Linear stability analysis of a solidification process with convection in a bounded region of space

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Дәйексөз келтіру

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Аннотация

The morphological/dynamic instability of crystallization process in a bounded region in the presence of intense convection in liquid is studied. The paper considers a linear theory of morphological instability with a flat solid-liquid interface on the example of molten metal and magma. The mathematical model includes heat transfer equations and convective type boundary conditions at the interface. The equations for perturbations of the temperature field and interfacial boundary are found, allowing to obtain the dispersion relation. Its analysis has shown the existence of morphological instability of the flat interfacial boundary for a wide range of wavenumbers. Dynamic perturbations (perturbations of the quasi-stationary crystallization velocity) were also analyzed and two solutions for the perturbation frequency were obtained. One of them is stable and the other one is unstable. The system selects one of them depending on the action of convection. The result of morphological and dynamic instability is the appearance of a two-phase region in front of a flat solid-liquid interface. Therefore, the paper also considers the dynamic instability of stationary crystallization with a two-phase region replaced by a discontinuity surface. In this case, the dynamic instability was also found for a wide range of crystallization velocities.

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Авторлар туралы

E. Makoveeva

Ural Federal University named after the First President of Russia B.N. Yeltsin

Email: dmitri.alexandrov@urfu.ru
Ресей, Yekaterinburg

I. Koroznikova

Ural Federal University named after the First President of Russia B.N. Yeltsin

Email: dmitri.alexandrov@urfu.ru
Ресей, Yekaterinburg

A. Glebova

Ural Federal University named after the First President of Russia B.N. Yeltsin

Email: dmitri.alexandrov@urfu.ru
Ресей, Yekaterinburg

A. Ivanov

Ural Federal University named after the First President of Russia B.N. Yeltsin

Email: dmitri.alexandrov@urfu.ru
Ресей, Yekaterinburg

D. Alexandrov

Ural Federal University named after the First President of Russia B.N. Yeltsin

Хат алмасуға жауапты Автор.
Email: dmitri.alexandrov@urfu.ru
Ресей, Yekaterinburg

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Жүктеу
2. Fig. 1. a Scheme of morphologically unstable crystallization with a flat solid-liquid interface; b scheme of dynamically unstable crystallization with a rupture surface.

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3. Fig. 2. a Dispersion curves constructed in accordance with expression (15), m/s; b the frequency of disturbances depending on the crystallization rate at (dynamic disturbances), shown in accordance with expression (15). There are two solutions for each temperature: one solution is stable, at , and the second is unstable, at . The physical parameters are taken for the molten metal from [34] (Table 1).

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4. Fig. 3. a Dispersion curves constructed in accordance with expression (15), m/s; b the frequency of perturbation depending on the crystallization rate at (dynamic perturbations), shown in accordance with expression (15). There are two solutions for each temperature, one solution is stable, at , the second unstable, at . The physical parameters are taken for magma [34] (Table 1).

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5. Fig. 4. Dependence of the frequency of dynamic disturbances on the crystallization rate in accordance with expression (27) for magma. The physical parameters of magma are given in Table 1 according to the data of [34, 46].

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