Dipole-flow disturbed by a circular inclusion of conductivity different from the background: From deterministic to a self-consistent analytical solution

Steady dipole-flow through a porous medium, and disturbed by a circular inclusion Omega(0) of conductivity different from the background, is solved analytically. The solution is achieved by means of the circle theorem, which is reformulated to account for the entry/leave of mass and energy through the boundary partial derivative Omega(0). It is shown that the governing potential is that which one would consider in absence of the disturbance supplemented with an ad hoc (fictitious) dipole laying inside Omega(0). Besides the theoretical interest, the analytical solution is used to compute the effective conductivity K-eff, by means of the self-consistent approximation. Overall, K-eff is found to depend upon the flow configuration, and therefore it cannot be sought as a medium's property (nonlocality). In particular, K(eff )depends upon the joint probability density function f of the conductivity and the distribution/size of the inclusions. Results, analyzed for a fairly general model of f, demonstrate that the coefficient of correlation rho between the involved random fields is the key parameter characterizing the structure of K-eff. Indeed, the latter results larger or smaller than that of the background, depending on whether rho is negative or positive, respectively. For rho = 0, the effective conductivity is a local property and, in this case, one can apply the superposition principle with the homogeneous conductivity replaced by the geometric mean.