Axisymmetry breaking instabilities of natural convection in a vertical bridgman growth con"guration

A study of the three-dimensional axisymmetry-breaking instability of an axisymmetric convective #ow associated with crystal growth from bulk of melt is presented. Convection in a vertical cylinder with a parabolic temperature pro"le on the sidewall is considered as a representative model. The main objective is the calculation of critical parameters corresponding to a transition from the steady axisymmetric to the three-dimensional non-axisymmetric (steady or oscillatory) #ow pattern. A parametric study of the dependence of the critical Grashof number Gr #3 on the Prandtl number 0)Pr)0.05 (characteristic for semiconductor melts) and the aspect ratio of the cylinder 1)A)4 (A"height/radius) is carried out. The stability diagram Gr #3 (Pr,A) corresponding to the axisymmetric * threedimensional transition is reported for the "rst time. The calculations are done using the spectral Galerkin method allowing an e!ective and accurate three-dimensional stability analysis. It is shown that the axisymmetric #ow in relatively low cylinders tends to be oscillatory unstable, while in tall cylinders the instability sets in due to a steady bifurcation caused by the Rayleigh}Benard mechanism. The calculated neutral curves are non-monotonous and contain hysteresis loops. The strong dependence of the critical Grashof number and the azimuthal periodicity of the resulting threedimensional #ow indicate the importance of a comprehensive parametric stability analysis in di!erent crystal growth con"gurations. In particular, it is shown that the "rst instability of the #ow considered is always three-dimensional. ( 2000 Elsevier Science B.V. All rights reserved.