Three dimensional simulations of flow effects on dendritic spacing: a study of the scaling laws
The fundamental problem of the selection of the primary spacing of dendritic arrays in directional solidification is reasonably well understood in the absence of convection. The theoretical model by Warren and Langer  treats the transition from a planar solidification front to stable array growth as a ripening process. Recently the present author demonstrated the dependence of the range of stable spacing on interface energy anisotropy by performing phase-field simulations . Sophisticated space experiments [3,4] have been employed to exclude the influence of convection and to test the respective theories. Direct comparison between solidification structures, grown under terrestrial and microgravity conditions, clearly demonstrate the importance of convection [5-7].
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.
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