Numerical Issues Associated With the Full-Zone Model of the Optically Heated Floating-Zone Used in Semiconductor Processing

Recent models of the thermocapillary driven liquid bridge or full-zone (FZ) have highlighted numerical difficulties in the system, associated with the large velocity gradients near the free surface and the geometric singularity at r = 0. High resolution spectral solutions have been developed to account for these issues. These result in complex representations and highly specialized numerical procedures. After a brief review of these methods, a simplified formulation for the FZ model with strong form boundary conditions is proposed and discussed. Comparisons are made using base flows and stability analyses. Existing solutions have overcome the geometric singularity either by moving the grid away from the r = 0 axis, or by maintaining the correct Taylor series expansion in the representation of each dependent variable. The former has the weakness that an important constraint is not applied. The later formulation is rigorous, but results in complex expressions for the governing equations. To decrease the load associated with the mathematical manipulation and numerical implementation of this method, here a more general Chebyshev polynomial representation of the stream function is applied to the axisymmetric base flow. This removes the need to maintain the proper expansion and instead offers a set of equations in strong form by treating the axis as a boundary known from the spatial symmetry of the model. However, this does not guarantee that momentum is conserved at the internal symmetry boundaries. Various applications of the other boundary conditions are also studied. In the most accurate representation, all boundary conditions except the thermocapillary condition are cast in the strong form via orthogonality. These strong equations must be chosen carefully to avoid introducing redundant conditions. However, the result is a mathematically simpler representation that mimics the accuracy of previous methods.