Large-amplitude solutions to the Sivashinsky and Riley–Davis equations for directional solidification

Abstract

Directional solidification of a dilute binary alloy is often characterized by the appearance of deep-root, finger-type interfaces. As a model of this phenomenon, we investigate the large-amplitude behavior of long-wave evolution equations previously derived for directional solidification. In this paper we conduct a detailed asymptotic and numerical study of the periodic solutions of the Sivashinsky and Riley–Davis equations. The Sivashinsky equation, which describes the limit of small segregation coefficient, is shown to form deep-root solutions which can be described in terms of elliptic functions. However, this equation is known to be ill-posed in the sense that the solution branches are unstable, and large classes of initial conditions lead to finite-time blow-up. By considering asymptotically large surface energy, Riley and Davis derived an evolution equation which may be considered a regularizing perturbation of the Sivashinsky equation. This equation has steady, large-amplitude solutions which are linearly stable. In the absence of solutal segregation, we may describe these solutions asymptotically; in this limit, amplitudes and wavelengths may become arbitrarily large. A stability analysis suggests that with periodic boundary conditions the solution may coarsen until a single finger fills the domain. If segregation effects are included, both the amplitude and wavelength are bounded independent of domain size. This wavelength bound gives the system a finger-width selection mechanism associated with a tip-splitting bifurcation that has also been observed in experiments.