Effect of Modulated Taylor-Couette Flows on Crystal-Melt Interfaces: Theory and Initial Experiments

  • G. B. McFadden
  • B. T. Murray
  • S. R. Coriell
  • M. E. Glicksman
  • M. E. Selleck
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 43)

Abstract

An important problem in the process of crystal growth from the melt phase is to understand the interaction of the crystal-melt interface with fluid flow in the melt. This area combines the complexities of the Navier-Stokes equations for fluid flow with the nonlinear behavior of the free boundary representing the crystal-melt interface. Some progress has been made by studying explicit flows that allow a base state corresponding to a one-dimensional crystal-melt interface with solute and/or temperature fields that depend only on the distance from the interface. This allows the strength of the interaction between the flow and the interface to be assessed by a linear stability analysis of the simple base state. The case of a time-periodic Taylor-Couette flow interacting with a cylindrical crystalline interface is currently being investigated both experimentally and theoretically; some preliminary results are given here. The results indicate that the effect of the crystal-melt interface in the two-phase system is to destabilize the system by an order of magnitude relative to the single-phase system with rigid walls.

Keywords

Vortex Furnace Convection Assure Vorticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. T. J. Hurle and E. Jakeman, Introduction to the techniques of crystal growth, PCH Physico Chem. Hydrodyn. 2 (1981), pp. 237–244.Google Scholar
  2. [2]
    M. E. Glicksman, E. Winsa, R. C. Hahn, T. A. Lograsso, S. H. Tirmizi, and M. E. Selleck, Isothermal dendritic growth - a proposed microgravity experiment, Metall. Trans. 19A (1988), pp. 1945–1953.Google Scholar
  3. [3]
    R. A. Brown, Theory of transport processes in single crystal growth from the melt, AIChE J. 34 (1988), pp. 881–911.CrossRefGoogle Scholar
  4. [4]
    S. R. Coriell, G. B. Mcfadden and R. F. Sekerka, Cellular growth during directional solidification,Ann. Rev. Mater. Sci. 15 (1985), pp. 119–145.CrossRefGoogle Scholar
  5. [5]
    M. E. Glicksman, S. R. Coriell and G. B. Mcfadden, Interaction of flows with the crystal-melt interface, Annu. Rev. Fluid Mech. 18 (1986), pp. 307–335.CrossRefGoogle Scholar
  6. [6]
    W. W. Mullins and R. F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, J. Appl. Phys. 35 (1964), pp. 444–451.CrossRefGoogle Scholar
  7. [7]
    S. R. Coriell, M. R. Cordes, W. J. Boettinger, and R. F. Sekerka, Convective and interfacial instabilities during unidirectional solidification of a binary alloy, J. Crystal Growth 49 (1980), pp. 13–28.CrossRefGoogle Scholar
  8. [8]
    S. R. Coriell and R. F. Sekerka, Effect of convective flow on morphological stability, PCH Physico Chem. Hydrodyn. 2 (1981), pp. 281–293.Google Scholar
  9. [9]
    S. H. Davis, Hydrodynamic interactions in directional solidification, J. Fluid. Mech. 212 (1990) pp. 241–262.MathSciNetCrossRefGoogle Scholar
  10. [10]
    R. T. Delves, Theory of Interface Stability, in Crystal Growth, B. R. Pamplin, ed., Pergamon, Oxford, 1974, pp. 40–103.Google Scholar
  11. [11]
    S. R. Coriell, G. B. Mcfadden, R. F. Boisvert, and R. F. Sekerka, Effect of a forced Collette flow on coupled convective and morphological instabilities during unidirectional solidification, J. Crystal Growth 69 (1984), pp. 15–22.CrossRefGoogle Scholar
  12. [12]
    S. R. Coriell and G. B. Mcfadden, Buoyancy effects on morphological instability during directional solidification,J. Crystal Growth 94 (1989), pp. 513–521.CrossRefGoogle Scholar
  13. [13]
    K. Brattkus, and S. H. Davis, Flow induced morphological instability: stagnation point flows, J. Crystal Growth 89 (1988), pp. 423–427.CrossRefGoogle Scholar
  14. [14]
    G. B. Mcfadden, S. R. Coriell, and J. I. D. Alexander, Hydrodynamic and free boundary instabilities during crystal growth: the effect of a plane stagnation flow, Comm. Pure and Appl. Math 41 (1988), pp. 683–706.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    K. Brattkus and S. H. Davis, Flow induced morphological instability: the rotating disk, J. Crystal Growth 87 (1988), pp. 385–396.CrossRefGoogle Scholar
  16. [16]
    S. A. Forth and A. A. Wheeler, Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model, J. Fluid Mech. 202 (1989), pp. 339–366.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    S. H. Davis, U. Moller, and C. Dietsche, Pattern selection in single-component systems coupling Be’nard convection and solidification,J. Fluid. Mech. 144 (1984) pp. 133–151.CrossRefGoogle Scholar
  18. [18]
    B. Caroli, C. Caroli, C. Misbah, and B. Roulet, Solutal convection and morphological instability in directional solidification of binary alloys,J. Phys. (Paris) 46 (1985), pp. 401–413.CrossRefGoogle Scholar
  19. [19]
    G. W. Young and S. H. Davis, Directional solidification with buoyancy in systems with small segregation coefficient, Phys. Rev. B34 (1986), pp. 3388–3396.Google Scholar
  20. [20]
    G. B. Mcfadden, S. R. Coriell, R. F. Boisvert, M. E. Glicksman, and Q. T. Fang, Morphological stability in the presence of fluid flow in the melt, Metall. Trans. 15A (1984), pp. 2117–2124.Google Scholar
  21. [21]
    Q. T. Fang, M. E. Glicksman, S. R. Coriell, G. B. Mcfadden, and R. F. Boisvert, Convective influence on the stability of a crystal-melt interface,J. Fluid Mech. 151 (1985) pp. 121–140.CrossRefGoogle Scholar
  22. [22]
    G. B. Mcfadden, S. R. Coriell, M. E. Glicksman, and M. E. Selleck, Instability of a Taylor-Couette flow interacting with a crystal-melt interface, PCH Physico Chem. Hydrodyn. 11 (1989), pp. 387–409.Google Scholar
  23. [23]
    G. B. Mcfadden, S. R. Coriell, B. T. Murray, M. E. Glicksman, and M. E. Selleck, Effect of a crystal-melt interface on Taylor-vortex flow, Phys. Fluids A 2 (1990), pp. 700–705.CrossRefGoogle Scholar
  24. [24]
    G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Phil. Trans. Roy. Soc. A 223 (1923), 289–343.Google Scholar
  25. [25]
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, (Clarendon Press, Oxford, 1961).MATHGoogle Scholar
  26. [26]
    P. G. Draztn and W. H. Reid, Hydrodynamic Stability, (Cambridge University Press, New York, 1981).Google Scholar
  27. [27]
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, (McGraw Hill, New York, 1955).Google Scholar
  28. [28]
    S. Carmi and J. I. Tustaniwskyj, Stability of modulated finite-gap cylindrical Couette flow: linear theory, Journal of Fluid Mech. 108 (1981) 19–42.MATHCrossRefGoogle Scholar
  29. [29]
    P. Hall, The stability of unsteady cylinder flows, J. Fluid Mech. 67 (1975) 29–63.MATHCrossRefGoogle Scholar
  30. [30]
    G. Seminara and P. Hall, Centrifugal instability of a Stokes layer: linear theory,Proc. R. Soc. Lond. A 350 (1976), 299–316.MATHCrossRefGoogle Scholar
  31. [31]
    B. T. Murray, G. B. Mcfadden, and S. R. Coriell, Stabilization of Taylor-Couette flow due to time-periodic outer cylinder oscillation, Phys. Fluids A, 2 (1990), 2147–2156.MATHCrossRefGoogle Scholar
  32. [32]
    B. T. Murray, S. R. Coriell, and G. B. Mcfadden, The effect of gravity modulation on solutal convection during directional solidification, J. Crystal Growth, in press.Google Scholar
  33. [33]
    S. H. Davis, The stability of time-periodic flows, Annu. Rev. Fluid Mech. 8 (1976), 57–74.CrossRefGoogle Scholar
  34. [34]
    M. R. Scott and H. A. Watts, Computational solution of linear two-point bound-ary value problems via orthonormalization, SIAM J. Numer. Anal. 14 (1977), 40–70.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    SLATEC Common Math Library, National Energy Software Center, Ar-gonne National Laboratory, Argonne, IL.Google Scholar
  36. [36]
    H. B. Keller, Numerical Solution of Two Point Boundary Value Problems, Regional Conference Series in Applied Mathematics 24, SIAM, Philadelphia, 1976.Google Scholar
  37. [37]
    The routine SNSQ is part of the SLATEC Common Math Library [35], and was written by K. L. Hiebert (1980); it is based on Powell [38].Google Scholar
  38. [38]
    M. J. D. Powell, in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon and Breach, New York, 1970), pp. 87–161.Google Scholar
  39. [39]
    C. Canuto, M. Y. Hussaini, A. Quarteront, and T. A. Zang, Spectral Methods in Fluid Mechanics,(Springer, New York, 1988).Google Scholar
  40. [40]
    D. Gottlieb, M. Y. Hussaini and S. A. Orszag, in Spectral Methods for Partial Differential Equations, R. G. Voigt, D. Gottlieb, and M. Y. Hussaini, eds., (SIAM, Philadelphia, 1984) pp. 1–54.Google Scholar
  41. [41]
    J. J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, UNPACK User’s Guide, (SIAM, Philadelphia, 1979).Google Scholar
  42. [42]
    D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software,(Prentice Hall, New Jersey, 1989) p. 295.MATHGoogle Scholar
  43. [43]
    G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd Ed., (Johns Hopkins University Press, Baltimore, 1989).MATHGoogle Scholar
  44. [44]
    G. B. Mcfadden, B. T. Murray, S. R. Coriell, M. E. Glicksman, and M. E. Selleck, Effect of a crystal-melt interface on Taylor-Couette flow with buoyancy, in Proceedings of the 5th International Colloquium on Free Boundary Problems: Theory and Applications, J. Chadam, ed., held in Montreal, Canada, June 13–22, 1990.Google Scholar
  45. [45]
    G. B. Mcfadden, S. R. Coriell, R. F. Boisvert, and M. E. Glicksman, Asymmetric instabilities in buoyancy-driven flow in a tall vertical annulus, Phys. Fluids 27 (1984), pp. 1359–1361.CrossRefGoogle Scholar
  46. [46]
    L. Quartapelle, Vorticity conditioning in the computation of two-dimensional viscous flows, J. Comp. Phys. 40 (1981) pp. 453–477.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    C. R. Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows, J. Comp. Phys. 80 (1989) 72–97.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • G. B. McFadden
  • B. T. Murray
  • S. R. Coriell
    • 1
  • M. E. Glicksman
  • M. E. Selleck
    • 2
  1. 1.National Institute of Standards and TechnologyGaithersburgUSA
  2. 2.Department of Materials EngineeringRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations