Three dimensional simulations of flow effects on dendritic spacing: a study of the scaling laws
Despite its practical importance, the problem of pattern formation during solidification in the presence of convection remains almost unexplored theoretically. Melt flow introduces a new length scale, the wavelength of convective rolls, and breaks the symmetry of transport depending on the direction of the vector of gravity relative to the direction of growth. Self-organizing pattern formation in solidification begins to compete with self-organization of convection patterns. On the one hand, convective transport of solute significantly alters the growth conditions. On the other hand, the magnitude of convection depends critically on solute gradients due to growth and on the friction of the convective liquid melt between the dendrite trunks. Recently, simple scaling relations have been obtained for the case of a two dimensional system under buoyancy driven flow . The goal of this project is to extend these studies to the case of three dimensions. A question of interest here is whether ideas similar to those proposed in  can be used to find scaling relations in 3D. This will allow working out similarities and differences between the 2D and 3D cases and test then via hybrid phase field-lattice Boltzmann simulations.
 Warren JA, Langer JS. Phys Rev E 1993;47:2702–12.
 Steinbach I. Acta Mater 2008:49:65–71.
 Winsa EA, Glicksman ME, Koss MB. Phys Rev Lett 1994;73:573.
 Lee Y-W, Smith RN, Glicksman ME, Koss MB. Effect of buoyancy on the growth of dendritic crystals. In: Tien CL, editor. Annual reviews in heat transfer. New York: Begell House; 1996. p. 59.
 Dupouy MD, Drevet B, Camel D. J Cryst Growth 1997;181:145–59.
 Drevet B, Nguyen-Thi H, Camel D, Billia B, Dupouy MD. J Cryst Growth 2003;281:419–33.
 Nguyen H, Dabo Ti Y, Dupouy MD, Camel D, Billia B, Hunt JD, et al. J Cryst Growth 2005;281:654–68.
 Steinbach I, Acta Mater 2009; 57:2640-2645.