  We simulate solidification in a narrow channel through the use of a
phase-field model with an adaptive grid. In different regimes, we find that the
solid can grow in fingerlike steady-state shapes, or become unstable,
exhibiting unsteady growth. At low melt undercoolings, we find good agreement
between our results, theoretical predictions, and experiment. For high
undercoolings, we report evidence for a new stable steady-state finger shape
which exists in experimentally accessible ranges for typical materials.
