Pattern formation in crystal growth under parabolic shear flow

Morphological instability of the solid-liquid interface occurring in a crystal growing from an undercooled thin liquid bounded on one side by a free surface and flowing down inclined plane, is investigated by a linear stability analysis under shear flow. It is found that restoring forces due to gravity and surface tension is an important factor for stabilization of the solid-liquid interface on long length scales. This is a stabilizing effect different from the Gibbs-Thomson effect. A particular long wavelength mode of about 1 cm of wavy pattern, observed on the surface of icicles covered with a thin layer of flowing water is obtained from the dispersion relation, including the effect of flow and restoring forces.