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.

Abstract. Little is known about morphological instability of a solidification front during the crystal growth of a thin film of flowing supercooled liquid with a free surface: for example, the ring-like ripples on the surface of icicles. The length scale of the ripples is nearly 1 cm. Two theoretical models for the ripple formation mechanism have been proposed. However, these models lead to quite different results because of differences in the boundary conditions at the solid-liquid interface and liquid-air surface. The validity of the assumption used in the two models is numerically investigated and some of the theoretical predictions are compared with experiments.

Wavy pattern of ice with a specific wavelength occurs during ice growth from a thin layer of undercooled water flowing down the surface of icicles or inclined plane. In the preceding paper [K.Ueno, Phys. Rev. E 68, 021603 (2003)], we have found that restoring forces due to gravity and surface tension is a factor for stabilization of morphological instability of the solid-liquid interface.However, the mechanism for the morphological instability and stability of the solid-liquid interface has not been well understood. In the present paper, it is shown that a phase difference between fluctuation of the solid-liquid interface and distribution of heat flux at the deformed solid-liquid interface, which depends on the magnitude of the restoring forces, is a cause of the instability and stability of the interface. This mechanism is completely different from the usual Mullins-Sekerka instability due to diffusion and stabilization due to the Gibbs-Thomson effect.