Statistical mechanics with three-dimensional particle tracking velocimetryexperiments in the study of anomalous dispersion. I. Theory

Eulerian models developed to simulate dispersion in fluid mechanics often c onsider the flux of the contaminant species to be proportional to the conce ntration gradient via a constant or time-dependent dispersion coefficient. These models are crude approximations for systems with velocity fluctuation s evolving over a hierarchy of scales on the scale of observation. We say a system behaves in a Fickian fashion if the dispersion coefficient is const ant, it is quasi-Fickian if the dispersion coefficient is time dependent, a nd it is convolution-Fickian if the flux is a convolution. The fractional f lux in the sense of fractional derivatives is a special case of a convoluti on-Fickian flux. More general forms of the flux are possible, and in any ca se we call all fluxes anomalous if there is not a constant coefficient of p roportionality between the flux and the gradient of concentration. In paper I of this two-part sequence we present a theory with statistical mechanica l origins for simulating anomalous dispersion. Under appropriate limiting c onditions the theory gives rise to Fickian, quasi-Fickian, convolution-Fick ian, and fractional-Fickian fluxes. The primary result is a dispersive flux of integral type which in its most general form is not a convolution on ti me (it is non-Markovian however), but it is always a convolution in space. The concentration is represented by the inverse Fourier transform of the se lf-part of the intermediate scattering function. In paper II we present an experimental procedure that uses this theory to examine if and when the Fic kian limit is reached in porous media homogeneous on the Darcy-scale but he terogeneous on the pore-scale. (C) 2001 American Institute of Physics.