Hydrodynamic Instability During Non-uniform Growth of a Helium Crystal

Abstract

The Rayleigh-Taylor instability (Taylor in Proc. R. Soc. Lond. Ser. A 201:192, 1950) is an instability of the interface which can be realized when the lighter fluid is propelling the heavier one. We analyze an analog of the hydrodynamic Rayleigh-Taylor instability for the superfluid-solid ⁴He interface under non-uniform growth rate of the solid phase. The development of the interface instability starts with an accelerated interface growth rate and provided that the magnitude of interface acceleration exceeds some critical value independent of the surface stiffness. The plane and spherical shapes of the liquid-solid interface are considered. For the Richtmyer-Meshkov (Meshkov in Izv. Akad. Nauk SSSR 5:151, 1969) limiting case of an impulsively accelerated interface, the onset of instability does not depend on the sign of the interface acceleration. The investigation of the surface instability can explain the features observed in the experiment (Chavanne et al. in Phys. Rev. Lett. 86:5506–5509, 2001) where the crystal ⁴He nucleation is triggered by focusing an ultrasonic wave of the large amplitude. The onset and development of the instability are expected at the surface of helium crystals in the course of their abnormal fast growth under large magnitudes of overpressures (Tsymbalenko in J. Low Temp. Phys. 138:795, 2005).