Parabolic periodic solutions of precipitation-modified ternary diffusion equations

Closed-form generalizations of Zener's binary solutions to the diffusion equations in one and two semi-inifite dimensions are found for the case where unstable dilute precipitation accompanies directed ternary diffusion. It has been argued elsewhere that instabilities caused by fine precipitates, subject to local conservation of the independent diffusion species and capillarity in a semi-infinite ternary medium, are evidenced by the appearance of a negative determinant of the effective diffusion matrix and a negative eigenvalue as in spinodal decomposition. In the unstable directed-diffusion planar case the eigenfunction corresponding to the negative coefficient is a spatially periodic Kummerian function of the parabolic coordinate [Formula: see text]. In the important asymptotic limit λ → ∞ this solution implies the exact Jablczynski-scaling relation for the time-independent position of the periodic precipitate bands, which is a well-known characteristic of the Liesegang phenomenon. This contrasts with the Belousov–Zhabotinskii instability where travelling waves are the norm. The cylindrical case exhibits an analogous character but is much richer in pattern, including rings, spirals of multifold character, and broken rings, all of which are observed. A free-boundary degeneracy as in the Saffman–Taylor problem is identified.