Liesegang patterned solutions of ternary diffusion equations with precipitate sinks

Generalizations of Zener's solutions to the diffusion equations in one and two dimensions have been found for the case where dilute precipitation accompanying ternary diffusion goes unstable on account of a negative eigenvalue. In the planar case the eigenfunction corresponding to the negative coefficient D is a periodic Kummerian function of the parabolic coordinate In the important asymptotic limit this solution implies pure spatial periodicity in accord with the Jablczynski scaling relation which is a well-known characteristic of the Liesegang Phenomenon. The cylindrical case exhibits an analogous character but is much richer in pattern, including rings, spirals of multifold character, broken rings and spirals and rotating and expanding versions of the same. All predicted patterns exhibit a univariate or multivariate degeneracy stemming from the physically indefinite initial condition and this calls for the application of a principle of pattern wave number selection. A heuristic principle equivalent to maximizing the path probability is used to remove the main degeneracy thus specifying a unique column or radial wave number