Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T15:17:05.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity

Published online by Cambridge University Press:  21 April 2006

R. W. Griffiths
R. W. Griffiths
Affiliation:
Research School of Earth Sciences, The Australian National University, G.P.O. Box 4, Canberra A.C.T. 2601, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 166 , May 1986 , pp. 115 - 138
Copyright
© 1986 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Brignell, A. S. 1975 Solute extraction from an internally circulating spherical liquid drop. Intl J. Heat Mass Transfer 18, 61–68.Google Scholar
Chu, T. Y. & Goldstein, R. J. 1973 Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141–159.Google Scholar
Elder, J. W. 1968 The unstable thermal interface. J. Fluid Mech. 32, 69–96.Google Scholar
Escudier, M. P. & Maxworthy, T. 1973 On the motion of turbulent thermals. J. Fluid Mech. 61, 541–552.Google Scholar
Fox, D. G. 1972 Numerical simulation of three-dimensional, shape-preserving convective elements. J. Atmos. Sci. 29, 322–341.Google Scholar
Griffiths, R. W. 1986a Particle motions induced by spherical convective elements in Stokes flow. J. Fluid Mech. 166, 139–159.Google Scholar
Griffiths, R. W. 1986b The differing effects of compositional and thermal buoyancies on the evolution of mantle diapirs. Phys. Earth Planet. Inter. (submitted).
Griffiths, R. W. 1986c Dynamics of mantle thermals with constant buoyancy or anomalous internal heating. Earth Planet. Sci. Lett. in press.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics, pp. 298–341. Prentice-Hall.
Howard, L. N. 1964 Convection at high Reynolds numbers. In Proc. 11th Intl Congress Appl. Mech., Munich (ed. H. Gortler), pp. 1109–1115. Springer.
Jarvis, G. T. 1984 Time-dependent convection in the Earth's mantle. Phys. Earth Planet. Inter. 36, 305–327.Google Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.Google Scholar
Kronig, R. & Brink, J. C. 1951 On the theory of extraction from falling droplets. Appl. Sci. Res. A2, 142–151.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.
Loper, D. E. & Stacey, F. D. 1983 The dynamical and thermal structure of deep mantle plumes. Phys. Earth Planet. Inter. 3, 304–317.Google Scholar
Marsh, B. D. 1982 On the mechanics of igneous diapirism, stoping and zone melting. Am. J. Sci. 282, 808–855.Google Scholar
Marsh, B. D. 1984 Mechanics and energetics of magma formation and ascension. In Explosive Volcanism: Inception, Evolution and Hazards, pp. 67–83. Washington: National Academy Press.
Marsh, B. D. & Kantha, L. H. 1978 On the heat and mass transfer from an ascending magma. Earth Planet. Sci. Lett. 39, 435–443.Google Scholar
Morgan, W. J. 1971 Convection plumes in the lower mantle. Nature 230, 42–43.Google Scholar
Morris, S. 1982 The effects of a strongly temperature dependent viscosity on slow flow past a hot sphere. J. Fluid Mech. 124, 1–26.Google Scholar
Morton, B. R. 1960 Weak thermal vortex rings. J. Fluid Mech. 9, 107–118.Google Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Olson, P. & Singer, H. 1985 Creeping plumes. J. Fluid Mech. 158, 511–531.Google Scholar
Pan, F. Y. & Acrivos, A. 1968 Heat transfer at high Peclet number in regions of closed streamlines. Intl J. Heat Mass Transfer 11, 439–444.Google Scholar
Ribe, N. M. 1983 Diapirism in the Earth's mantle: experiments on the motion of a hot sphere in a fluid with temperature dependent viscosity. J. Volcanol. Geothermal Res. 16, 221–245.Google Scholar
Schlien, D. J. & Thompson, D. W. 1975 Some experiments on the motion of an isolated laminar thermal. J. Fluid Mech. 72, 35–47.Google Scholar
Scorer, R. S. 1957 Experiments on convection of isolated masses of buoyant fluid. J. Fluid Mech. 2, 583–594.Google Scholar
Scorer, R. S. 1978 Environmental Aerodynamics. John Wiley.
Sparrow, E. M., Husar, R. B. & Goldstein, R. J. 1970 Observations and other characteristics of thermals. J. Fluid Mech. 41, 793–800.Google Scholar
Tritton, D. J. 1985 Experiments on turbulence in geophysical fluid dynamics. 2. Convection in a very viscous fluid. In Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, pp. 193–199. Bologna: Soc. Italiana di Fisica.
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. Lond. A 239, 61–75.Google Scholar
Turner, J. S. 1964 The dynamics of spheroidal masses of buoyant fluid. J. Fluid Mech. 19, 481–490.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Whitehead, J. A. & Luther, D. S. 1975 Dynamics of laboratory diapir and plume models. J. Geophys. Res. 80, 705–717.Google Scholar
Yuen, D. A. & Schubert, G. 1976 Mantle plumes; a boundary layer approach for Newtonian and non-Newtonian temperature-dependent rheologies. J. Geophys. Res. 81, 2499–2510.Google Scholar