Shear stabilization of a solidifying front: Weakly nonlinear analysis in a long-wave limit

The manufacturing of single crystals of multi-component materials with uniform material properties is frequently hampered by the presence of morphological instabilities during the solidification. In this paper we extend into the nonlinear regime our previous work on the influence of shear flows on the linear stability of the solid/liquid interface during the directional solidification of binary alloys. The flows are generated by unidirectional or nonplanar harmonic oscillations of the crystal parallel to the mean interface position, and oscillations with physically realizable amplitudes and frequencies are found to be useful for stabilization purposes. A strongly nonlinear equation which governs the evolution of the interface in the limit of high surface energy, a weak flow and thermodynamic equilibrium is derived, and a weakly nonlinear analysis of this equation is performed. For the unidirectional case, it is found that oscillations with sufficiently large amplitude will change the initial bifurcation from super- to subcritical. For the nonplanar case, it is found that subcritical instability of roll, square and hexagonal cells is favored as the amplitude of the flow is increased. Thus, some of the stabilization due to the flow may be lost at finite amplitude, but substantial stabilization can be retained.