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# Abstracts

## Modelling of a phase transition liquid-crystal at the application of an outer impact

Kabanov P.G.

Moscow State University of Technology STANKIN, The Department of Applied mathematics, Russia, 127994, Moscow, 1, Vadkovsky lane

2 pp.

The problem of mathematical modelling of liquid metal crystallization under pressure is considered in the present work. The application of pressure in the process of crystallization allows to eliminate many drawbacks inherent in other technologies of foundry industry. How-ever, it is known from the process of application in the production that this process isn’t al-ways stable; the problem of an optimal choice of deformation parameters hasn’t been solved yet. These circumstances seriously hamper broad introduction of casting with crystallization under pressure.

The growth of crystal from flux is a complex physical and chemical process. Stefan model is a classical model for description of crystallization in mathematical physics. [1].

Alongside with Stefan problem the model of a phase field based on the theory of Lan-dau-Ginsburg phase transitions has been intensively developed recently. [2-3]. Within the frames of this model an additional parameter - the order parameter ξ - alongside with temperature and density is used for description of thermodynamic condition of a two-phase system. The order parameter in a crystal is equal to a unit, in liquid is equal to zero, and it takes intermediate values in the sphere of a phase transition.

The density of thermodynamic potential is given by the Landau-Ginsburg functional.

Evolutionary equations for the process of crystallization with account for compressibil-ity of the medium have been received in the work. As an example, temperature field and posi-tion of an interphase area in the cylinder at the height of 11 cm and the radius of 5 cm., which is a typical size of casting used in the industry, have been modelled in the work.

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