Thermal effects in rapid directional solidification: weakly-nonlinear analysis of oscillatory instabilities

Abstract

Huntley and Davis have performed a linear stability analysis on a model for the rapid directional solidification of a dilute binary alloy. This model has a velocity-dependent segregation coefficient and liquidus slope, a linear form of attachment kinetics, and the effects of latent heat and the full temperature distribution. The analysis revealed two modes of instability: (i) a steady cellular instability and (ii) an oscillatory instability driven by disequilibrium effects. The oscillatory instability has either a zero or nonzero critical wavenumber, depending on the thermal properties of the system. In this paper, we study the nonlinear behavior of the oscillatory instability with critical wavenumber zero through a weakly-nonlinear analysis. The analysis leads to a complex-coefficient Ginzburg-Landau equation (GLE) governing the amplitude of the fundamental mode of instability to the planar solid/liquid interface. A bifurcation analysis is applied to demark the parametric regions of supercritical (smooth) and subcritical (jump) transitions to the pulsatile mode. Asymptotic limits for low and high pulling speeds are applied by means of a double-expansion procedure to simply the GLE. With these simplified results, stability of the two-dimensional supercritical solutions against sideband disturbances are calculated. We find several regions of parameter space which give either sharp wavenumber selection of the fundamental mode or no stable two-dimensional solutions. The results are then applied to several physical systems.