THERMAL EFFECTS IN RAPID DIRECTIONAL SOLIDIFICATION - WEAKLY-NONLINEAR ANALYSIS OF OSCILLATORY INSTABILITIES

Huntley and Davis have performed a linear stability analysis on a mode l for the rapid directional solidification of a dilute binary alloy. T his model has a velocity-dependent segregation coefficient and liquidu s slope, a linear form of attachment kinetics, and the effects of late nt heat and the full temperature distribution. The analysis revealed t wo modes of instability: (i) a steady cellular instability and (ii) an oscillatory instability driven by disequilibrium effects. The oscilla tory instability has either a zero or nonzero critical wavenumber, dep ending on the thermal properties of the system. In this paper, we stud y the nonlinear behavior of the oscillatory instability with critical wavenumber zero through a weakly-nonlinear analysis. The analysis lead s to a complex-coefficient Ginzburg-Landau equation (GLE) governing th e amplitude of the fundamental mode of instability to the planar solid /liquid interface. A bifurcation analysis is applied to demark the par ametric regions of supercritical (smooth) and subcritical (jump) trans itions to the pulsatile mode. Asymptotic limits for low and high pulli ng speeds are applied by means of a double-expansion procedure to simp ly the GLE. With these simplified results, stability of the two-dimens ional supercritical solutions against sideband disturbances are calcul ated. We find several regions of parameter space which give either sha rp wavenumber selection of the fundamental mode or no stable two-dimen sional solutions. The results are then applied to several physical sys tems.