Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-25T05:27:38.027Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluxes through steady chimneys in a mushy layer during binary alloy solidification

Published online by Cambridge University Press:  02 January 2013

David W. Rees Jones  and
M. Grae Worster
David W. Rees Jones*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
M. Grae Worster
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: dwr29@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Solute transport within solidifying binary alloys occurs predominantly by convection from narrow liquid chimneys within a porous mushy layer. We develop a simple model that elucidates the dominant structure and driving forces of the flow, which could be applied to modelling brine fluxes from sea ice, where a cheaply implementable approach is essential. A horizontal density gradient within the mushy layer in the vicinity of the chimneys leads to baroclinic torque which sustains the convective flow. In the bulk of the mushy layer, the isotherms are essentially horizontal. In this region, we impose a vertically linear temperature field and immediately find that the flow field is a simple corner flow. We determine the strength of this flow by finding a similarity solution to the governing mushy-layer equations in an active region near the chimney. We also determine the corresponding shape of the chimney, the vertical structure of the solid fraction and the interstitial flow field. We apply this model first to a periodic, planar array of chimneys and show analytically that the solute flux through the chimneys is proportional to a mush Rayleigh number. Secondly we extend the model to three dimensions and find that an array of chimneys can be characterized by the average drainage area alone. Therefore we solve the model in an axisymmetric geometry and find new, sometimes nonlinear, relationships between the solute flux, the Rayleigh number and the other dimensionless parameters of the system.

JFM classification

Convection: Convection in porous media Geophysical and Geological Flows: Sea ice Phase change: Solidification/melting
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 127 - 151
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beckermann, C. & Wang, C. Y. 1995 Multiphase/-scale modelling of alloy solidification. In Annual Reviews of Heat Transfer (ed. Tien, C. L.), vol. 6, pp. 115–198. Begell House.Google Scholar
Boettinger, W. J., Warren, J. A., Beckermann, C. & Karma, A. 2002 Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32 (1), 163–194.CrossRefGoogle Scholar
Chung, C. A. & Worster, M. G. 2002 Steady-state chimneys in a mushy layer. J. Fluid Mech. 455, 387–411.CrossRefGoogle Scholar
Copley, S. M., Giamei, A. F., Johnson, S. M. & Hornbecker, M. F. 1970 The origin of freckles in unidirectionally solidified castings. Metall. Trans. 1, 2193–2204.CrossRefGoogle Scholar
Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. 2006 Sea ice is a mushy layer. Geophys. Res. Lett. 33 (14).CrossRefGoogle Scholar
Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35 (2), 159–174.CrossRefGoogle Scholar
Hills, R. N., Loper, D. E. & Roberts, P. H. 1983 A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Mech. 36 (4), 505–540.CrossRefGoogle Scholar
Hunke, E. C., Notz, D., Turner, A. K. & Vancoppenolle, M. 2011 The multiphase physics of sea ice: a review for model developers. Cryosphere 5 (4), 989–1009.CrossRefGoogle Scholar
Huppert, H. E. 1990 The fluid mechanics of solidification. J. Fluid Mech. 212, 209–240.CrossRefGoogle Scholar
Huppert, H. E. & Worster, M. G. 1985 Dynamic solidification of a binary melt. Nature 314 (6013), 703–707.CrossRefGoogle Scholar
Huppert, H. E. & Worster, M. G. 2012 Flows involving phase change. In Handbook of Environmental Fluid Dynamics (ed. H. J. Fernando). Taylor and Francis (in press).Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44 (4), 508–521.CrossRefGoogle Scholar
Notz, D. & Worster, M. G. 2008 In situ measurements of the evolution of young sea ice. J. Geophys. Res. 113, doi:10.1029/2007JC004333.Google Scholar
Notz, D. & Worster, M. G. 2009 Desalination processes of sea ice revisited. J. Geophys. Res. 114, doi:10.1029/2008JC004885.Google Scholar
Oertling, A. B. & Watts, R. G. 2004 Growth of and brine drainage from NaCl–H2O freezing: a simulation of young sea ice. J. Geophys. Res. 109, doi:10.1029/2001JC001109.Google Scholar
Peppin, S. S. L., Aussillous, P., Huppert, H. E. & Worster, M. G. 2007 Steady-state mushy layers: experiments and theory. J. Fluid Mech. 570, 69–77.CrossRefGoogle Scholar
Petrich, C., Langhorne, P. & Eicken, H. 2011 Modelled bulk salinity of growing first-year sea ice and implications for ice properties in spring. In Proc. 21st Intl Conf. on Port and Ocean Engineering under Arctic Conditions (POAC), Montreal, Canada, pp. 1–10. POAC11–187 (electronic).Google Scholar
Schulze, T. P. & Worster, M. G. 1998 A numerical investigation of steady convection in mushy layers during the directional solidification of binary alloys. J. Fluid Mech. 356, 199–220.CrossRefGoogle Scholar
Schulze, T. P. & Worster, M. G. 1999 Weak convection, liquid inclusions and the formation of chimneys in mushy layers. J. Fluid Mech. 388, 197–215.CrossRefGoogle Scholar
Untersteiner, N. 1968 Desalination processes of sea ice revisited. J. Geophys. Res. 74, 1251–1257.CrossRefGoogle Scholar
Vancoppenolle, M., Goosse, H., de Montety, A., Fichefet, T., Tremblay, B. & Tison, J.-L. 2010 Modelling brine and nutrient dynamics in Antarctic sea ice: the case of dissolved silica. J. Geophys. Res. 115, doi:10.1029/2009JC005369.Google Scholar
Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2010 Maximal potential energy transport: a variational principle for solidification problems. Phys. Rev. Lett. 105, 254502.CrossRefGoogle ScholarPubMed
Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. 1997 Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316.CrossRefGoogle Scholar
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.CrossRefGoogle Scholar
Worster, M. G. 1992a Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.CrossRefGoogle Scholar
Worster, M. G. 1992b The dynamics of mushy layers. In Interactive Dynamics of Convection and Solidification, NATO ASI Series , vol. E219,. pp. 113–138. Kluwer.CrossRefGoogle Scholar
Worster, M. G. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29 (1), 91–122.CrossRefGoogle Scholar
Worster, M. G. 2000 Solidification of fluids. In Perspectives in Fluid Dynamics: a Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.). pp. 393–446. Cambridge University Press.Google Scholar
Supplementary material: PDF

Rees Jones supplementary material

Supplementary data

Download Rees Jones supplementary material(PDF)
PDF 609.1 KB