Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow


A theory of stable dendrite growth in an undercooled binary melt is developed for the case of intense convection. Conductive heat and mass transfer boundary conditions are replaced by convective conditions, where the flux of heat (or solute) is proportional to the temperature or concentration difference between the surface of the dendrite and far from it. The marginal mode of perturbation wavelengths is calculated using the linear morphological stability analysis. Combining this analysis with the solvability theory, we have derived a selection criterion that represents the first condition to define a combination of dendrite tip velocity and tip diameter. The second condition—the undercooling balance—is derived for intense convection. The theory under consideration determines the dendrite tip velocity and tip diameter for low undercooling. This convective theory is combined with the classical theory of dendritic growth (conductive boundary conditions), which is valid for moderate and high undercooling. Thus, the entire range of melt undercooling is covered. Our results are in good agreement with experiments on Al–Ge crystallization. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.Ministry of Education and Science of the Russian Federation, Minobrnauka: 075-02-2021-1387; Russian Science Foundation, RSF: 21-19-00279; Foundation for the Advancement of Theoretical Physics and Mathematics: 21-1-3-11-1L.V.T. acknowledges financial support from the Ministry of Science and Higher Education of the Russian Federation (project 075-02-2021-1387 for the development of the regional scientific and educational mathematical center “Ural Mathematical Center”) for the linear stability analysis. Moreover, she is grateful to the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (project No. 21-1-3-11-1) for the development of solvability theory. P.K.G. and D.V.A. acknowledge the Russian Science Foundation (Project No. 21-19-00279) for the stitching of selection criteria, computer simulations, and comparison with experimental data. Open Access funding enabled and organized by Projekt DEAL

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