Effects of Boundary Conditions and Relative Dimensions of a Solution System on Scaling Laws

A two-dimensional hydrodynamic model is employed to analyze the characteristics of crystal growth from solution with only the variation of the solution density caused by the temperature change taken into account. In that case all the characteristics of the solution system only depend on three numbers: the Rayleigh number Ra, the Prandtl number Pr and the Schemidt number Sc. In certain regions of the parameter (Ra, Pr and Sc) spaces, some scaling laws are generated: the scales of the temperature distribution index S_θ, the concentration distribution index S_ø (see text), the fluid velocity and the growth rate of crystal are given by power functions of Ra, Pr and Sc. The effects of the geometrical shape and the boundary condition of the solution system on the scaling laws are studied. When the ratio X of the height to the length of the solution system changes, the scaling laws are still valid and only the coefficients of power functions are changed, which are also power functions of A. The scaling laws are valid both under the isothermal temperature boundary condition and the adiabatic boundary condition at the surfaces of the top and the bottom sides of the solution system. The only difference is that the ratio of S_θ to Ra is greater for the latter than for the former. In certain ranges of Ra, there are no differences between the other power functions for the two cases.