532.51 Modeling of thermal convection processes under unidirectional crystallization of alloys with liquid bridges motion

Dimitrienko Y. I. (Bauman Moscow State Technical University), Leontieva S. V. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2018-4-324

The paper considers the problem of thermal convection in the melt zone during unidirectional crystallization of a metal axisymmetric sample with a free surface boundary (liquid bridge) under microgravity. The mathematical problem includes a system of Navier-Stokes equations in the Boussinesq approximation with an equation for the mass transfer of impurity particles in a liquid, as well as equations for the motion of the liquid free surface.
A numerical algorithm for solving the problem based on the vortex and current function method, linearization of the problem, and finite-difference approximation using the variable direction method to solve the difference system of linear equations is developed. The physical parameters of thermal convection processes in the melt zone are calculated. It is shown that taking into account the motion of the free boundary near the crystallizing liquid phase leads to a change in the distribution of impurities near the curing surface, which in turn causes a change in the characteristics of the cured material.

[1] Volkov P.K. Priroda (Nature), 2001, no. 11, pp. 35–43
[2] Avduevsky V.S., ed. Matematicheskoe modelirovanye konvektivnogo tep-lomassoobmena na osnove uravneniy Navye-Stoksa [Mathematical modeling of convective heat and mass transfer based on the Navier-Stokes equations]. Mos-cow, Nauka Publ., 1987, 258 p.
[3] Shevtsova V.M., Melnikov D.E., Legros J.C., Yan Y., Saghir Z., Lyubimova T., Sedel G. Physics of Fluids, 2007, vol. 19 (1), 017111.
[4] Shevtsova V.M., Ermakov M.K., Ryabitskii E., Legros J.C. Acta. Astronautica, 1997, no. 41, pp. 471–479.
[5] Melnikov D.E., Pushkin D., Shevtsova V. The European Physical Journal Spe-cial Topics, 2011, vol. 192, no.1, pp. 29–39.
[6] Ermakov M.K., Nikitin S.A., Polezhaev V.I. Fluid Dynamics, 1997, vol. 23,
no. 3, pp. 338–350.
[7] Martínez I., Perales J.M., Meseguer J. Lecture Notes in Physics, 1996, vol. 464, pp. 271–282.
[8] Kulikov V., Briesen H., Marquardt W. Chemical Engineering Research and De-sign, 2005, vol. 83, pp. 706–717.
[9] Nishino K., Yano T., Kawamura H., Matsumoto S., Ueno I., Ermakov M.K. Journal of Crystal Growth, 2015, vol. 420, pp. 57–63.
[10] Dimitrienko Yu. I. Mekhanika Sploshnoy Sredy. Tom 1. Tensornyy Analiz [Con-tinuum Mechanics. Vol. 1. Tensor Analysis]. Moscow, BMSTU Publ., 2011, 463 p.
[11] Dimitrienko Yu.I. Nonlinear Continuum Mechanics and Large Inelastic Defor-mations. Springer Publ., 2002, 721 p.
[12] Dimitrienko Yu. I. Mekhanika Sploshnoy Sredy. Tom 2. Universalniye zakony mekhaniki i elektrodinamiki sploshykh sred [Continuum Mechanics. Vol. 2. Universal lows of continuum mechanics and electrodynamics]. Moscow, BMSTU Publ., 2011, 560 p.
[13] Samarsky A.A. Teoriya raznostnykh skhem. [Theory of difference schemes]. Moscow, Nauka Publ., 1989, 616 p.
[14] Dimitrienko Y. I., Li S. Matematicheskoe modelirovanie i chislennye menody – Mathematical Modeling and Computational Methods, 2018, no. 2, pp. 70–95.

Димитриенко Ю.И., Леонтьева С.В. Моделирование термоконвективных процессов при однонаправленной кристаллизации сплавов с учетом движения свободных границ. Математическое моделирование и численные методы. 2018. № 4. с. 3–24.

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