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Analytical solutions of compacting flow past a sphere

  • John F. Rudge (a1)


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.


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Acheson, D. J. 1990 Elementary Fluid Dynamics. Oxford University Press.
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.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Griffiths, D. J. 1999 Introduction to Electrodynamics. 3rd edn. Prentice-Hall.
Guazzelli, E. & Morris, J. F. 2012 A Physical Introduction to Suspension Dynamics. Cambridge University Press.
Hewitt, I. J. & Fowler, A. C. 2009 Melt channelization in ascending mantle. J. Geophys. Res. 114, B06210.
Katz, R. F. 2010 Porosity-driven convection and asymmetry beneath midocean ridges. Geochem. Geophys. Geosyst. 11, Q0AC07.
Katz, R. F., Spiegelman, M. & Holtzman, B. 2006 The dynamics of melt and shear localization in partially molten aggregates. Nature 442, 676679.
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.
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, 561593.
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, 313338.
Manga, M. 2005 Deformation of flow bands by bubbles and crystals. Geol. Soc Am. Spec. Pap. 396, 4753.
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713765.
McKenzie, D. & Holness, M. 2000 Local deformation in compacting flows: development of pressure shadows. Earth Planet. Sci. Lett. 180, 169184.
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.
Neuber, H. 1934 Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. J. Appl. Math. Mech. 14, 203212.
Papkovich, P. F. 1932 Solution générale des équations differentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques. C. R. Acad. Sci. Paris 195, 513515.
Phan-Thien, N. & Kim, S. 1994 Microstructures in Elastic Media. Oxford University Press.
Poritsky, H. 1938 Generalizations of the Gauss law of the spherical mean. Trans. Am. Math. Soc. 43 (2), 199225.
Qi, C., Zhao, Y.-H. & Kohlstedt, D. L. 2013 An experimental study of pressure shadows in partially molten rocks. Earth Planet. Sci. Lett. 382, 7784.
Ricard, Y. 2007 Physics of mantle convection. In Treatise on Geophysics (ed. Schubert, G.), chap. 7.02, pp. 3187. Elsevier.
Rudge, J. F., Bercovici, D. & Spiegelman, M. 2011 Disequilibrium melting of a two phase multicomponent mantle. Geophys. J. Intl 184, 699718.
Sabelfeld, K. K. & Shalimova, I. A. 1997 Spherical Means for PDEs. VSP.
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, 641664.
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2010 A multiscale model of partial melts: 1. Effective equations. J. Geophys. Res. 115, B04410.
Spiegelman, M. 1993 Flow in deformable porous media. Part 1. Simple analysis. J. Fluid Mech. 247, 1738.
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Analytical solutions of compacting flow past a sphere

  • John F. Rudge (a1)


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