Similarity solutions in the theory of curvature driven diffusion along planar curves: I. Symmetric curves expanding in time

Abstract

A numerical method is developed for obtaining similarity solutions of the differential equations governing the evolution of a planar curve in the theory of diffusion along curves. The method is applied to cases in which the solution can be interpreted as describing the subsequent evolution (by curvature driven surface diffusion) of the boundary of a three-dimensional body that in the limit t→0+ has a form close to that of a wedge with angle of aperture 2Φ. In the theory of curvature driven evaporation, the analogous problem can be solved analytically, and hence the relation between t, Φ, and the retraction of the wedge tip can be rendered explicit. Although the differential equations of the two theories are of different orders and have solutions that differ in such qualitative properties as preservation of convexity and conservation of volume, it is found that the explicit expressions obtained for the retraction of a wedge tip by curvature driven evaporation can be transformed by rescaling into expressions that appear to be in perfect agreement with numerical results for retraction of the tip by curvature driven diffusion.