Crystalline Saffman-Taylor Fingers

We study the existence and structure of steady-state ngers in two-dimensional solidiication, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wull shape and hence the solid-liquid interface are polygons, and in the one-sided quasi-static limit so that the diiusion eld satisses Laplace's equation in the liquid. In a channel of nite width, this problem is the crystalline analog of the classic Saaman-Taylor smooth nger in Hele-Shaw ow. By a combination of analysis and numerical Schwarz-Christooel mapping methods, we show that, as for solutions of the smooth problem, for each choice of Wull 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 Wull shape approaches a circle. We also consider the open dendrite geometry, and show that there are no steady-state solutions having a nite number of sides for any crystalline surface energy. This is in striking contrast to the smooth case, and is an indication that the time-dependent behavior may be more complicated for crystalline surface energies.