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Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

Published online by Cambridge University Press:  30 January 2013

Andrew J. Wells
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA
J. S. Wettlaufer
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA Department of Physics, Yale University, New Haven, CT 06520-8109, USA Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
Steven A. Orszag
Affiliation:
Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA

Abstract

We model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

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Papers
Copyright
©2013 Cambridge University Press

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Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes
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