CAHN-HILLIARD EQUATION IN THE YAKOVLEV-LUKASHEVICH-RADKEVICH CRYSTALLIZATION MODEL

The report will focus on the numerical implementation and some mathematical issues regarding the non-traditional crystallization model that appeared in the work "N.N. Yakovlev, E.A. Lukashev, E.V. Radkevich. Problems of Reconstruction of the Directional Crystallization Process // DAN RAS, 2008, v. 421, no. 5, pp. 625-629." The numerical implementation in question was performed in 2011-2012, i.e. more than ten years ago, but is probably still of interest, since The model incorporates non-traditional theoretical concepts. The model is based on the concept of the space of a crystallizing alloy as a porous medium, the propagation of disturbances in which is described by Biot-type equations. A modified Cahn-Hilliard equation is used to describe the process of nucleation. A numerical scheme is constructed and its convergence in the one-dimensional and two-dimensional cases is demonstrated. The possibility of obtaining various crystallization regimes with a change in parameters is shown. An attempt to rigorously substantiate the basic principles of the model is also given. A simple model of the flow of a two-component, two-speed continuous medium, which is exposed to small-scale fluctuations, is formulated. It is shown that in this case, regimes arise in the model in which so-called non-classical jumps are formed (viscous terms are naturally omitted). The evolution of such jumps is not determined by the Rankine-Hugoniot relations.