Flow-induced morphological instabilities due to temporarily-modulated stagnation-point flow

In order to determine how unsteady, non-parallel flows affect the growth of single crystals, we consider the effect of periodically-modulated planar stagnation-point flow on the morphological stability of a directionally-solidifying interface. Far from the interface, velocity components in the melt are proportional to 1+σ cos ωt. The system is modelled by assuming that the thickness of the viscous boundary layer is much larger than the thickness of the solute boundary layer and that the frequency of modulation is much smaller than the strength of plane stagnation-point flow. The linear-stability problem for long-wave disturbances is solved by perturbation theory for small σ and by a Galerkin method for σ=O(1). The result of the Galerkin method implies that the small-σ expansion is actually valid for σ as large as unity. The solidifying interface is either stabilized or destabilized depending on the ratio of the period of modulation to the solute-diffusion time. The unstable disturbances contain two frequencies -viz. ω given by the forcing and a second frequency, intrinsic to the travelling-wave instability associated with non-parallel flows, which is modified by the modulation.