Oscillatory convection in liquid metals in a horizontal temperature gradient

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Abstract
When growing semiconductor crystals from the melt, the free convection induced in the liquid phase by the imposed thermal boundary conditions can become periodic in time. Recently, the onset of this oscillatory convection has been simulated directly through solution of the time‐dependent governing equations for the particular case of an imposed horizontal temperature gradient. In this paper a new approach for detecting the oscillatory instability is presented in which the transition from steady to periodic flow is identified with a Hopf bifurcation in the solution of the steady equations. The critical Grashof number and frequency are predicted by solving an extended system of steady equations that locates exactly the Hopf bifurcation point, and the variation with aspect ratio and Prandtl number of the threshold for oscillations is obtained through continuation methods. By introducing a homotopy parameter into the boundary conditions the variation of the critical Grashof number is computed as the thermal and viscous conditions on the upper surface vary.