Physica 21D (1986) 371-380 North-Holland, Amsterdam THE GEOMETRICAL MODEL OF DENDRITIC GROWTH: THE SMALL VELOCITY LIMIT Roger F. DASHEN,t David A. KESSLER,* Herbert LEVINE** and Robert SAVIT* Received 19 December 1985 Revised manuscript received 31 March 1986 We present a systematic analysis of the geometrical model of dendritic growth in the small velocity lim2t. Velocity selection is demonstrated analytically and the allowed velocities are explicitly calculated as a function of anisotropy. 1. Introduction A number of non-equilibrium systems give rise asymptotically to complex, yet reproducible, pat- terns independent of the initial conditions. For example, the dendritic tips of growing snowflakes have a parabolic shape with fixed width and rate of growth (dependent, of course, on the external undercooling and the material parameters) [1]. Similarly, in a Hele-Shaw call, when a viscous fluid is displaced by an inviscid one, the system evolves into a single finger of reproducible shape and velocity [2]. The problem of understanding these patterns has received increasing attention in recent years. In both systems mentioned above, it has been known for some time that, in the absence of surface tension, the equations of motion possess a continuum family of steady-state solutions char- acterized by arbitrary velocity [2, 3]. Furthermore, these solutions, for the experimentally correct velocity, well approximated the actual patterns seen. It had long been conjectured that the surface tension acted in some fashion to select a unique tlnstitute for Advanced Study, Princeton, NJ 08540, USA *Department of Physics, The University of Michigan, Ann Arbor, MI 48109, USA **Schlumberger-Doll Research, Old Quarry Road, Kidgefield, CT 06877, USA velocity, but no mechanism had been elucidated. In particular, perturbative analyses gave no infor- mation on the velocity selection problem [1, 8]. Recently, though, much progress has been made on this question on a number of fronts. One involves a set of simplified models developed to explore the dendritic growth problem. The first is the geometrical model [4], in which the dynamics of an interfacial curve is controlled by the local geometry of the curve. The model, for a wide range of parameters, was found to evolve into complex dendritic shapes reminescent of snow- flakes, with dendritic tips which grew out at a rate independent of initial conditions. The analysis of the model at zero surface tension yielded a con- tinuous family of solutions with arbitrary velocity. It was demonstrated numerically, however, that upon the introduction of surface tension, this family breaks down to a discrete set of solutions [5]. This breakdown is nonperturbative, arising from a singularity exponentially small in the surface tension. The requirement that this singu- larity be absent then forms a non-linear eigen- value problem for the velocity. The final selection of a unique pattern from this discrete set is a dynamical question. In general, only the fastest moving (if any) of this set is linearly stable and thus seen in the simulations [6]. Similar conclusions were drawn from an analy- sis of another simplified model of dendritic growth, 0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)