Dendritic growth of an elliptical paraboloid with forced convection in the melt

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Abstract
All experimental observations of the growth of fully developed dendritic ice crystals indicate that the shape of the tip region is an elliptical paraboloid. Therefore, moving-boundary solutions of the three-dimensional Navier-Stokes and energy equations are obtained here for the shape-preserving growth of isothermal elliptical paraboloids by using the Oseen approximation which is valid for the low-Reynolds-number viscous flows which prevail in dendritic growth. Explicit expressions for the flow and the temperature fields are derived in a simple way using Ivantsov's method. It is shown that the growth Péclet number, PG, is a function of the aspect ratio A, the Stefan number St, the Reynolds number Re, and the Prandtl number Pr. As the Reynolds number increases PG becomes linear in St, less dependent on A and ultimately varies roughly as Re½.A comparison between the exact solutions given here and the experiments of Kallungal (1974) indicate that A decreases as Re increases. This result agrees qualitatively with the experiments of Kallungal (1974) and Chang (1985). The differences between theory and experiments for Re > 10−3 may be due to attachment kinetic resistance to growth along the c-axis and capillary effects at the tip which make ice dendrites non-isothermal and create conduction in the solid phase. However, more accurate simultaneous measurements of R1 and R2 are needed to determine definitively the mechanisms responsible for these deviations between theory and experiment.