Computation of Transcendental Effects in Growth Problems: Linear Solvability Conditions and Nonlinear Methods-The Example of the Geometric Model

Abstract

It is now well understood that in many moving interface problems, surface tension plays a singular role⁽¹⁾. The basic reason is that surface tension multiplies a term that contains the highest number of derivatives. This realization⁽²⁾ has made possible the analysis of several problems, including the velocity selection for a parabolic needle crystal moving in an undercooled melt⁽³⁾ and the shape selection of viscous fingers moving in a linear Hele-Shaw cell⁽⁴⁾. The first general analytic method for treating this kind of singular perturbation problems was found by J. Langer⁽⁵⁾ in a simple case. He proposed to linearize the equations around a zero-surface tension solution while keeping in the linear equation terms with the highest number of derivatives although they were formally of higher order in the expansion parameter. Exponentially small terms could then be obtained from a linear solvability condition. It was however clear since its very proposal that Langer’s method could not be completely correct since the magnitude of the exponentially small term was about half the result obtained by solving the equation numerically⁽⁵⁾. A more satisfactory way of handling the problem was described by M. Kruskal and H. Segur⁽⁶⁾ who generalized to nonlinear ODE a well-known idea⁽⁷⁾ in the linear case. It consists in extending the equation into the complex plane and analyzing it carefully in the neighborhhood of a singularity of the terms of the regular perturbation expansion. Contrary to Langer’s treatment it is not entirely analytic but reduces to solving a parameter free nonlinear problem numerically. Therefore the method of ref. (5) has continued to be utilized under one form or another by some authors, in spite of its somewhat unclear basis.