Three-dimensional numerical simulation of spoke pattern in oxide melt

Abstract

In order to reveal the mechanism of the well-known surface spoke patterns, three-dimensional numerical simulations of LiNbO 3 melt flow in an open crucible (47 mm φ×46 mm h) were performed by means of the finite difference 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 surface to the ambient was at a temperature of T a. It was found that the Rayleigh effect alone could not reproduce the spoke pattern. However, if the Marangoni effect is taken into account, the numerical results could semi-quantitatively explain the spoke pattern. A series of simulations with various values of the temperature coefficient of surface tension suggests that the spoke patterns are caused by the Marangoni instability in the thin thermal boundary layer near the melt surface. Incipience of the spoke pattern is approximately predicted by a critical Marangoni number value Ma c= γ T Δ Tδ/ μα=57.6, which corresponds to the critical Marangoni number for the incipience of the Marangoni instability in an adiabatic horizontal layer with a free slip bottom at a constant temperature, where γ T =−∂ γ/∂ T is the temperature coefficient of surface tension, δ the depth of the thermal boundary layer beneath the melt surface, Δ T the temperature drop in the boundary layer, μ the viscosity and α the thermal diffusion coefficient of the melt, respectively. A series of simulations suggests that a constant temperature boundary condition at the crucible side wall stabilizes the melt surface and the spoke pattern becomes very difficult to appear.