Formation of deep grooves in directional solidification

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Abstract
We study a simple model of two-dimensional directional solidification of a binary mixture, controlled by chemical diffusion. In the limit of small partition coefficient, Sivashinsky has reduced the model to a single nonlinear differential equation for the position of the solid-liquid interface. We investigate the behavior of this equation and find that sufficiently far beyond the onset of instability of the straight-line interface, it has stationary solutions which have a periodic spatial dependence. However, these solutions are themselves linearly unstable. We also argue that if this equation is correct, then the straight-line interface will also have finite-amplitude instabilities whether or not it is linearly stable, and the interface profile will penetrate infinitely far into the bulk solid region in finite time. This latter behavior, of course, invalidates the assumptions under which the equation was originally derived. We obtain higher-order corrections to the equation which cut off the growth of the liquid protrusions at a large but finite amplitude.