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We report progress in three areas of investigation related to dendritic crystal growth. Those items include: 1. Selection of tip features dendritic crystal growth; 2) Investigation of nonlinear evolution for two-sided model; and 3) Rigorous mathematical justification.

We discuss the framework and issues brought forth in the recent work of Kunka, Foster & Tanveer, which incorporates small but nonzero surface energy effects in the nonlinear dynamics of a conformal mapping function z(zeta,t) that maps the upper-half zeta plane into the exterior of a dendrite. In this paper, surface energy effects on the singularities of z(zeta,t) in the lower-half plane were examined, as they move toward the real axis from below. In particular, the dynamics of complex singularities manifests itself in predictions on nature and growth rate of disturbances, as well as of coarsening.

An investigation of a more complete theoretical understanding of convection effects in a vertical Bridgman apparatus is described. The aim is to develop a clear understanding of scalings of various features of dendritic crystal growth in the case that both the surface energy and undercooling are small.

Asymptotic results obtained by Foster and Smith (1989) for inviscid instability modes of the Type-II Long's vortex is extended to account for the effects of finite Reynolds number. It is shown that the nonparallelism of the flow is more important than the viscous terms in determining the finite-Re behavior due to the radial velocity scales with Re exp -1 M. A critical layer of the three-layer structrue of the parallel-flow instability modes is considerably modified by radial velocity. It is found that for azimuthal wavenumber n greater than 1, the nonparallelism stabilizes the unstable inertial modes, leading to determination of neutral curves. For n less than -1, the nonparallel effects always destabilize the vortex to these helical modes.

Long's self-similar vortex is known to have two solutions for each supercritical value of the flow force. Each of those solutions is shown to have a double structure if the flow force is large. The inertial instabilities of one of those large-flow-force limit solutions are investigated, showing that they are related to the instabilities of the Bickley jet in one regime. However, the swirl in the vortex becomes important for long waves, very strongly modifying the sinuous and varicose, Bickley modes. The asymptotic results obtained agree well with the numerical solutions for the sinuous mode, but not for the varicose mode, the difficulty in the latter case being apparently due to mode jumping.