Global feedback control of a long-wave morphological instability

Nonlinear dynamics of a long-wave morphological instability of a crystallization front during directional solidification in the presence of feedback control is investigated. Feedback control is applied by varying the pulling speed and the temperature gradient according to the measured maximum depressions of the crystal–melt interface. For the case of small segregation coefficient, a long-scale nonlinear evolution equation for the shape of the interface is derived in the frozen-temperature approximation. The equation contains a nonlocal (global) control term and describes the near-threshold nonlinear dynamics of the morphological instability in the presence of the feedback control. This equation is studied analytically and numerically, both for 1 + 1 and 2 + 1 interfaces. It is shown that the chosen global feedback control can prevent the blow-up of subcritical and supercritical solutions, i.e. the formation of deep cells. It leads to the formation of stationary periodic patterns or spatially-localized structures.