Geometrical models of interface evolution

We introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion. The intended application is to systems for which competing stabilizing and destabilizing forces act on the phase boundary to produce irregular or patterned structures, such as those which occur in solidification. We discuss the kinematics of moving interfaces in two or more dimensions in terms of their intrinsic geometric properties. We formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems. Special solutions for dendritic crystal growth and their stability are analyzed in some detail.

[1]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  Martin E. Glicksman,et al.  Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth , 1981 .

[3]  H. Müller-Krumbhaar,et al.  Theory of dendritic growth—II. Instabilities in the limit of vanishing surface tension☆ , 1978 .

[4]  Martin E. Glicksman,et al.  Overview 12: Fundamentals of dendritic solidification—II development of sidebranch structure☆ , 1981 .

[5]  Richard C. Brower,et al.  Geometrical Approach to Moving-Interface Dynamics , 1983 .

[6]  H. Müller-Krumbhaar,et al.  Theory of dendritic growth—III. Effects of surface tension☆ , 1978 .

[7]  H. Müller-Krumbhaar,et al.  Mode selection in a dendritelike nonlinear system , 1983 .

[8]  M. Glicksman,et al.  Capillaritl-limited steadl-state dendritic grolth—I. Theoretical development☆ , 1974 .

[9]  J. Langer.  Instabilities and pattern formation in crystal growth , 1980 .

[10]  J. Koplik,et al.  Percolation and capillary fluid displacement , 1983 .

[11]  D. M. Kroll,et al.  A generalized kinetic equation for crystal growth , 1977 .

[12]  R. Sekerka,et al.  Stability of a Planar Interface During Solidification of a Dilute Binary Alloy , 1964 .

[13]  H. Müller-Krumbhaar,et al.  THEORY OF DENDRITIC GROWTH—I. ELEMENTS OF A STABILITY ANALYSIS , 1978 .

[14]  Eshel Ben-Jacob,et al.  Dynamics of Interfacial Pattern Formation , 1983 .

[15]  Jeffrey Smith.  Shape instabilities and pattern formation in solidification: A new method for numerical solution of the moving boundary problem , 1981 .

[16]  G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1977 .

[17]  James S. Langer,et al.  Propagating pattern selection , 1983 .

[18]  P. van Meurs,et al.  The Instability of Slow, Immiscible, Viscous Liquid-Liquid Displacements in Permeable Media , 1959 .