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Bi-directional flows in constrained systems

Published online by Cambridge University Press:  26 April 2007

HERBERT E. HUPPERT  and
MARK A. HALLWORTH
HERBERT E. HUPPERT
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: heh1@esc.cam.ac.uk; hallwort@esc.cam.ac.uk
MARK A. HALLWORTH
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: heh1@esc.cam.ac.uk; hallwort@esc.cam.ac.uk
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Abstract

We consider the exchange flow of relatively dense, viscous fluid in a container connected by a vertical pipe to a container beneath it, initially full of relatively light fluid. A non-dimensional value for the flux of dense fluid down the tube is determined experimentally as a function of the ratio of the two viscosities and the Reynolds number. The experimental data are satisfactorily collapsed using dimensional analysis and balancing buoyancy, inertial and viscous forces as appropriate. A theoretical analysis, assuming steady, axisymmetric motion, captures a considerable part, but not all of the processes involved. The paper discusses quantitative applications of the results to the movement of magma in volcanic conduits. The concepts indicate how bi-directional convection in the conduit between a lava lake and a magma reservoir deep in the crust is the essential ingredient in the explanation of the long-standing problem that the amount of degassing of sulphur dioxide from a lava lake in a volcanic crater can exceed by many orders of magnitude that consistent with the amount of lava solidified in the crater. Movies are available with the online version of the paper.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 95 - 112
Copyright
Copyright © Cambridge University Press 2007

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References

REFRENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Clanet, C. 2000 From Galilei to Torricelli. Phys. Fluids 12, 2743–2751.CrossRefGoogle Scholar
Clanet, C. & Searby, G. 2004 On the glug-glug of ideal bottles. J. Fluid Mech. 510, 145–168.CrossRefGoogle Scholar
Francis, P., Oppenheimer, C. & Stevenson, D. 1993 Endogenous growth of persistently active volcanoes. Nature 366, 554–557.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.Google Scholar
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251–262.CrossRefGoogle Scholar
Jaupart, C. 2000 Magma ascent at shallow levels. In Encylopedia of Volcanoes (ed. H. Sigurdson). Academic.Google Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Fluid Mech. 29, 65–90.CrossRefGoogle Scholar
Kazahaya, K., Shinohara, H. & Saito, G. 1994 Excessive degassing of Izu–Oshima volcano: Magma convection in a conduit. Bull. Volc. 56, 2078–216.Google Scholar
Oppenheimer, C., McGonigie, A. J. S. Allard, P. Wooster, M. J. & Tsanev, V. 2004 Sulphur, heat, and magma budget of Erta 'Ale lava lake, Ethiopia. Geol. Soc. Am. 32, 509–512.Google Scholar
Scoffoni, J., Lajeunesse, E. & Homsy, G. M. 2001 Interface instabilities during displacements of two miscible fluids in a vertical pipe. Phys. Fluids 13, 553–556.CrossRefGoogle Scholar
Sparks, R. S. J. 2003 Dynamics of magma degassing. In Volcanic Degassing (ed. Oppenheimer, C., Pyle, D. M. & Barclay, J.). Geol. Soc. London Spec. Pub. 213, pp. 5–22.Google Scholar
Stevenson, D. S. & Blake, S. 1998 Modelling the dynamics and thermodynamics of volcanic degassing. Bull. Volc. 60, 307–317.CrossRefGoogle Scholar
Wallace, P. J. 2001 Volcanic SO2 emissions and the abundance and distribution of exsolved gas in magma bodies. J. Volcanol. Geotherm. Res. 108, 85–106.CrossRefGoogle Scholar

Huppert and Hallworth supplementary movie

Movie 1. Water draining from a sealed cylinder of height 16.9cm and radius 7.9cm, through a tube of internal radius 0.55cm and length 100cm (not all in view). The emptying of the chamber passes through three distinct regimes, characterised by their accoustic signature as popping, glugging and slugging.

Download Huppert and Hallworth supplementary movie(Video)
Video 6.2 MB

Huppert and Hallworth supplementary movie

Movie 1. Water draining from a sealed cylinder of height 16.9cm and radius 7.9cm, through a tube of internal radius 0.55cm and length 100cm (not all in view). The emptying of the chamber passes through three distinct regimes, characterised by their accoustic signature as popping, glugging and slugging.

Download Huppert and Hallworth supplementary movie(Video)
Video 1.6 MB
Supplementary material: PDF

Huppert and Hallworth Appendix

Appendix.pdf

Download Huppert and Hallworth Appendix(PDF)
PDF 66.9 KB

Huppert and Hallworth supplementary movie

Movie 2. Bi-directional flow between an upper reservoir containing pure glycerine and a lower reservoir containg 65 wt.% glycerine, connected by a tube of internal radius 0.55cm and length 20cm, (gamma = 83, Re = 0.45).

Download Huppert and Hallworth supplementary movie(Video)
Video 7.5 MB

Huppert and Hallworth supplementary movie

Movie 2. Bi-directional flow between an upper reservoir containing pure glycerine and a lower reservoir containg 65 wt.% glycerine, connected by a tube of internal radius 0.55cm and length 20cm, (gamma = 83, Re = 0.45).

Download Huppert and Hallworth supplementary movie(Video)
Video 6.9 MB

Huppert and Hallworth supplementary movie

Movie 3. Bi-directional flow between an upper reservoir containing pure water and a lower reservoir containing corn oil, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 0.015, Re = 1700).

Download Huppert and Hallworth supplementary movie(Video)
Video 848.6 KB

Huppert and Hallworth supplementary movie

Movie 3. Bi-directional flow between an upper reservoir containing pure water and a lower reservoir containing corn oil, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 0.015, Re = 1700).

Download Huppert and Hallworth supplementary movie(Video)
Video 724.4 KB

Huppert and Hallworth supplementary movie

Movie 4. Bi-directional flow between an upper reservoir containing pure glycerine and a lower reservoir containing pure water, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 1000, Re = 3.7).

Download Huppert and Hallworth supplementary movie(Video)
Video 4.1 MB

Huppert and Hallworth supplementary movie

Movie 4. Bi-directional flow between an upper reservoir containing pure glycerine and a lower reservoir containing pure water, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 1000, Re = 3.7).

Download Huppert and Hallworth supplementary movie(Video)
Video 3.3 MB

Huppert and Hallworth supplementary movie

Movie 5. Bi-directional flow between an upper reservoir containing 13.7 wt.% salt water (density 1.10 g cm-3) and a lower reservoir containing pure water, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 1.3, Re = 1500).

Download Huppert and Hallworth supplementary movie(Video)
Video 3.7 MB

Huppert and Hallworth supplementary movie

Movie 5. Bi-directional flow between an upper reservoir containing 13.7 wt.% salt water (density 1.10 g cm-3) and a lower reservoir containing pure water, connected by a tube of internal radius 1.5cm and length 20cm, (gamma = 1.3, Re = 1500).

Download Huppert and Hallworth supplementary movie(Video)
Video 3.2 MB