By G. Müller (auth.)
1 The content material ofthis article relies on a German booklet model ) which seemed on the finish of the yr 1986. the writer attempted to include - so far as attainable - new very important effects released within the final yr. however the literature within the box of "convection and inhomogeneities in crystal progress from the soften" has elevated lots meanwhile that the reader and the collegues should still make allowance for any incompleteness, additionally within the case that their very important contributions haven't been stated. this might for instance carry for difficulties with regards to the Czochralski progress. yet particularly for this subject the reader can be refered to the coming near near quantity of this sequence, which includes precise contributions on "Surface rigidity pushed circulate in Crystal progress Melts" through D. Schwabe and on "Convection in Czochralski Melts" by means of M. Mihelcic, W. Uelhoff, H. Wenzl and okay. Wingerath. The practise of this manuscript has been supported through numerous girls whose assistance is gratefully said via the autor: Mrs. Gisela Neuner for the sort writing, Mrs. Abigail Sanders, Mrs. Fiona Eels and particularly Prof. Nancy Haegel for his or her assist in questions of the English language and Mrs. Christa Weber for examining corrections. additionally the great cooperation with the Springer Verlag, specially Mrs. Bohlen and with the dealing with editor of Crystals, Prof. H. C. Freyhardt, who seriously learn the manuscript, is acknowledged.
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T "- \ ~ ~ , , , \ , ... - -- - -. ---/ /' \ \ ".. \ \ '-... ------~ . , ...... --~ ~ - \ , \\ \ \ \ 1 r 1 1 / I / ! t I ! - - - - - - - - - - , . ___ .. -------- N Fig. 41. 023 and schematic representation of the flow (according to Fig. 40). 2 Two-Dimensional Numerical Models of Radial Segregation Brown and co-workers85 ,86) have developed a numerical method for axisymmetric solutions of the Stefan problem. They combine the finite element method with a Newton method for solution of the coupled field variables and take into consideration a deformable moving phase boundary.
This profile can clearly be changed by buoyancy convection. This example demonstrates that the model of Favier and de Goer can be used for a qualitative modeling of convective influence on the longitudinal segregation. 5 Determination of Boundary Layer Thicknesses from Solutions of the Basic Equations BPS have already given in their treatment of the effective distribution coefficient63) in the Czochralski method a relation describing the dependence of the boundary layer thickness from the rotation frequency of the crystal.
31. Further details of the apparatus are given in46), and48). In parallel to these experiments flows have been calculated at our institute by Neumann52), using the method described in Sect. 2. These experimental and numerical results will be presented together in the following sections. 42 Lateral Segregation vibrating Fig. 30. 1 Insulating (Adiabatic) Side Walls Results obtained for insulating side walls and water are combined in the representation given in Fig. 32. Typical specific results of experimental and calculated flows are depicted in Figs.