Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.
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Victor Roytburd, Michael L. Frankel, Laura Kathryn Gross, Thermo-kinetically controlled pattern selection. Interfaces Free Bound. 2 (2000), no. 3, pp. 313–330