On the method of moments for solute dispersion

A cloud of solute injected into a pipe or channel is known to spread out by a dispersion process based on cross-sectional diffusion across a velocity shear. The original description of the process is due to Taylor (1953, 1954), and an important subsequent contribution was by Aris (1956), who framed and partially solved equations for the integral moments of the cloud of contaminant. The present work resolves some technical difficulties that occur when Aris’ solution method (separation of variables) is pursued in depth. In particular, it is shown that Aris’ technique has to be modified to give the moments at short and moderate times after the injection of solute into the flow. The paper is concerned with dispersion in those parallel flows for which an associated eigenvalue problem has a discrete spectrum of eigenvalues; fortunately, this case appears to be the rule rather than the exception. Expressions are obtained for the second and third moment about the mean, and the theory is applied to three cases of interest.