Nonlinear transport processes in disordered media

Abstract

AbstractNonlinear transport processes in disordered systems such as porous media and heterogeneous solids are studied, which are represented by two‐ or three‐dimensional networks of interconnected bonds, by a Bethe network (a branching network with no closed loops) of a given coordination number, or by a continuum in which circular or spherical inclusions have been inserted at random. The bonds represent the pores of the pore space, or the conducting and insulating regions of a disordered solid, to which we assign effective properties (radii or conductances) selected at random from a probability density function. Three types of nonlinear transport processes are considered. (1) The relation between the current q and the potential gradient v is of power‐law type (as in, for example, flow of power‐law fluids or the electric current in doped polycrystalline semiconductors). (2) The relation between q and v is piecewise linear, characterized by a threshold (as in flow of Bingham fluids or in mechanical or dielectric breakdown of composite solids). (3) A large v is imposed on the system, so that a linear transport theory is not valid. The behavioral study of the effective transport and topological properties of the system, such as the permeability, conductivity, diffusivity, and the shape of the samplespanning cluster of conducting paths shows that in all cases the concepts of percolation theory play a prominent role, even if the system is well connected and percolation may seem not to play any role. For most cases, new effective‐medium approximations (EMAs) are derived for estimating effective transport properties. Compared to the case of linear transport, new EMAs are considerably more accurate in predicting the scaling properties of the transport coefficients near a critical point such as the percolation threshold. For a power‐law transport process, an exact solution is also derived for the Bethe networks. Using the concepts of percolation theory, scaling laws relating the effective properties to various regimes of transport and to topological properties of the system are also given. A relation between the volumetric flow rate of a power‐law fluid in porous media and the macroscopic pressure drop is derived, which contains no adjustable parameter and is valid at any porosity. To test the accuracy of our analytical predictions, Monte Carlo simulations are carried out for several cases. In most cases, good agreement is found between the simulation results and predictions. The extension of the results to other types of nonlinearities is also discussed.