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.

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.

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].

We consider a phenomenological model proposed by Barenblatt [1] for non-equilibrium two-phase flow in porous media. In the case of zero total flow it reduces to a pair of equations:

where Φ(σ) is a non-decreasing (not necessarily increasing) smooth function defined in the interval 0 ≤ σ ≤ 1. We consider initial-boundary problems for this system in which the initial data are given only for s, and the boundary data only for ς (which corresponds to the physical sense of the model). The degenerate character of the system allows us to apply simple topological methods. We show that the boundary problem is well-posed, in the sense that there exits a unique (weak) solution which satisfies the maximum principle and depends continuously on the initial data. The solution is no less smooth than the initial and boundary data.

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.

The purpose of this paper is to study a mathematical model of lubricating flow between elastic surfaces obeying the linear Hertzian theory when cavitation takes place. Cavitation is a free boundary phenomenon that is described in this paper by the New Elrod–Adams model. This model introduces the concentration of fluid as well as the pressure as unknown functions and is suggested in preference to the classical variational inequality due to its ability to describe inflow and outflow. This leads to a nonlinear variational and nonlocal equation. Herein, an existence theorem is proved by means of two different techniques.

A general method is presented to facilitate the solution of a class of polydisperse spray problems in which a cloud of droplets can be described using a sectional or group model. The procedure involves replacing the original coupled droplet sectional variables conservation equations by a set of uncoupled sectional equations for auxiliary variables. The form of these latter equations is identical to that of the single spray equation for a quasi-monodisperse spray, solutions of which are more readily attainable even for multidimensional spray problems. Thus, these ready-made solutions can be exploited directly for the auxiliary variables, from which solutions can then be constructed in a straightforward manner for the desired original sectional variables. Three illustrative examples for spray diffusion flames with different features of complexity highlight the potential applicability of the proposed method, and indicate the sensitivity of flame characteristics to initial spray conditions and in-spray related phenomena.