Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T18:23:34.789Z Has data issue: false hasContentIssue false
  • English
  • Français

The combined effects of shear and buoyancy on phase boundary stability

Published online by Cambridge University Press:  17 April 2019

S. Toppaladoddi  and
J. S. Wettlaufer Open the ORCID record for J. S. Wettlaufer [Opens in a new window]
S. Toppaladoddi
Affiliation:
University of Oxford, Oxford OX2 6GG, UK Yale University, New Haven, CT 06520, USA
J. S. Wettlaufer*
Affiliation:
University of Oxford, Oxford OX2 6GG, UK Yale University, New Haven, CT 06520, USA Nordita, Royal Institute of Technology and Stockholm University, Stockholm SE-10691, Sweden
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the effects of externally imposed shear and buoyancy driven flows on the stability of a solid–liquid interface. A linear stability analysis of shear and buoyancy-driven flow of a melt over its solid phase shows that buoyancy is the only destabilizing factor and that the regime of shear flow here, by inhibiting vertical motions and hence the upward heat flux, stabilizes the system. It is also shown that all perturbations to the solid–liquid interface decay at a very modest shear flow strength. However, at much larger shear-flow strength, where flow instabilities coupled with buoyancy might enhance vertical motions, a re-entrant instability may arise.

JFM classification

Convection: Buoyancy-driven instability Phase change: Morphological instability Phase change: Solidification/melting
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 868 , 10 June 2019 , pp. 648 - 665
Copyright
© 2019 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

Camporeale, C. & Ridolfi, L. 2012 Ice ripple formation at large Reynolds numbers. J. Fluid Mech. 694, 225–251.Google Scholar
Chandrasekhar, S. 2013 Hydrodynamic and Hydromagnetic Stability. Dover Publications.Google Scholar
Chung, C. A. & Chen, F. 2001 Morphological instability in a directionally solidifying binary solution with an imposed shear flow. J. Fluid Mech. 436, 85–106.Google Scholar
Claudin, P., Durán, O. & Andreotti, B. 2017 Dissolution instability and roughening transition. J. Fluid Mech. 832, R2.Google Scholar
Coriell, S. R., Cordes, M. R., Boettinger, W. J. & Sekerka, R. F. 1980 Convective and interfacial instabilities during unidirectional solidification of a binary alloy. J. Cryst. Growth 49 (1), 13–28.Google Scholar
Coriell, S. R., McFadden, G. B., Boisvert, R. F. & Sekerka, R. F. 1984 Effect of a forced couette flow on coupled convective and morphological instabilities during unidirectional solidification. J. Cryst. Growth 69 (1), 15–22.Google Scholar
Davies Wykes, M. S., Huang, J. M., Hajjar, G. A. & Ristroph, L. 2018 Self-sculpting of a dissolvable body due to gravitational convection. Phys. Rev. Fluids 3 (4), 043801.Google Scholar
Davis, S. H. 1990 Hydrodynamic interactions in directional solidification. J. Fluid Mech. 212, 241–262.Google Scholar
Davis, S. H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.Google Scholar
Delves, R. T. 1968 Theory of stability of a solid–liquid interface during growth from stirred melts. J. Cryst. Growth 3, 562–568.Google Scholar
Delves, R. T. 1971 Theory of the stability of a solid–liquid interface during growth from stirred melts II. J. Cryst. Growth 8 (1), 13–25.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Driscoll, T. A., Bornemann, F. & Trefethen, L. N. 2008 The chebop system for automatic solution of differential equations. BIT 48 (4), 701–723.Google Scholar
Epstein, M. & Cheung, F. B. 1983 Complex freezing-melting interfaces in fluid flow. Annu. Rev. Fluid Mech. 15 (1), 293–319.Google Scholar
Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. 2006 Sea ice is a mushy layer. Geophys. Res. Lett. 33 (14), L14501.Google Scholar
Feltham, D. L. & Worster, M. G. 1999 Flow-induced morphological instability of a mushy layer. J. Fluid Mech. 391, 337–357.Google Scholar
Feltham, D. L., Worster, M. G. & Wettlaufer, J. S. 2002 The influence of ocean flow on newly forming sea ice. J. Geophys. Res.-Oceans 107 (C2), 1–9.Google Scholar
Forth, S. A. & Wheeler, A. A. 1989 Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model. J. Fluid Mech. 202, 339–366.Google Scholar
Gilpin, R. R., Hirata, T. & Cheng, K. C. 1980 Wave formation and heat transfer at an ice-water interface in the presence of a turbulent flow. J. Fluid Mech. 99 (3), 619–640.Google Scholar
Glicksman, M. E., Coriell, S. R. & McFadden, G. B. 1986 Interaction of flows with the crystal-melt interface. Annu. Rev. Fluid Mech. 18 (1), 307–335.Google Scholar
Huppert, H. E. 1990 The fluid mechanics of solidification. J. Fluid Mech. 212, 209–240.Google Scholar
Liu, Y., Ning, L. & Ecke, R. E. 1996 Dynamics of surface patterning in salt-crystal dissolution. Phys. Rev. E 53 (6), R5572.Google Scholar
McPhee, M. G. 2008 Air-Ice-Ocean Interaction: Turbulent Ocean Boundary Layer Exchange Processes. Springer.Google Scholar
Meakin, P. & Jamtveit, B. 2009 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. Lond. A 659–694.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. Dover Publications.Google Scholar
Neufeld, J. A. & Wettlaufer, J. S. 2008a An experimental study of shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 363–385.Google Scholar
Neufeld, J. A. & Wettlaufer, J. S. 2008b Shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 339–361.Google Scholar
Neufeld, J. A., Wettlaufer, J. S., Feltham, D. L. & Worster, M. G. 2006 Corrigendum to flow-induced morphological instability of a mushy layer. J. Fluid Mech. 549, 442–443.Google Scholar
Ramudu, E., Hirsh, B. H., Olson, P. & Gnanadesikan, A. 2016 Turbulent heat exchange between water and ice at an evolving ice–water interface. J. Fluid Mech. 798, 572–597.Google Scholar
Schulze, T. P. & Davis, S. H. 1994 The influence of oscillatory and steady shears on interfacial stability during directional solidification. J. Cryst. Growth 143 (3–4), 317–333.Google Scholar
Schulze, T. P. & Davis, S. H. 1995 Shear stabilization of morphological instability during directional solidification. J. Cryst. Growth 149 (3–4), 253–265.Google Scholar
Schulze, T. P. & Davis, S. H. 1996 Shear stabilization of a solidifying front: weakly nonlinear analysis in a long-wave limit. Phys. Fluids 8 (9), 2319–2336.Google Scholar
Solari, L. & Parker, G. 2013 Morphodynamic modeling of the basal boundary of ice cover on brackish lakes. J. Geophys. Res. Earth Surf. 118 (3), 1432–1442.Google Scholar
Sreenivasan, K. R. 1989 The turbulent boundary layer. In Frontiers in Experimental Fluid Mechanics, pp. 159–209. Springer.Google Scholar
Toppaladoddi, S. & Wettlaufer, J. S. 2018 Penetrative convection at high Rayleigh numbers. Phys. Rev. Fluids 3, 043501.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Untersteiner, N. & Badgley, F. I. 1965 Roughness Parameters of Sea Ice. J. Geophys. Res. 70 (18), 4573–4577.Google Scholar
Veronis, G. 1963 Penetrative convection. Astrophys. J. 137, 641–663.Google Scholar
Wettlaufer, J. S. 1991 Heat flux at the ice-ocean interface. J. Geophys. Res. 96 (C4), 7215–7236.Google Scholar
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.Google Scholar
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.Google Scholar
Worster, M. G. 1992 Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.Google Scholar
Worster, M. G. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29 (1), 91–122.Google 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