Directional crystallization with a mushy region. Part 2: nonlinear analysis of dynamic stability

In this paper, we develop a nonlinear theory of self-oscillatory solidification mode during directional crystallization in the presence of a quasi-equilibrium two-phase region of constitutional supercooling. This study is based on the linear stability theory (Part 1), where we demonstrated that the indicated regime can be formed due to the oscillatory instability at certain values of physical and operating parameters of the system. The development of oscillatory instability is based on a new frontal model of crystallization with a two-phase region, the main feature of which is the replacement of real two-phase region by a discontinuity surface between purely solid and liquid phases. We derive a nonlinear system of equations for determining frequencies and amplitudes of perturbations responsible for the development of oscillatory instability. The solution of this system allows one to analytically determine the fundamental and secondary harmonics of perturbations and calculate the resulting self-oscillations of the crystallization velocity and impurity distribution. The impurity concentration and period of its layered distribution in the solid phase are in good agreement with experimental data.