strip unchanged, with no breakdown, but if it encounters stationary blobs of fluid placed within the strip then, modulo multiply-connected complications, these are first absorbed into the advancing front of the rectangular blob and then disgorged from its retreating rear, leaving behind stationary blobs of exactly the same form in exactly the same place as those originally present, but consisting of different fluid particles. This soliton-like interaction involves no phase change: with a given pressure difference driving the motion, the rectangular blob is in the same position at a given time after the interaction as it would have been had no intervening blobs been present.

We consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap. A blob of Newtonian fluid is sandwiched between the planes, and we suppose its plan-view to occupy a bounded, multiply-connected domain; physically, we have a viscous fluid with the holes giving rise to the multiple connectivity occupied by relatively inviscid air. The relevant free boundary condition is taken to be one of constant pressure, but we allow different pressures to act within the different holes, and at the outer boundary. The motion is driven either by injection of further fluid into the blob at certain points, or by injection of air into the holes to change their area, or by a combination of these; suction, instead of injection, is also contemplated. A general mathematical theory of the above class of problems is developed, and applied to the particular situation that arises when fluid is injected into an initially empty gap bounded by two straight, semi-infinite barriers meeting at right-angles: injection into a quarterplane. For a range of positions of the injection point, air is trapped in the corner and, invoking images, the problem is equivalent to one involving a doubly-connected blob. When there is an air vent in the corner, so that the pressure is the same on the two free boundaries in these circumstances, the air hole rapidly disappears, as might be expected. If, however, there is no air vent and we suppose the air to be incompressible, so that the area of the region occupied by the air in the plan-view remains constant, we find there to be no solution within the framework of our model. Other scenarios within this same geometry, involving both suction and injection of fluid at the injection point, and air at the corner, are also examined.

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.

Oscillatory flow over a circular cylinder, or part-cylinder, placed on a plane boundary, when the Strouhal and streaming Reynolds numbers are large, is considered. The solution is developed in matching inner and outer boundary layers. A steady streaming motion in the outer layer can lead to a net flow away from the cylinder along the plane boundary. A simple experiment substantiates this prediction, and the implications for bed-scouring are examined.

In Part I of this paper (Terrill & Byatt-Smith, 1993) the problem of the flow between an obstacle in the form of a wedge and a porous flexible tow was modelled using lubrication theory. In the region close to the wedge where bending is important the following equation

was derived for the non-dimensional gap width, H(X), between the tow and the wedge.

The simplified magnetic Bénard; equations (SMB) are derived from the two-dimensional magnetic Bénard system and extend the well-known Lorenz equations (L). (SMB) contains a parameter Q associated with the magnetic field. When Q = 0, the long time dynamics of (SMB) agrees with that of (L). In this paper, we investigate the long time behaviour of (SMB) as Q→0. We prove analytically that the global attractor of (SMB) converges to the one of (L) as Q→0 upper semicontinuously in the Hausdorff sense. However, numerical computation indicates that generically there is no continuity in the dimension of the attractor.

A one-dimensional free boundary problem arising in the modelling of internal oxidation of binary alloys is studied in this paper. The free boundary of this problem is determined by the equation u = 0, where u is the solution of a parabolic partial differential equation with discontinuous coefficients across the free boundary. Local existence, uniqueness and the regularity of the free boundary are established. Global existence is also studied.

Plane, quasi-steady, free-boundary flows of an incompressible viscous fluid with surface tension in a container are considered. The mathematical problem is decomposed into an auxiliary elliptic problem for the Stokes system in a fixed flow domain, whose solution leads to the Cauchy problem for the free boundary with the so-called ‘normal velocity’ operator. By introducing the complex stress-stream function and applying time-dependent conformal mapping, the auxiliary problem is reduced to a boundary integral equation via consideration of two Hilbert problems for analytic functions in a unit disc. As an application, plane capillary flow with moving contact points is investigated asymptotically for small capillary numbers. We prove that in the case when a dynamic contact angle is equal to π, this problem is well-posed for a filling regime, and ill-posed for a drying one.

We investigate the local behaviour at the boundary singularity for the following moving boundary problems: (i) Hele-Shaw flows in which the interface is initially non-analytic; (ii) power-law Hele-Shaw flows in which the interface contains a corner; (iii) Stefan problems in which the interface contains a corner. Both well-posed (‘injection’) and ill-posed (‘suction’) problems are considered. Related results for corner development in the presence of an impermeable boundary are also noted.

This paper deals with the mathematical characterization of microstructure in elastic solids. We formulate our ideas in terms of rank-one convexity and identify the set of probability measures for which Jensen's inequality for this type of functions holds. This is the set of laminates. We also introduce generalized convex hulls of sets of matrices and investigate their structure.

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.

We study radially symmetric stationary points of the functional

where u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and ε accounts for surface energy. Under the further assumption of small energy, that is

for small ε, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation. Each of these solutions is monotone in the radial direction and converges as ε→0 to one of two possible radially symmetric single interface minimizers of E0. Our main tool is the method of matched asymptotic expansions from which we construct exact solutions.

In this paper we study the question of existence and uniqueness of solutions to the equation

which occurs in the modelling of hard contact lenses. Using ‘shooting’ arguments, we show that, depending on the parameters in the problem, there are cases where there is precisely one solution, others where there is more than one, and others where there are none.

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.

This paper deals with Maxwell's equations in a quasi-stationary electromagnetic field subject to the effects of temperature. This model is encountered in the penetration of a magnetic field in substances where the electrical conductivity depends on the temperature. Similar phenomena also occur in some industrial problems such as the thermistor. Taking the effect of Joule heating into the consideration, we obtain a strongly coupled nonlinear system. Global solvability is established for this system.