| Particulate materials (suspensions) comprise particles (the "filler") randomly dispersed throughout a continuous phase of different material properties. The practically important issue is to the find expressions for the effective transport coefficients. In this work, for the case of heat conduction in polydisperse spherical suspension, the method of Random Point functions (RPA) is used based on truncated Volterra-Wiener Expansion (VWE) with basis function which is a random point function of perfect-disorder type. We show that the effect of the filler is related to the 'one-sphere' and 'two-sphere' solutions in a field with a constant gradient at infinity.;For finding the two-sphere solution, bi-spherical coordinates are used. A transformation of the dependent variable is used that leads to separation of variables allowing the use of Legendre's series with exponential convergence. The latter is confirmed by the computations. The high efficiency of the method allows an accuracy of 10-10 to be achieved with as few as 20 terms, which is a major advantage over the method of iterative reflections. Results have been obtained even for relative distance between the spheres of 10-4 which show that even then, the contribution of the pure interaction between two spheres does not exceed 20% of the combined contribution of the two non-interacting spheres. This outlines the quantitative importance of the second order terms in the WVE and allows us to use the first-order terms only, when practical issues are concerned.;The first-order VWE method is applied for identifying the response of the effective heat flux to temporal changes of the average temperature gradient. The boundary value problem for the time dependent first-order kernel is derived and solved by the Laplace transform method. The statistical average for the flux turns out to be a memory integral of the spatially averaged time-dependent temperature gradient. Thus, a novel result is obtained showing that the constructive relationship between the average flux and the averaged temperature gradient is not local in time, but rather involves a convolution integral representing the memory due to the heterogeneity of the system. This gives a rigorous justification of the usage of generalizations of the heat conduction law involving fractional time derivatives. |
