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  • 1
    Electronic Resource
    Electronic Resource
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 10 (1998), S. 1329-1343 
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Stability and bifurcation analyses of a partially melted or solidified material heated from below and cooled from above in a cavity, the so-called two-phase Rayleigh–Benard problem, are conducted by a finite-volume/Newton's method. Bifurcation analysis techniques using a numerical Jacobian and an iterative matrix solver suitable to this large complicated system are adopted. The onset and evolution of melt flows coupling with the heat conduction in the solid and a deformable melt/solid interface are illustrated through detailed bifurcation diagrams, and the linear stability of each flow family is carefully examined. Some comparison with the one-phase system is performed. Results are presented for a variety of parameters of interest, including the Rayleigh number, aspect ratio, and tilt angle. Although most calculations are presented for the melt with a Prandtl number of one, the effects of Prandtl number on the onset of cellular convection and the sensitivity of symmetry breaking by tilting are examined. Furthermore, the dynamic responses of an unstable static state to stable solutions after small disturbances are illustrated, and the effect of heat of fusion is discussed. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 12 (1991), S. 59-80 
    ISSN: 0271-2091
    Keywords: Thermocapillary flow ; Natural convection ; Melt/solid interface ; Grashof number ; Marangoni number ; Prandtl number ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A vertical melt column set up between an upper heating rod and a lower sample rod, i.e. the so-called halfzone system, is a convenient experimental tool for studying convection in the melt in floating-zone crystal growth. In order to help understand the convection observed in the melt column, a computer model has been developed to describe steady state, axisymmetrical thermocapillary flow and natural convection in the melt. The governing equations and boundary conditions are expressed in general non-orthogonal curvilinear co-ordinates in order to accurately treat the unknown melt/solid interface as well as all other physical boundaries in the system. The effects of key dimensionless variables on the following items are discussed: (1)convection and temperature distribution in the melt; (2) the shape of the melt/solid interface; (3) the height of the melt column. These dimensionless variables are the Grashof, Marangoni and Prandtl numbers.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 19 (1994), S. 41-65 
    ISSN: 0271-2091
    Keywords: Newton's method ; Interface ; Floating zone ; Thermocapillary flow ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier-Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 621-636 
    ISSN: 0029-5981
    Keywords: three-dimensional ; finite volume method ; Newton's method ; free boundaries ; thermal-capillary ; floating zone ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A three-dimensional finite-volume/Newton method is developed for solving thermal-capillary problems in materials processing. The conductive heat transfer, melt-solid interfaces, the melt-gas free surface, and the shape of grown material are calculated simultaneously. The implementation of interface and free surface boundary conditions, as well as co-ordinate transformation, is described in detail. During the Newton iterations, due to the complexity of the problem, the Jacobian matrix is estimated by finite differences, and the linear equations are solved by the ILU(0) preconditioned GMRES iterative method. Nearly quadratic convergence of the scheme is achieved. Sample calculations for floating-zone and Stepanov crystal growth are illustrated. © 1997 by John Wiley & Sons, Ltd.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
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