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Analytical solutions of compacting flow past a sphere

Published online by Cambridge University Press:  03 April 2014

John F. Rudge
John F. Rudge*
Affiliation:
Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: rudge@esc.cam.ac.uk
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Abstract

A series of analytical solutions are presented for viscous compacting flow past a rigid impermeable sphere. The sphere is surrounded by a two-phase medium consisting of a viscously deformable solid matrix skeleton through which a low-viscosity liquid melt can percolate. The flow of the two-phase medium is described by McKenzie’s compaction equations, which combine Darcy flow of the liquid melt with Stokes flow of the solid matrix. The analytical solutions are found using an extension of the Papkovich–Neuber technique for Stokes flow. Solutions are presented for the three components of linear flow past a sphere: translation, rotation and straining flow. Faxén laws for the force, torque and stresslet on a rigid sphere in an arbitrary compacting flow are derived. The analytical solutions provide instantaneous solutions to the compaction equations in a uniform medium, but can also be used to numerically calculate an approximate evolution of the porosity over time whilst the porosity variations remain small. These solutions will be useful for interpreting the results of deformation experiments on partially molten rocks.

JFM classification

Geophysical and Geological Flows: Magma and lava flow Multiphase and Particle-laden Flows: Multiphase flow Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 746 , 10 May 2014 , pp. 466 - 497
Copyright
© 2014 Cambridge University Press 

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References

Acheson, D. J. 1990 Elementary Fluid Dynamics. Oxford University Press.Google Scholar
Alisic, L., Rudge, J. F., Wells, G. N., Katz, R. F. & Rhebergen, S.2013 Shear banding in a partially molten mantle. In SIAM Conference on Mathematical and Computational Issues in the Geosciences, Padova, Italy. CP 18, pp. 87–88.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Griffiths, D. J. 1999 Introduction to Electrodynamics. 3rd edn. Prentice-Hall.Google Scholar
Guazzelli, E. & Morris, J. F. 2012 A Physical Introduction to Suspension Dynamics. Cambridge University Press.Google Scholar
Hewitt, I. J. & Fowler, A. C. 2009 Melt channelization in ascending mantle. J. Geophys. Res. 114, B06210.Google Scholar
Katz, R. F. 2010 Porosity-driven convection and asymmetry beneath midocean ridges. Geochem. Geophys. Geosyst. 11, Q0AC07.Google Scholar
Katz, R. F., Spiegelman, M. & Holtzman, B. 2006 The dynamics of melt and shear localization in partially molten aggregates. Nature 442, 676–679.Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Kohlstedt, D. L. & Holtzman, B. K. 2009 Shearing melt out of the Earth: an experimentalist’s perspective on the influence of deformation on melt extraction. Annu. Rev. Earth Planet. Sci. 37, 561–593.CrossRefGoogle Scholar
Lister, J. R., Kerr, R. C., Russell, N. J. & Crosby, A. 2011 Rayleigh–Taylor instability of an inclined buoyant viscous cylinder. J. Fluid Mech. 671, 313–338.Google Scholar
Manga, M. 2005 Deformation of flow bands by bubbles and crystals. Geol. Soc Am. Spec. Pap. 396, 47–53.Google Scholar
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713–765.CrossRefGoogle Scholar
McKenzie, D. & Holness, M. 2000 Local deformation in compacting flows: development of pressure shadows. Earth Planet. Sci. Lett. 180, 169–184.Google Scholar
Mittelstaedt, E., Ito, G. & van Hunen, J. 2011 Repeat ridge jumps associated with plume–ridge interaction, melt transport, and ridge migration. J. Geophys. Res. 116, B01102.Google Scholar
Neuber, H. 1934 Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. J. Appl. Math. Mech. 14, 203–212.Google Scholar
Papkovich, P. F. 1932 Solution générale des équations differentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques. C. R. Acad. Sci. Paris 195, 513–515.Google Scholar
Phan-Thien, N. & Kim, S. 1994 Microstructures in Elastic Media. Oxford University Press.Google Scholar
Poritsky, H. 1938 Generalizations of the Gauss law of the spherical mean. Trans. Am. Math. Soc. 43 (2), 199–225.Google Scholar
Qi, C., Zhao, Y.-H. & Kohlstedt, D. L. 2013 An experimental study of pressure shadows in partially molten rocks. Earth Planet. Sci. Lett. 382, 77–84.Google Scholar
Ricard, Y. 2007 Physics of mantle convection. In Treatise on Geophysics (ed. Schubert, G.), chap. 7.02, pp. 31–87. Elsevier.Google Scholar
Rudge, J. F., Bercovici, D. & Spiegelman, M. 2011 Disequilibrium melting of a two phase multicomponent mantle. Geophys. J. Intl 184, 699–718.CrossRefGoogle Scholar
Sabelfeld, K. K. & Shalimova, I. A. 1997 Spherical Means for PDEs. VSP.Google Scholar
Schiemenz, A., Liang, Y. & Parmentier, E. M. 2011 A high-order numerical study of reactive dissolution in an upwelling heterogeneous mantle – I. Channelization, channel lithology and channel geometry. Geophys. J. Intl 186, 641–664.CrossRefGoogle Scholar
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2010 A multiscale model of partial melts: 1. Effective equations. J. Geophys. Res. 115, B04410.Google Scholar
Spiegelman, M. 1993 Flow in deformable porous media. Part 1. Simple analysis. J. Fluid Mech. 247, 17–38.Google Scholar