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It seems likely that the fundamental relation eqn. (12) was recognized much earlier since the mathematical equivalence between the flow of inviscid fluids and electrical conduction is well known [58,64,102]. Recently, Brown [133] pointed out a non-perturbative generalization of eqn. (12). He showed that the conductivity σ of a non-conducting rigid matrix filled with a conducting fluid of conductivity σo is related to the change in the average 'effective mass' ('effective fluid density') of the corresponding inviscid fluid at arbitrary volume fractions. Johnson and Sen [134] showed that this analogy also implies that the acoustic index of refraction n of an ideal fluid in a rigid matrix equals,
where 1 − φ is the 'porosity'. This relation, like so many others, was apparently known to Rayleigh [135]. Acoustic index of refraction measurements for superfluid He4 ('fourth sound') in a porous medium [136] and salt water in a sintered glass bead pack [24] confirm (65) to a good approximation. Recently, there has been a non-perturbative generalization of (3) [137],
where the 'electrical tortuosity' αE is defined as
E = Do/Dp, where Dp is the
diffusion coefficient of particles in the fluid region such that the diffusing particles obey a
reflecting boundary condition when they encounter the matrix.
Do is the diffusion coefficient of the
particle in the fluid in the absence of the rigid matrix.
The diffusion coefficient observed in a macroscopic diffusion measurement on a porous medium with insulating rigid inclusions, however, is not equal to the pore diffusion coefficient, Dp. Rather, the diffusion coefficient D measured in a macroscopic measurement is related to Dp as,
so that (66) reduces to the generalized Einstein relation [138],
For low concentrations (φ → 0+ ) the non-perturbative relation (68) reduces to the known virial expansion [16] for fixed hard sphere inclusions having a reflecting boundary condition,
From a conductivity standpoint (69) corresponds to insulating inclusions. Eqn. (69) shows that the average effective mass <M> of the rigid inclusions determines the leading concentration dependence of the diffusion coefficient in a porous medium in the absence of particle interaction. We note that the insensitivity of [σ]o to particle shape for extended particles (not platelets) means that as particles aggregate at higher concentration then (69) should remain a good approximation. Experiment, indeed, shows that the leading linear concentration dependence in (69) holds to a good approximation over a wide concentration range [33,34].