Physical experiments indicate that when an expanding fluid flows through a porous rock then the boundary between the wet and the dry region can be very irregular (e.g. see [OMBAFJ] and the references therein). In fact, it has been conjectured that this boundary is a fractal with Hausdorff dimension about 2.5. The (one-phase) fluid flow in a porous medium can be modelled mathematically by a system of partial differential equations, which, under some simplifying assumptions, can be reduced to a family of semi-elliptic boundary value problems involving the (unknown) pressure p(x) of the fluid (at the point x and at t) and the (unknown) wet region Ut at time t. (See equations (1.5)–(1.7) below). This set of equations, called the moving boundary problem involves the permeability matrix K(x) of the medium at x. A question which has been debated is whether this relatively simple mathematical model can explain such a complicated fractal nature of ∂Ut. More precisely, does there exist a symmetric non-negative definite matrix K(x) such that the solution Ut of the corresponding (expanding) moving boundary problem has a fractal boundary? The purpose of this paper is to prove that this is indeed the case. More precisely, we show that a porous medium which produce fractal wet boundaries can be obtained by distorting a completely homogeneous medium by means of a quasiconformal map.

A model describing the evolving shape of a growing pile is considered, and is shown to be equivalent to an evolutionary quasi-variational inequality. If the support surface has no steep slopes, the inequality becomes a variational one. For this case existence and uniqueness of the solution are proved.

This work treats the injection of certain thermoplastics into a planar mould cavity. The problem is to determine the filling pattern. It is assumed that the thermoplastic can be modelled as a non-Newtonian fluid of power-law type whose power-law exponent is relatively small (the pseudo-plastic case). The dependence of the viscosity on thermal variations is neglected. The mathematical description leads to a moving boundary problem, for which an asymptotic solution is found. According to this solution, the expansion of the polymer melt follows the level sets of an interior distance function, which is determined by the geometry of the mould, and the position of the injection point. The solution is easily computed and results of numerical experiments are given.

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length-and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalized to give new models for fully three-dimensional sheets.

A mean-field model for the motion of rectilinear vortices in the mixed state of a type-II superconductor is formulated. Steady-state solutions for some simple geometries are examined, and a local existence result is proved for an arbitrary smooth geometry. Finally, a variational formulation of the steady-state problem is given which shows the solution to be unique.

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.

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 two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

In this paper we consider a superconductor free boundary problem. Under isothermal conditions, a superconductor material (of ‘type I’) will develop two phases separated by a sharp interface Γ(t). In the ‘normal’ conducting phase the magnetic field is divergence free and satisfies the heat equation, whereas on the interface Γ(t), curl , where n is the normal and Vn is the velocity of Γ(t) in the direction of n; further, (constant) on Γ(t). Existence and uniqueness of a classical solution locally in time are established by Newton's iteration method under assumptions which enable us to reduce the 3-dimensional problem to a problem depending on essentially two space variables.

We study a system of two second-order differential equations with cubic nonlinearities which model a film of superconductor material subjected to a tangential magnetic field. We verify some recent conjectures of one of the authors about multiplicity of solutions. We show that for an appropriate range of parameter values the relevant boundary value problem has at least two symmetric solutions. It is also proved that a second range of parameters exists for which there are three symmetric solutions.

This paper discusses the initial value problem

where A, Bi and Ci are d × d complex matrices, pi, qi ∈ (0, 1), i = 1, 2, …, and y0 is a column vector in ℂd. By using ideas from the theory of ordinary differential equations and the theory of functional equations, we give a comprehensive analysis of the asymptotic behaviour of analytic solutions of this initial value problem.

We consider phase transition processes in which the thermodynamic variables are the temperature and an order parameter. Various classes are identified and many specific examples are illustrated. In this framework the question of the range of applicability of the so-called ‘additivity rules’ is investigated, showing that they apply only to a very special type of processes.

Spontaneous potential well-logging is an important technique in petroleum exploitation. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on interfaces. At the joint points of the interfaces, the jumps of the spontaneous potential do not, in general, satisfy the compatibility condition. It turns out that it is impossible to find a piecewise H1 solution to the problem, and the standard finite element method cannot be applied to get an approximate solution. In this paper, by means of a new method, we prove that the problem exists a unique piecewise solution for some p < 2. We give an estimate for the solution as well. This allows us to prove the convergence of a numerical scheme proposed in [3].

In this paper we examine the movement of hard contact lenses on the eye. In so doing, we take into account hydrodynamic forces underneath the lens, as well as surface tension forces at the lens periphery. This involves solving for the free surface of the tear film away from the lens in order to determine the magnitudes of the pressure and surface tension forces on the lens. The analysis, which assumes quasi-steady motion, is carried out in both two and three dimensions.

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limt↑Ts(t) > 0 and lim inft↑T⋅(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

This paper deals with the spherically symmetric Stefan problem in three space dimensions. The melting temperature satisfies the Gibbs–Thomson law. The solution is obtained as a limit of solutions of similar problems containing a small additional kinetic term in the melting temperature. Under some structural assumptions we show that the phase-change boundary has at most one discontinuity point t = T0 (see the corresponding result for the planar Stefan problem in Götz & Zaltzman (1995)). In the one-phase problem the discontinuity point always exists. At the time T0 the whole solid phase melts instantaneously. We study also the asymptotical stability (t → ∞) of stationary solutions satisfying boundary conditions of thermostat type.

In this paper we present a new formulation of a large class of phase-field models, which describe solidification of a pure material and allow for both surface energy and interface kinetic anisotropy, in terms of the Hoffman–Cahn ξ-vector. The ξ-vector has previously been used in the context of sharp interface models, where it provides an elegant tool for the representation and analysis of interfaces with anisotropic surface energy. We show that the usual gradient-energy formulations of anisotropic phase-field models are expressed in a natural way in terms of the ξ-vector when appropriately interpreted. We use this new formulation of the phase-field equations to provide a concise derivation of the Gibbs–Thomson–Herring equation in the sharp-interface limit in three dimensions.

In the focussing problem for the porous medium equation, one considers an initial distribution of material outside some compact set K. As time progresses material flows into K, and at some finite time T first covers all of K. For radially symmetric flows, with K a ball centred at the origin, it is known that the intermediate asymptotics of this focussing process is described by a family of self-similar solutions to the porous medium equation. Here we study the postfocussing regime and show that its onset is also described by self-similar solutions, even for nonsymmetric flows.

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.