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Which effective viscosity?

  • N. Petford (a1)

Abstract

Magmas undergoing shear are prime examples of flows that involve the transport of solids and gases by a separate (silicate melt) carrier phase. Such flows are called multiphase, and have attracted much attention due to their important range of engineering applications. Where the volume fraction of the dispersed phase (crystals) is large, the influence of particles on the fluid motion becomes significant and must be taken into account in any explanation of the bulk behaviour of the mixture. For congested magma deforming well in excess of the dilute limit (particle concentrations >40% by volume), sudden changes in the effective or relative viscosity can be expected. The picture is complicated further by the fact that the melt phase is temperature- and shear-rate-dependent. In the absence of a constitutive law for the flow of congested magma under an applied force, it is far from clear which of the many hundreds of empirical formulae devised to predict the rheology of suspensions as the particle fraction increases with time are best suited. Some of the more commonly used expressions in geology and engineering are reviewed with an aim to home in on those variables key to an improved understanding of magma rheology. These include a temperature, compositional and shear-rate dependency of viscosity of the melt phase with the shear-rate dependency of the crystal (particle) packing arrangement. Building on previous formulations, a new expression for the effective (relative) viscosity of magma is proposed that gives users the option to define a packing fraction range as a function of shear stress. Comparison is drawn between processes (segregation, clustering, jamming), common in industrial slurries, and structures seen preserved in igneous rocks. An equivalence is made such that congested magma, viewed in purely mechanical terms as a high-temperature slurry, is an inherently nonequilibrium material where flow at large Pe´clet numbers may result in shear thinning and spontaneous development of layering.

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Anikeenko, A.V. and Medvedev, N.N. (2007) Polytetrahedral nature of dense disordered packings of hard spheres. Physical Review Letters, 98, 235504(4).
Bagnold, R.A. (1954) Experiments on a gravity-free dispersion. Proceedings o f the Royal Society, London, 225, 49—63.
Barnes, H.A. (1997) Thixotropy — a review. Journal of Non-Newtonian Fluid Mechanics, 70, 1 —33.
Batchelor, G.K. and Green, J.T. (1972a) The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. Journal of Fluid Mechanics, 56, 375—400.
Batchelor, G.K. and Green, J.T. (1972b) The determination of the bulk stress in a suspension of spherical particles to order c2. Journal of Fluid Mechanics, 56, 401—427.
Best, M.G. and Christiansen, E.H. (2001) Igneous Petrology. Blackwell Science Inc, Malden, USA, 458 pp.
Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena. Wiley, New York.
Blundy, J. and Cashman, K. (2001) Ascent-driven crystallisation of dacite magmas at Mount St Helens, 1980—1986. Contributions to Mineralogy and Petrology, 140, 631—650.
Brady, J.F. and Bossis, G. (1988) Stokesian dynamics. Annual Review of Fluid Mechanics, 20, 111 — 157.
Carmichael, I.S.E., Francis, J.T. and Verhoogen, J. (1974) Igneous Petrology. McGraw-Hill, New York, 739 pp.
Caricchi, L., Burlinia, L., Ulmer, P., Geryaa, T., Vassalli, M. and Papale, P. (2007) Non-Newtonian rheology of crystal-bearing magmas and implications for magma ascent dynamics. Earth and Planetary Science Letters, 264, 402—419.
Cates, M.E., Wittmer, J.P., Bouchauad, J.-P. and Claudin, P. (1998) Jamming, force chains and fragile matter. Physics Review Letters, 81, 1841 — 1844.
Chapman, S. and Cowling, T.G. (1990) The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.
Chhabra, R.P and Richardson, J. (1999) Non-Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications. Butterworth-Heinemann, UK, 436 pp.
Cordonnier, B., Hess, K.-U., Lavallée, Y. and Dingwell, D.B. (2008) Rheological properties of dome lavas: Case study of Unzen volcano. Earth and Planetary Science Letters, 279, 263—272.
Costa, A. (2005) Viscosity of high crystal content melts: dependence on solid fraction, Geophysical Research Letters, 32, L22308, doi:10.1029/2005GL024303.
Da Cunha, F.R. and Hinch, E.J. (1996) Shear-induced dispersion in a dilute suspension of rough spheres. Journal of Fluid Mechanics, 309, 211—223.
Davis, M. and Koenders, M. (2007) Oscillated densely packed granular media immersed in a fluid. Journal of Sound and Vibration, 308, 526—540.
Davis, M., Koenders, M.A. and Petford, N. (2007) Vibro-agitation of chambered magma. Journal of Volcanological and Geothermal Research, 167, 24—36.
Davis, M., Koenders, M. and Vahid, S. (2008) Granular temperature model for oscillated slurries: a cell model. Institution of Mechanical Engineers. Proceedings. Part C: Journal of Mechanical Engineering Science, 222, 1995—2006.
Davis, R.H. (1992) Effects of surface roughness on a sphere sedimenting through a dilute suspension of neutrally buoyant spheres. Physics of Fluids, A4, 2607—2619.
Dingwell, D.B. (1995) Relaxation in silicate melts: some applications. Pp. 21—66 in: Structure Dynamics and Properties of Silicate Melts (J. Stebbins, P.F. McMillan and D.B. Dingwell, editors). Reviews in Mineralogy 32, Mineralogical Society of America, Chantilly, Virginia, USA.
Dingwell, D.B. and Webb, S.L. (1990) Relaxation in silicate melts. European Journal of Mineralogy, 2, 427—449.
Dingwell, D., Bagdassarov, N., Bussod, G. and Webb, S.L. (1993) Magma Rheology. Short Course on Experiments at High Pressure and Applications to the Earth's Mantle, pp. 131 — 196, Mineralogical Association of Canada.
Eilers, H. (1943), Die Viskosität-Konzentrations-abhängigkeit kolloider systeme in organischen Lbsungsmitteln. Kolloid-Zeitschrift, 102, 154—169.
Einstein, A. (1906) Eine neue Bestimmung der Molekuldimensionen. Annalen der Physik, 19, 289 (see also, 1911 Annalen der Physik, 34, 591). Also reprinted in Einstein, A. (1956) Investigations of the Theory of Brownian Movement (R. Fiirth, editor; translated by A.D. Cowper). Dover, New York.
Emmons, R.C. (1940) The contribution of differential pressures to magmatic differentiation. American Journal of Science, 238, 1—21.
Fernandez, A.G. and Gasguet D.R. (1994) Relative rheological evolution of chemically contrasted coeval magmas: examples from the Tickka plutonic complex (Morocco). Contributions to Mineralogy and Petrology, 116, 316—326.
Foss, D.R. and Brady, J.F. (2000) Brownian Dynamics simulation of hard sphere colloidal dispersions. Journal of Rheology, 44, 629—651.
Frankel, N.A. and Acrivos, A. (1967) On the viscosity of a concentrated suspension of solid spheres. Chemical Engineering Science, 22, 847—853.
Gidaspow, D. (1994) Multiphase Flow and Fluidization. Academic Press, San Diego, CA, USA.
Giordano, D., Russell, K.K. and Dingwell, D.B. (2008) Viscosity of magmatic liquids: a model. Earth and Planetary Science Letters, 271, 23 — 134
Gonnermann, H.M. and Manga, M. (2003) Explosive volcanism may not be a consequence of magma fragmentation. Nature, 426, 432—435.
Gourlay, C.M. and Dahle, A.K. (2007) Dilatant shear bands in solidifying metals. Nature, 445, 70—73, doi:10.1038/nature05426.
Hale, A.J., Wadge, G. and Mühlhaus, H.-B. (2007) The influence of viscous and latent heating in crystal-rich magma. Geophysical Journal International, 171, 1406—1429.
Hallot, E., Davy, P., Bremond, d’Ars J., Martin, B. and Van Damme, H. (1996) Non—Newtonian effects during injection in partially crystallised magmas. Journal of Volcanology and Geothermal Research, 71, 31—44.
Happel, J. and Brenner, H. (1965) Low Reynolds Number Hydrodynamics. Prentice-Hall, New Jersey, USA.
He, Y.B., Laskowski, J.S. and Klein, B. (2001) Particle movement in non-Newtonian slurries: the effect of yield stress on dense medium separation. Chemical and Engineering Science, 56, 2991—2998.
Hess, K.-U., Cordonnier, B., Lavallée, Y. and Dingwell, D.B. (2008) Viscous heating in rhyolite: an in situ experimental determination. Earth and Planetary Science Letters, 275, 121 — 128.
Heyes, D.M., Kim, J.J., Montrose, C.J. and Litovitz, T.A. (1980) Time dependent nonlinear shear stress effects in simple liquids: A molecular dynamics study. Journal of Chemical Physics, 73, 3987 —3996.
Higgins, M.D. (2000) Measurement of crystal size distributions. American Mineralogist, 85, 1105 — 1116.
Iverson, R.M. (1997) The physics of debris flows. Reviews of Geophysics, 35, 245—296.
Jeffrey, D.J. and Acrivos, A. (1996) The rheological properties of suspensions of rigid particles. AIChE Journal, 22, 417—432.
Jeffrey, D.J. and Onishi, Y. (1984) Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. Journal of Fluid Mechanics, 139, 261—290.
Jenkins, J.T. and Koenders, A.A. (2005) Hydrodynamic interaction of rough spheres. Granular Matter, 7, 13—18.
Jenkins, J.T. and Mancini, F. (1989) Kinetic theory for smooth, nearly elastic spheres. Physics of Fluids, A1, 2050—2057.
Kargel, J.S., Croft, S.K., Lunine, J.I. and Lewis, J.S. (1991) Rheological properties of ammonia-water liquids and crystal-liquid slurries: Planetological applications. Icarus, doi:10.1016/0019- 1035(91)90090-G
Kerr, C.K. and Lister, J.R. (1991) The effects of shape on crystal settling and on the rheology of magmas. Journal of Geology, 99, 457—467
Koenders, M.A. and Petford, N. (2005) Shear-induced pressure changes and seepage phenomena in a deforming porous layer-II. Geophysics Journal International, 163, 385—402.
Koenders, M.A. and Petford, N. (2007) Shear-induced pressure changes and seepage phenomena in a deforming porous layer-III. Geophysics Journal International, 171, 943—953.
Koh, C.J., Hookham, P. and Leal L.G. (1994) An experimental investigation of concentrated suspension flows in a rectangular channel. Journal of Fluid Mechanics, 266, 1—32.
Komar, P.D. (1972) Flow differentiation in igneous dykes and sills: profiles of velocity and phenocryst concentration. Geological Society of America Bulletin, 83, 3443—3448.
Krieger, I.M. and Dougherty, T.J. (1959) A mechanism for non-Newtonian flow in suspensions of rigid spheres. Transactions of the Society of Rheology, 3, 137—152.
Krishnan, G.P., Beimfohr, S. and Leighton, D.T. (1996) Shear-induced radial segregation in bidisperse suspensions. Journal of Fluid Mechanics, 321, 371—393.
Lavallée, Y., Hess, K-U., Cordonnier, B and Dingwell, D.B. (2007) Non-Newtonian rheological law for highly crystalline dome lavas. Geology, 35, 843—846, DOI: 10.1130/G23594A.1.
Lavallée, Y., Meredith, P.G., Dingwell, D.B., Hess, K.- U., Wassermann, J., Cordonnier, B., Gerik, A. and Kruhl, J.H. (2008) Seismogenic lavas and explosive eruption forecasting. Nature, 453, 507—510.
Leat, P.T. (2008) On the long-distance transport of Ferrar magmas. Pp. 45—61 in Structure and Emplacement of High-Level Magmatic Systems (K. Thomson and N. Petford, editors). Special Publication, 302, Geological Society of London.
Leighton, D. and Acrivos, A. (1987) The shear-induced migration of particles in concentrated suspensions. Journal of Fluid Mechanics, 177, 109—131.
Lejeune, A.M. and Richet, P. (1995) Rheology of crystal-bearing silicate melts: An experimental study at high viscosities. Journal ofGeophysical Research, 100, 4215—4229.
Lewis, J.S. (1972) Low temperature condensation from the solar nebula. Icarus, 16, 241.
Manga, M. and Wang, C.Y. (2007) Pressurized oceans and the eruption of liquid water on Europa and Enceladus. Geophysical Research Letters, 34, L0702.
Marsh, D.B. (1981) On the crystallinity, probability of occurrence, and rheology of lava and magma. Contributions to Mineralogy and Petrology, 78, 85—98.
Marsh, D.B. (1996) Solidification fronts and magmatic evolution. Mineralogical Magazine, 60, 5—40.
Marsh, D.B. (2004) A magmatic mush column Rosetta stone: the McMurdo Dry Valleys of Antarcica. EOS, 85, 497—502.
McBirney, A.R. and Murase, T. (1984) Rheological properties of magmas. Annual Reviews ofEarth and Planetary Sciences, 12, 337—357.
D.F, McTigue and J.T., Jenkins (1992) Channel flow of a concentrated suspension. Pp. 381—390 in: Advances in Micromechanics of Granular Materials (H.H. Shen et al., editors). Elsevier Science Publishers BV, Amsterdam.
Mead, W.J. (1925) The geologic role of dilatancy. Journal of Geology, 33, 685—698.
Melnik, O. and Sparks, R.S. (2005) Controls on conduit flow dynamics during lava dome building eruptions. Journal of Geophysical Research, 110, B02209.
Mitchell, K.L., Zhong, F., Hays, C.C., Barmatz, M., Hodyss, R., Castillo, J.C. and Robshaw, L.C. (2008) Preliminary viscometry of methanol-water as a Titan cryomagma analogue. Lunar and Planetary Science, XXXIX, 2131.
Mitri, G., Showman, A.P., Lunine, J.I. and Lopes, R.M. (2008) Resurfacing of Titan by ammonia- water cryomagma. Icarus, 196, 216—224 .
Mock, A. and Jerram, D.A. (2005) Crystal size distributions (CSD) in three dimensions: insights from the 3D reconstruction of a highly porphyritic rhyolite. Journal of Petrology, 46, 1525 — 1541 doi:10.1093/petrology/egi024.
Mooney, M. (1951) The viscosity of a concentrated suspension of spherical particles. Journal of Colloidal Science, 6, 162—170.
M0llera, P.C., Mewisb, J. and Bonn, D. (2006) Yield stress and thixotropy: on the difficulty of measuring yield stresses in practice. Soft Matter, 2, 274—283, DOI: 10.1039/b517840a.
Nguyen, Q.D. and Boger, D.V. (1992) Measuring the flow properties of yield stress fluids. Annual Review of Fluid Mechanics, 24, 47—88.
Nott, P.R. and Brady, J.F. (1994) Pressure driven flow of suspensions: Simulation and theory. Journal ofFluid Mechanics, 275, 157—199.
Okumura, S., Nakamura, M., Tsuchiyama, A., Nakano, T. and Uesugi, K. (2008) Evolution of bubble microstructure in sheared rhyolite: formation of a channel-like bubble network. Journal of Geophysical Research, 113, B07208, DOI:10.1029/ 2007JB005362.
Pal, R. (2003) Rheological behaviour of bubble-bearing magmas. Earth and Planetary Science Letters, 207, 165 — 179.
Patir, N. and Cheng, H.S. (1978) An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. Transactions ASME, 100, 12—17.
Parsons, I. (editor) (1987) Origins of Igneous Layering. Reidel, Boston, USA, 666 pp.
Petford, N. (2003) Rheology of granitic magmas during ascent and emplacement. Annual Reviews of Earth and Planetary Sciences, 31, 339—427.
Petford, N. (2005) Rheology and multiphase flow in congested ammonia-water-ice slurries. Lunar and Planetary Science, XXXV, #1043.
Petford, N. and Koenders, M.A. (1998) Granular flow and viscous fluctuations in low Bagnold number granitic magmas. Journal of the Geological Society, London, 155, 873—881.
Petford, N. and Koenders, M.A. (2003) Shear-induced pressure changes and seepage phenomena in a deforming porous layer — I. Geophysics Journal International, 155, 857—869.
Petford, N. and Marsh, D.B. (2008) Image-based modelling of lateral magma flow: the Basement Sill, Antarctica. LASI, III, 65—66.
A.R., Philpotts (1990) Principles of Igneous and Metamorphic Petrology. Prentice Hall, New Jersey, USA, 498 pp.
Picard, D., Arbaret, L., Pichavant, M., Champallier, R. and Launeau, P. (2008) Experimental investigation at high pressure-high temperature of structures and rheology of crystal-bearing magmas deformed in simple shear. Geophysical Research Abstracts, EUG2008-A-07441.
Pinkerton, H. and Stevenson, R.J. (1992) Methods of determining the rheological properties of magmas at sub liquidus temperatures. Journal of Volcanology and Geothermal Research, 53, 47—66.
Reese, C.Cand Solomatov, V.S. (2006) Fluid dynamics of local martian magma oceans. Icarus, doi:10.1016/ j.icarus.2006.04.008.
Reynolds, O. (1885) On the dilatancy of media composed of rigid particles in contact with experimental illustration. Philosophical Magazine, 20, 469—481.
Rosato, A.D., Blackmore, D.L., Zhang, N. and Lan, Y. (2002) A perspective on vibration-induced size segregation of granular materials. Chemical and Engineering Science, 57, 265—275.
Rust, A. and Manga, M. (2002) Effects of bubble deformation on the viscosity of dilute suspensions. Journal of Non Newtonian Fluid Mechanics, 104, 53—63.
Rutgers, I.R. (1962) Relative viscosity and concentration. Rheologica Acta, 2, 305—348.
Ryerson, F.J., Weed, H.C. and Piwinskii, A.J. (1988) Rheology of subliquidus magmas 1. Picritic compositions. Journal of Geophysical Research, 93, 3421—3436.
Santra, S.B., Schwarzer, S. and Herrmann, H. (1996) Fluid-induced particle-size segregation in sheared granular assemblies. Physics Reviews, E54, 5066—5072.
Scarfe, C.M. and Cronin, D.J. (1986) Viscosity temperature relationships of melts at 1 atm in the system Diopside—Albite. American Mineralogist, 71, 767—771.
Shapley, N.C., Armstrong, R.C., and Brown, R.A. (2002) Laser Doppler Velocimetry measurements of particle velocity fluctuations in a concentrated suspension. Journal of Rheology, 46, 241—272.
Shaw, H.R. (1965) Comments on viscosity, crystal settling and convection in granitic magmas. American Journal of Science, 263, 120—152.
Shaw, H.R. (1972) Viscosities of magmatic silicate liquids: an empirical method of prediction. American Journal ofScience, 272, 870—893.
Sherman, P. (1968) Emulsion Science. Academic Press, New York, 351 pp.
Smith, J.V. (2000) Textural evidence for dilatant (shear thickening) rheology of magma at high crystal concentrations. Journal of Volcanology and Geothermal Research, 99, 1—7.
Spera, F.J. (2000) Physical properties of magmas. Encylopedia of Volcanoes, Academic Press, New York, pp. 171 — 190.
Stickel, J.J. and Powell, R.L. (2005) Fluid mechanics and rheology of dense suspensions. Annual Review of Fluid Mechanics, 37, 129—149. doi:10.1146/ annurev.fluid.36.050802.122132.
Sumita, I. and Manga, M. (2008) Suspension rheology under oscillatory shear and its geophysical implications. Earth and Planetary Science Letters, 269, 467—476
Thomas, D.G. (1965) Transport characteristics of suspensions VII: a note on the viscosity Newtonian suspensions of uniform spherical particles. Journal of Colloid Science, 20, 267—277.
Torquato, S., Truskett, T.M. and Debenedetti, P.G. (2000) Is random close packing of spheres well defined? Physics Review Letters, 84, 2064—2067.
Tuffen, H. and Dingwell, D.B. (2005) Fault textures in volcanic conduits: evidence for seismic trigger mechanisms during silicic eruptions. Bulletin of Volcanology, 67, 370—387.
Van der Molen, I. and Paterson, M. (1979) Experimental deformation of partially molten granite. Contributions to Mineralogy and Petrology, 70, 299—318.
Vigneresse, J.L. (2008) Granite batholiths: from pervasive and continuous melting in the lower crust to discontinuous and spaced plutonism in the upper crust, in plutons and batholiths (N. Petford R.S. Sparks and D.H.W. Hutton, editors). Transactions of the Royal Society Edinburgh, 97, 311—324.
Wadsworth, W.J. (1985) Layered intrusions — a fluid situation? Geology Today, 1, 50—54.
Wager, L.R. and Brown, G.M. (1968) Layered Igneous Rocks. Freeman, San Francisco, 588 pp.
Webb, S.L. and Dingwell, D.B. (1990) Non-Newtonian rheology of igneous melts at high stresses and strain rates: experimental results for rhyolite, andesite, basalt and nephelinite. Journal of Geophysical Research, 95, 15695 — 15701.
Wickham, S.M. (1987) The segregation and emplacement of granitic magmas. Journal of the Geological Society, 144, 281—297.
Wildemuth, C.R. and Williams, M.C. (1984) Viscosity of suspensions modeled with a shear-dependent maximum packing fraction. Rheologica Acta, 23, 627—635.
Whittington, A.G., Hellwig, B.M., Behrens, H., Joachim, B., Stechern, A. and Vetere, F. (2009) The viscosity of hydrous dacitic liquids: implications for the rheology of evolving silicic magmas. Bulletin of Volcanology, 71, 185 — 199.
Zapryanov, Z. and Tabakova, S. (1999) Bubbles, Drops and Particles in Non Newtonian Fluids. Kluwer Academic Publishers, Dordrecht, The Netherlands, 540 pp.
Zeng, S., Kerns, E.T. and Davis, R.H. (1996) The nature of particle contacts in sedimentation. Physics of Fluids, 8, 1389—1396.

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Which effective viscosity?

  • N. Petford (a1)

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