Transverse dispersion in ordered pillar arrays as a Markov chain: extension of the Galton-board model

An extension of the Galton-board model of the transverse solute dispersion in laminar flow through ordered arrays of non-porous cylindrical pillars was proposed. In contrast to the original model, which describes the dispersion process as a one-dimensional random walk with independent, equally probable steps, the extended model treats the process as a Markov chain, namely as a random walk with such correlated steps that the velocity-dependent probability to make a step in the same direction as the preceding step is smaller than the probability to reverse the direction of motion. The relationship between the average squared transverse displacement of the solute and the number of steps in the chain was used to find the expression for the velocity dependence of the transverse dispersion coefficient. The obtained equation differs from the one in the Galton-board model by the multiplier that accounts for the leveling-off of the experimental dependences at high reduced velocities. Although this modified Galton-board model cannot be directly applied to low velocities, a few additional assumptions lead to the expression that fits the whole range of the recent simulated dispersion data well.