Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-16T09:35:15.486Z Has data issue: false hasContentIssue false

Solidification of pressure-driven flow in a finite rigid channel with application to volcanic eruptions

Published online by Cambridge University Press:  26 April 2006

John R. Lister
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Paul J. Dellar
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

Abstract

Competition between conductive cooling and advective heating occurs whenever hot fluid invades a cold environment. Here the solidification of hot viscous flow driven by a fixed pressure drop through an initially planar or cylindrical channel embedded in a cold rigid solid is analysed. At early times, or far from the channel entrance, the flow starts to solidify and block the channel, thus reducing the flow rate. Close to the channel entrance, and at later times, the supply of new hot fluid starts to melt back the initial chill. Eventually, either solidification or meltback becomes dominant throughout the channel, and flow either ceases or continues until the source is exhausted. The evolution of the dimensionless system, which is characterized by the initial Péclet number Pe, the Stefan number S and the dimensionless solidification temperature Θ, is calculated numerically and by a variety of asymptotic schemes. The results show the importance of variations along the channel and caution against models based on a single ‘representative’ width. The critical Péclet number Pec, which marks the boundary between eventual solidification and eventual meltback, is determined for a wide range of parameters and found to be much larger for cylindrical channels than for planar channels, owing to the slower rate of decay of the heat flux into the solid in a cylindrical geometry. For a planar channel Pec is given by the simple algebraic result Pec ∼ 0.46[Θ2/(1 - Θ)(S + 2/π)]3 when (1 - Θ)−1 [Gt ] S [Gt ] 1, but in general it requires numerical solution. Similar analyses, in which there is a spatially varying and time-dependent interaction between the rates of solidification and flow, have a range of applications to geological and industrial processes.

Type
Research Article
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bruce, P. M. 1989 Thermal convection within the Earth's crust. PhD thesis, University of Cambridge.
Bruce, P. M. & Huppert, H. E. 1989 Thermal control of basaltic fissure eruptions. Nature 342, 665667 (referred to herein as BH).Google Scholar
Bruce, P. M. & Huppert, H. E. 1990 Solidification and melting in dykes by the laminar flow of basaltic magma. In Magma Transport and Storage (ed. M.P. Ryan). Wiley (referred to herein as BH).
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.
Crank, J. 1984 Free and Moving Boundary Problems. Oxford University Press.
Delaney, P. T. & Pollard, D. D. 1982 Solidification of basaltic magma during flow in a dike. Am. J. Sci. 282, 856885 (referred to herein as DP).Google Scholar
Fedotov, S. A. 1978 Ascent of basic magmas in the crust and the mechanism of basaltic fissure eruptions. Intl Geol. Rev. 20, 3348.Google Scholar
Graetz, L. 1885 Über die Wärmeleltungsfähigkeit von Flüssigkeiten. Ann. Phys. Chem. 25, 337357.Google Scholar
Helfrich, K. R. 1995 Thermo-viscous fingering of flow in a thin gap: application to magma emplacement. J. Fluid Mech. 305, 219238.Google Scholar
Leveque, M. A. 1928 Les lois de la transmission de chaleur par convection. Ann. Mines Mem. 13, 201299.Google Scholar
Lister, J. R. 1994a The solidification of buoyancy-driven flow in a flexible-walled channel. Part 1. Constant-volume release. J. Fluid Mech. 272, 2144.Google Scholar
Lister, J. R. 1994b The solidification of buoyancy-driven flow in a flexible-walled channel. Part 2. Continual release. J. Fluid Mech. 272, 4565.Google Scholar
Lister, J. R. 1995 Fluid-mechanical models of the interaction between solidification and flow in dykes. In Physics and Chemistry of Dykes, (ed. G. Baer & A. Heimann), pp. 115124. Balkema.
Morris, S. J. S. 1996 Stability of thermoviscous Hele-Shaw flow. J. Fluid Mech. 308, 111128.Google Scholar
Newmann, J. 1969 Extension of the Leveque solution. J. Heat Transfer 91, 177178.Google Scholar
Ockendon, H. & Ockendon, J. R. 1977 Variable-viscosity flows in heated and cooled channels. J. Fluid Mech. 83, 177190.Google Scholar
Petford, N., Lister, J. R. & Kerr, R. C. 1994 The ascent of felsic magmas in dykes. Lithos 32, 161168.Google Scholar
Richardson, S. M. 1983 Injection moulding of thermoplastics: Freezing during mould filling. Rheol. Acta 22, 223236.Google Scholar
Richardson, S. M. 1986 Injection moulding of thermoplastics: Freezing of variable viscosity fluids. II. Developing flows with very low heat generation. Rheol. Acta 25, 308318.Google Scholar
Turcotte, D. L. & Schubert, G. 1982 Geodynamics. John Wiley
Wilson, L. & Head, J. W. 1981 Ascent and eruption of basaltic magma on the Earth and Moon. J. Geophys. Res. 86. 29713001.Google Scholar
Wylie, J. J. & Lister, J. R. 1995 The effect of temperature-dependent viscosity on flow in a cooled channel with application to basaltic fissure eruptions. J. Fluid Mech. 305, 239261.Google Scholar