TY - JOUR

T1 - Similarity solutions in the theory of curvature driven diffusion along planar curves

T2 - I. Symmetric curves expanding in time

AU - Asvadurov, Sergey

AU - Coleman, Bernard D.

AU - Falk, Richard S.

AU - Moakher, Maher

N1 - Funding Information:
We are grateful to Alberto Cuitifio and David Swigon for the valuable discussions. This research was supported by the National Science Foundation under Grants DMS-94-04580 and DMS-94-03552.

PY - 1998

Y1 - 1998

N2 - 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 rescating into expressions that appear to be in perfect agreement with numerical results for retraction of the tip by curvature driven diffusion.

AB - 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 rescating into expressions that appear to be in perfect agreement with numerical results for retraction of the tip by curvature driven diffusion.

KW - Similarity solutions

KW - Surface diffusion

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U2 - 10.1016/S0167-2789(98)00160-2

DO - 10.1016/S0167-2789(98)00160-2

M3 - Article

AN - SCOPUS:0039938046

SN - 0167-2789

VL - 121

SP - 263

EP - 274

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 3-4

ER -