Three-dimensional numerical simulation of spoke pattern in oxide melt

In order to reveal the mechanism of the well-known surface spoke patterns, three-dimensional numerical simulations of LiNbO3, melt flow in an open cru cible (47 mm(phi) x 46 mm(h)) were performed by means of the Finite differe nce method. The crucible side wall was heated at constant heat flux and the bottom was assumed to be adiabatic. Radiation heat loss from the melt surf ace to the ambient was at a temperature of T-a. It was found that the Rayle igh effect alone could not reproduce the spoke pattern. However, if the Mar angoni effect is taken into account, the numerical results could semi-quant itatively explain the spoke pattern. A series of simulations with various v alues of the temperature coefficient of surface tension suggests that the s poke patterns are caused by the Marangoni instability in the thin thermal b oundary layer near the melt surface. Incipience of the spoke pattern is app roximately predicted by a critical Marangoni number value Ma(c) = gamma(T)D elta T delta/mu alpha = 57.6, which corresponds to the critical Marangoni n umber for the incipience of the Marangoni instability in an adiabatic horiz ontal layer with a free slip bottom at a constant temperature, where gamma( T) = - gamma partial derivative/partial derivative T is the temperature coe fficient of surface tension, delta the depth of the thermal boundary layer beneath the melt surface, Delta T the temperature drop in the boundary laye r, mu the viscosity and a the thermal diffusion coefficient of the melt, re spectively. A series of simulations suggests that a constant temperature bo undary condition at the crucible side wall stabilizes the melt surface and the spoke pattern becomes very difficult to appear. (C) 1999 Elsevier Scien ce B.V. All rights reserved.