Crystalline Saffman–Taylor Fingers

Abstract

We study the existence and structure of steady-state fingers in two-dimensional solidification, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wulff shape and hence the solid–liquid interface are polygons, and in the one-sided quasi-static limit so that the diffusion field satisfies Laplace’s equation in the liquid. In a channel of finite width, this problem is the crystalline analog of the classic Saffman–Taylor smooth finger in Hele–Shaw flow. By a combination of analysis and numerical Schwarz–Christoffel mapping methods, we show that, as for solutions of the smooth problem, for each choice of Wulff shape there is a critical maximum value of the magnitude of surface tension above which no convex steady-state solutions exist. We then exhibit convergence of convex crystalline solutions to convex smooth solutions as the Wulff shape approaches a circle. We also consider the open dendrite geometry and show that there are no steady-state solutions having a finite number of sides for any crystalline surface energy. This is in striking contrast to the smooth case and an indication that the time-dependent behavior may be more complicated for crystalline surface energies.