Asymptotic solutions of the Erdogan-Chatwin equation

Erdogan & Chatwin (1967) derived a nonlinear diffusion equation \[ \partial_tc = \partial_z([D_0+(\partial_zc)^2D_2]\partial_zc) \] which models the effect of buoyancy upon the longitudinal dispersion of a solute in pipe flow. The same equation arises more widely as a limiting form in which only the first buoyancy correction is retained. In this paper long-term asymptotic solutions are obtained both for the smearing-out of a concentration jump and for the approach to normality of a finite discharge. A variant of the method provides an approximate solution to the initial-value problem, and a comparison is made with Prych's (1970) experimental results.