Pattern Selection in Realistic Models of Dendritic Solidification.

In this thesis I present an analytic approach to the problem of pattern selection in realistic models of dendritic solidification. In the first part of the work I consider steady-state properties of the needle-crystal solutions of the integro-differential equation which describes dendritic crystal growth. I analyze the different situations corresponding to dendritic growth from a pure substance, dendritic growth from a supersaturated solution, and dendritic growth from a dilute binary mixture. The analytic method employed exploits the singular nature of surface tension effects to derive a solvability condition whose solution determines the properties of the observed dendritic tip. Comparison with the available numerical and experimental data is also presented and discussed. In the second part of the thesis I consider the evolution of time-dependent deformations of the needle-crystal solution of the two-dimensional symmetric model of solidification. I show that small frequency perturbations are initially strongly amplified as they propagate from near the tip down the dendrite and I discuss the relevance of these results to sidebranching in dendrites.