Advertisement

Transport in Porous Media

, Volume 82, Issue 2, pp 385–399 | Cite as

On Perturbation and Marginal Stability Analysis of Magneto-Convection in Active Mushy Layer

  • Dambaru BhattaEmail author
  • Mallikarjunaiah S. Muddamallappa
  • Daniel N. Riahi
Article

Abstract

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush–liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy’s law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.

Keywords

Variable permeability Porous media Mushy layer Magneto-convection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amberg G., Homsy G.M.: Nonlinear analysis of buoyant convection in binary Solidification with application to channel formation. J. Fluid Mech. 252, 79–98 (1993)CrossRefGoogle Scholar
  2. Anderson D.M., Worster M.G.: Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J. Fluid Mech. 302, 307–331 (1995)CrossRefGoogle Scholar
  3. Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)Google Scholar
  4. Chen C.F.: Experimental study of convection in a mushy layer during directional solidification. J. Fluid Mech. 293, 81–98 (1995)CrossRefGoogle Scholar
  5. Chen C.F., Chen F.: Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586 (1991)CrossRefGoogle Scholar
  6. Cheney W., Kincaid D.: Numerical Mathematics and Computing. 6th edn. Thomson Brooks/Cole, Pacific Grove (2008)Google Scholar
  7. Copley S.M., Giamei A.F., Johnson S.M., Hornbecker M.F.: The origin of freckles in unidirectionally solidified castings. Metall. Mater. Trans. 1, 2193–2204 (1970)Google Scholar
  8. Davis S.H.: Theory of Solidification. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  9. Fowler A.C.: The formation of freckles in binary alloys. IMA J. Appl. Math. 35, 159–174 (1985)CrossRefGoogle Scholar
  10. Hills R., Loper D., Roberts P.: A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Math. 36, 505–539 (1983)CrossRefGoogle Scholar
  11. Muddamallappa M.S., Bhatta D., Riahi D.N.: Numerical investigation on marginal stability and convection with and without magnetic field in a mushy layer, Trans. Porous Media (2008, in press). doi: 10.1007/s11242-008-9319-4
  12. Okhuysen, B.S.: Analytical and computational studies of convection in solidifying binary media. Ph.D. thesis, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, USA (2005)Google Scholar
  13. Okhuysen B.S., Riahi D.N.: Nonlinear buoyant convection in mushy layers during alloy solidification of alloys. J. Porous Media 11, 291–303 (2008a)CrossRefGoogle Scholar
  14. Okhuysen B.S., Riahi D.N.: Flow instabilities of liquid and mushy regions during alloy solidification and under high gravity environment induced by rotation. Int. J. Eng. Sci. 46, 189–201 (2008b)CrossRefGoogle Scholar
  15. Riahi D.N.: Effects of a vertical magnetic field on chimney convection in a mushy layer. J. Cryst. Growth 216, 501–511 (2000)CrossRefGoogle Scholar
  16. Riahi D.N.: Effects of centrifugal and coriolis forces on a hydromagnetic chimney convection in a mushy layer. J. Cryst. Growth 226, 393–405 (2001)CrossRefGoogle Scholar
  17. Tait S., Jaupart C.: Compositional convection in a reactive crystalline mush and melt differentiation. J. Geophys. Res. 97, 6735–6756 (1992)CrossRefGoogle Scholar
  18. Vives C., Perry C.: Effects of magnetically damped convection during the controlled solidification of metals and alloys. Int. J. Heat Mass Transf. 30(3), 479–496 (1987)CrossRefGoogle Scholar
  19. Worster M.G.: Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359 (1991)CrossRefGoogle Scholar
  20. Worster M.G.: Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Dambaru Bhatta
    • 1
    Email author
  • Mallikarjunaiah S. Muddamallappa
    • 1
  • Daniel N. Riahi
    • 1
  1. 1.Department of MathematicsThe University of Texas-Pan AmericanEdinburgUSA

Personalised recommendations