20 04 Conformal Mapping Methods for Interfacial Dynamics

1 Microstructural evolution is typically beyond the reach of mathematical analysis, but in two dimensions certain problems become tractable by complex analysis. Via the analogy between the geometry of the plane and the algebra of complex numbers, moving free boundary problems may be elegantly formulated in terms of conformal maps. For over half a century, conformal mapping has been applied to continuous interfacial dynamics, primarily in models of viscous fingering and solidification. Current developments in materials science include models of void electro-migration in metals, brittle fracture, and viscous sintering. Recently, conformal-map dynamics has also been formulated for stochastic problems, such as diffusion-limited aggregation and dielectric breakdown, which has reinvigorated the subject of fractal pattern formation. Although restricted to relatively simple models, conformal-map dynamics offers unique advantages over other numerical methods discussed in this chapter (such as the Level-Set Method) and in Chapter 9 (such as the Phase Field Method). By absorbing all geometrical complexity into a time-dependent conformal map, it is possible to transform a moving free boundary problem to a simple, static domain, such as a circle or square, which obviates the need for front tracking. Conformal mapping also allows the exact representation of very complicated domains, which are not easily discretized, even by the most sophisticated adap-tive meshes. Above all, however, conformal mapping offers analytical insights for otherwise intractable problems. After reviewing some elementary concepts from complex analysis in §1, we consider the classical application of conformal mapping methods to continous-time interfacial free boundary problems in §2. This includes cases where the governing field equation is harmonic, bihar-monic, or in a more general conformally invariant class. In §3, we discuss the recent use of random, iterated conformal maps to describe analogous discrete-time phenonena of fractal growth. Although most of our examples involve planar domains, we note in §4 that interfa-cial dynamics can also be formulated on curved surfaces in terms of more general conformal maps, such as stereographic projections. We conclude in §5 with some open questions and an outlook for future research.