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Selective withdrawal from a viscous two-layer system

Published online by Cambridge University Press:  21 April 2006

John R. Lister
John R. Lister
Affiliation:
Research School of Earth Sciences, Australian National University, PO Box 4, Canberra 2601, A.C.T., Australia
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Abstract

When fluid is withdrawn from a body of stratified fluid the surfaces of constant density are deformed towards the region of withdrawal. The equations describing the flow caused by withdrawal through a point sink in a two-layer unbounded system in which viscous forces dominate are formulated using the boundary-integral representation of Stokes flow. It is shown by dimensional and analytic arguments that surface tension between the layers is a necessary condition for the stability of an interfacial equilibrium in which only one fluid is withdrawn. The critical flow rate above which both fluids are withdrawn is determined numerically as a function of the capillary number. When the flow is supercritical a small adaptation of the numerical scheme allows the proportion of fluid withdrawn from each layer to be found. The various analyses and conclusions further our understanding of the physical processes that determine the compositional output of volcanic eruptions that tap an underlying stratified reservoir of magma.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 198 , January 1989 , pp. 231 - 254
Copyright
1989 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A., 1965 Handbook of Mathematical Functions. Dover.
Bear, J. & Dagan, G., 1964 Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69, 1563–1572.Google Scholar
Blake, J. R. & Chwang, A. T., 1974 Fundamental singularities of viscous flow. Part 1: the image systems in the vicinity of a stationary no-slip boundary. J. Engng Maths 8, 23–29.Google Scholar
Blake, S. & Ivey, G. N., 1986 Magma-mixing and the dynamics of withdrawal from stratified reservoirs. J. Volcanol. Geoth. Res. 27, 153–178.Google Scholar
Craya, A.: 1949 Recherches théoriques sur l'écoulement de couches superposées de fluides de densités différentes. Huille Blanche 4, 44–55.Google Scholar
Freundt, A. & Tait, S. R., 1986 The entrainment of high-viscosity magma into low-viscosity magma in eruption conduits. Bull. Volcanol. 48, 325–339.Google Scholar
Geller, A. S., Lee, S. H. & Leal, L. G., 1986 The creeping motion of a spherical particle normal to a deformable interface. J. Fluid Mech. 169, 27–69.Google Scholar
Hickox, C. E.: 1971 Stability of two fluids in a pipe. Phys. Fluids 14, 251–262.Google Scholar
Hildreth, W.: 1981 Gradients in silicic magma chambers: Implications for lithospheric magmatism. J. Geophys. Res. 86, 10153–10192.Google Scholar
Huppert, H. E. & Sparks, R. S. J.1984 Double-diffusive convection due to crystallization in magmas. Ann. Rev. Earth Planet. Sci. 12, 11–37.Google Scholar
Imbergek, J.: 1980 Selective withdrawal: a review. In 2nd Int. Symp. on Stratified Flows, Trondheim, pp. 381–400. Tapir.
Jansons, K. M. & Lister, J. R., 1988 The general solution of Stokes flow in a half-space as an integral of the velocity on the boundary. Phys. Fluids (in press).Google Scholar
Joseph, D. D., Renardy, M. & Renardy, Y., 1984 Instability of the flow of two immiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141, 309–317.Google Scholar
Koyaguchi, T.: 1985 Magma-mixing in a conduit. J. Volcanol. Geoth. Res. 25, 365–369.Google Scholar
Ladyzhenskaya, O. A.: 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
Lee, S. H., Chadwick, R. S. & Leal, L. G., 1979 The motion of a sphere in the presence of a plane interface. J. Fluid Mech. 93, 705–726.Google Scholar
Lee, S. H. & Leal, L. G., 1982 The motion of a sphere in the presence of a deformable interface. II. A numerical study of the translation of a sphere normal to an interface. J. Colloid. Interface Sci. 87, 81–106.Google Scholar
Lorentz, H. A.: 1907 A general theory concerning the motion of a viscous fluid. Abhandl. Theoret. Phys. 1, 23.Google Scholar
Mikami, T., Cox, R. G. & Mason, S. G., 1975 Breakup of extending liquid threads. Int. J. Multiphase Flow 2, 113–138.Google Scholar
Rallison, J. M.: 1984 The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45–66.Google Scholar
Rallison, J. M. & Acrivos, A., 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
Sparks, R. S. J., Huppert, H. E. & Turner, J. S., 1984 The fluid dynamics of evolving magma chambers. Phil. Trans. R. Soc. Lond. A 310, 511–634.Google Scholar
Spera, F. J.: 1984 Some numerical experiments on the withdrawal of magma from crustal reservoirs. J. Geophys. Res. 89, 8222–8236.Google Scholar
Spera, F. J., Yuen, D. A., Greer, J. C. & Sewell, G., 1986 Dynamics of magma withdrawal from stratified magma chambers. Geology 14, 723–726.Google Scholar
Taylor, G. I.: 1964 Conical free surfaces and fluid interfaces. In Proc. 11th Intl Congr. Appl. Mech., Munich, pp. 790–796. Springer.
Tomotika, S.: 1936 Breaking up of a drop of viscous fluid immersed in another viscous fluid which is extending at a uniform rate. Proc. R. Soc. Lond. A 153, 302–318.Google Scholar
Tuck, E. O. & Vanden-Broeck, J. M.1984 A cusp-like free-surface flow due to a submerged source or sink. J. Austral. Math. Soc. B B25, 443–450.Google Scholar
Youngren, G. A. & Acrivos, A., 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377–403.Google Scholar