## Abstract

In this chapter we study a particular case of multiphase systems, namely two-phase materials in which one of the phases is randomly dispersed in the other, so that the composite can be viewed on a macroscale as an effective continuum, with well defined properties. In general, the theoretical determination of the parameter for an effective medium requires, as a rule, the solution of a corresponding transport problem at the microscale, which takes into account the morphology of the system and its evolution. As the mathematical problem is well-posed on a microscale, this can be accomplished using, for example, the multiple scale approach shown in Chap. 11; however, the task requires massive computations and is therefore difficult to implement from the practical standpoint. Here, instead, we focus on a deterministic approach to the problem, where the geometry and spatial configuration of the particles comprising the included phase are given and the solution to the microscale problem is therefore sought analytically. As examples, we study the effective thermal conductivity of solid reinforced materials (Sect. 10.1), the effective viscosity of non-colloidal suspensions (Sect. 10.2), the effective permeability of porous materials (10.3) and the effective self- and gradient diffusivities of colloidal suspensions (Sect. 10.4). Then, in Sect. 10.5, an alternative dynamic definition of the transport coefficients is considered, which can also serve as a basis to determine the effective properties of complex systems.

## Keywords

Effective Thermal Conductivity Brownian Particle Gradient Diffusivity Spherical Inclusion Momentum Diffusion## References

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