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Transport in Porous Media

, Volume 72, Issue 1, pp 121–138 | Cite as

On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

  • Daniel N. RiahiEmail author
Article
  • 56 Downloads

Abstract

We consider the problem of mixed oscillatory and steady modes of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of convection.

Keywords

Compositional convection Mushy layer Mixed convection Buoyant flow Nonlinear convection 

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References

  1. Amberg G. and Homsy G.M. (1993). Nonlinear analysis of buoyant convection in binary solidification with application to channel formation. J. Fluid Mech. 252: 79–98 CrossRefGoogle Scholar
  2. Anderson D.M. and Worster M.G. (1995). Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J. Fluid Mech. 302: 307–331 CrossRefGoogle Scholar
  3. Anderson D.M. and Worster M.G. (1996). A new oscillatory instability in a mushy layer during the solidification of binary alloys. J. Fluid Mech. 307: 245–267 CrossRefGoogle Scholar
  4. Busse F.H. (1967). The stability of finite amplitude convection and its relation to an extremum principal. J. Fluid Mech. 30: 625–649 CrossRefGoogle Scholar
  5. Busse F.H. (1975). Patterns of convection in spherical shells. J. Fluid Mech. 72: 67–85 CrossRefGoogle Scholar
  6. Chandrasekhar S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford Google Scholar
  7. Chung C.A. and Chen F. (2000). Onset of plume convection in mushy layers. J. Fluid Mech. 408: 53–82 CrossRefGoogle Scholar
  8. Fowler A.C. (1985). The formation of freckles in binary alloys. IMA J. Appl. Maths. 35: 159–174 CrossRefGoogle Scholar
  9. Riahi D.N. (1983). Nonlinear convection in a porous layer with finite conducting boundaries. J. Fluid Mech. 129: 153–171 CrossRefGoogle Scholar
  10. Riahi D.N. (2002). On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection. J. Fluid Mech. 467: 331–359 CrossRefGoogle Scholar
  11. Riahi D.N. (2004). On nonlinear convection in mushy layers Part 2 Mixed oscillatory and stationary modes of convection. J. Fluid Mech. 517: 71–101 CrossRefGoogle Scholar
  12. Schulze T.P. and Worster M.G. (1999). Weak convection, liquid inclusions and the formation of chimneys in mushy layers. J. Fluid Mech. 388: 197–215 CrossRefGoogle Scholar
  13. Tait S., Jahrling K. and Jaupart C. (1992). The planform of compositional convection and chimney formation in a mushy layer. Nature 359: 406–408 CrossRefGoogle Scholar
  14. Worster M.G. (1992). Instabilities of liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237: 649–669 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas-Pan AmericanEdinburgUSA

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