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Rayleigh–Taylor instability of an inclined buoyant viscous cylinder

Published online by Cambridge University Press:  01 February 2011

JOHN R. LISTER ,
ROSS C. KERR ,
NICK J. RUSSELL  and
ANDREW CROSBY
JOHN R. LISTER*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK
ROSS C. KERR
Affiliation:
Research School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia
NICK J. RUSSELL
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK
ANDREW CROSBY
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK
*
†Email address for correspondence: [email protected]
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Abstract

The Rayleigh–Taylor instability of an inclined buoyant cylinder of one very viscous fluid rising through another is examined through linear stability analysis, numerical simulation and experiment. The stability analysis represents linear eigenmodes of a given axial wavenumber as a Fourier series in the azimuthal direction, allowing the use of separable solutions to the Stokes equations in cylindrical polar coordinates. The most unstable wavenumber k∗ is long-wave if both the inclination angle α and the viscosity ratio λ (internal/external) are small; for this case, k∗ ∝ max{α, (λ ln λ−1)1/2} and thus a small angle in experiments can have a significant effect for λ ≪ 1. As α increases, the maximum growth rate decreases and the upward propagation rate of disturbances increases; all disturbances propagate without growth if the cylinder is sufficiently close to vertical, estimated as α ≳ 70°. Results from the linear stability analysis agree with numerical calculations for λ = 1 and experimental observations. A point-force numerical method is used to calculate the development of instability into a chain of individual plumes via a complex three-dimensional flow. Towed-source experiments show that nonlinear interactions between neighbouring plumes are important for α ≳ 20° and that disturbances can propagate out of the system without significant growth for α ≳ 40°.

JFM classification

Convection: Buoyancy-driven instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 313 - 338
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Ascoli, E. P. & Lagnado, R. R. 1992 The linear stability of a spherical drop migrating in a vertical temperature gradient. Phys. Fluids 4, 225–233.CrossRefGoogle Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.CrossRefGoogle Scholar
de Bremond d'Ars, J., Jaupart, C. & Sparks, R. S. J. 1995 Distribution of volcanoes in active margins. J. Geophys. Res. 100, 20421–20432.CrossRefGoogle Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.CrossRefGoogle Scholar
Ekiel-Jeżewska, M. L., Metzger, B. & Guazzelli, E. 2006 Spherical cloud of point particles falling in a viscous fluid. Phys. Fluids 18, 038104.CrossRefGoogle Scholar
Fedotov, S. A. 1975 Mechanism of magma ascent and deep feeding channels of island arc volcanoes. Bull. Volcanol. 39 (2), 1–14.CrossRefGoogle Scholar
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251–262.CrossRefGoogle Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Hinch, E. J. & Acrivos, A. 1980 Long slender drops in a simple shear flow. J. Fluid Mech. 98, 305–328.CrossRefGoogle Scholar
Huppert, H. E., Sparks, R. S. J., Whitehead, J. A. & Hallworth, M. A. 1986 The replenishment of magma chambers by light inputs. J. Geophys. Res. 91, 6113–6122.CrossRefGoogle Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.CrossRefGoogle Scholar
Kerr, R. C. & Lister, J. R. 1988 Island arc and mid-ocean ridge volcanism, modelled by diapirism from linear source regions. Earth Planet. Sci. Lett. 88, 143–152.CrossRefGoogle Scholar
Kerr, R. C. & Lister, J. R. 2008 Rise and deflection of mantle plume tails. Geochem. Geophys. Geosyst. 9, Q10004.CrossRefGoogle Scholar
Kerr, R. C. & Mériaux, C. 2004 Structure and dynamics of sheared mantle plumes. Geochem. Geophys. Geosyst. 5, Q12009.CrossRefGoogle Scholar
Kerr, R. C., Mériaux, C. & Lister, J. R. 2008 Effect of thermal diffusion on the stability of strongly tilted mantle plume tails. J. Geophys. Res. 113, B09401.Google Scholar
Koh, C. J. & Leal, G. 1989 The stability of drop shapes for translation at zero Reynolds number through a quiescent fluid. Phys. Fluids 1, 1309–1313.CrossRefGoogle 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.CrossRefGoogle Scholar
Lister, J. R. 1987 Long-wavelength instability of a line plume. J. Fluid Mech. 175, 413–428.CrossRefGoogle Scholar
Lister, J. R. 1989 Selective withdrawal from a viscous two-layer system. J. Fluid Mech. 198, 231–254.CrossRefGoogle Scholar
Lister, J. R. & Kerr, R. C. 1989 The effect of geometry on the gravitational instability of a buoyant region of viscous fluid. J. Fluid Mech. 202, 577–594.CrossRefGoogle Scholar
Machu, G., Meile, W., Nitsche, L. & Schaflinger, U. 2001 Coalescence, torus formation and breakup of sedimenting drops: experiments and computer simulations. J. Fluid Mech. 447, 299–336.CrossRefGoogle Scholar
Metzger, B., Nicolas, M. & Guazzelli, E. 2007 Falling clouds of particles in viscous fluids. J. Fluid Mech. 580, 283–301.CrossRefGoogle Scholar
Neuber, H. 1934 Ein neuer Ansatz zur Lösung räumblicher Probleme der Elastizitätstheorie. Z. Angew. Math. Mech. 14, 203–212CrossRefGoogle Scholar
Papkovich, P. F. 1932 Solution générale des équations differentielles fondamentales d'élasticité exprimé par trois fonctions harmoniques. C. R. Acad. Sci., Paris 195, 513–515.Google Scholar
Pignatel, F., Nicolas, M., Guazzelli, E. & Saintillan, D. 2009 Falling jets of particles in viscous fluids. Phys. Fluids 21, 123303.CrossRefGoogle Scholar
Pozrikidis, C. 1990 The instability of a moving viscous drop. J. Fluid Mech. 210, 1–21.CrossRefGoogle Scholar
Proudman, I. & Pearson, J. R. A. 1956 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237–262.CrossRefGoogle Scholar
Ramberg, H. 1968 Instability of layered systems in a field of gravity. Parts 1 and 2. Phys. Earth Planet. Inter. 1, 427–474.CrossRefGoogle Scholar
Richards, M. A. & Griffiths, R. W. 1988 Deflection of plumes by mantle shear flow: experimental results and a simple theory. Geophys. J. 94, 367–376.CrossRefGoogle Scholar
Richards, M. A. & Griffiths, R. W. 1989 Thermal entrainment by deflected mantle plumes. Nature 342, 900–902.CrossRefGoogle Scholar
Schouten, H., Klitgord, K. D. & Whitehead, J. A. 1985 Segmentation of mid-ocean ridges. Nature 317, 225–229.CrossRefGoogle Scholar
Selig, F. 1965 A theoretical prediction of salt dome patterns. Geophysics 30, 633–643.CrossRefGoogle Scholar
Sierou, A. & Lister, J. R. 2003 Self-similar solutions for viscous capillary pinch-off. J. Fluid Mech. 497, 381–403.CrossRefGoogle Scholar
Sigurdsson, H. & Sparks, R. S. J. 1978 Lateral magma flow within rifted Icelandic crust. Nature 274, 126–130.CrossRefGoogle Scholar
Skilbeck, J. N. & Whitehead, J. A. 1978 Formation of discrete islands in linear island chains. Nature 272, 499–501.CrossRefGoogle Scholar
Steinberger, B. 2000 Plumes in a convecting mantle: models and observations for individual hotspots. J. Geophys. Res. 105, 11 127–11 152.CrossRefGoogle Scholar
Steinberger, B. & O'Connell, R. J. 1998 Advection of plumes in mantle flow: implications for hotspot motion, mantle viscosity and plume distribution. Geophys. J. Intl 132, 412–434.CrossRefGoogle Scholar
Taylor, G. I. 1964 Conical free surfaces and fluid interfaces. In Proceedings of the 11th International Congress on Theoretical and Applied Mechanics, Munich (ed. Görtler, H.), pp. 790–796. Springer.Google Scholar
Whitehead, J. A. 1982 Instabilities of fluid conduits in a flowing earth – are plates lubricated by the asthenosphere? Geophys. J. R. Astron. Soc. 70, 415–433.CrossRefGoogle Scholar
Whitehead, J. A. & Luther, D. S. 1975 Dynamics of laboratory diapir and plume models. J. Geophys. Res. 80, 705–717.CrossRefGoogle Scholar
Whittaker, R. J. & Lister, J. R. 2008 a The self-similar rise of a buoyant thermal in Stokes flow. J. Fluid Mech. 606, 295–324.CrossRefGoogle Scholar
Whittaker, R. J. & Lister, J. R. 2008 b Slender-body theory for steady sheared plumes in viscous fluid. J. Fluid Mech. 612, 21–44.CrossRefGoogle Scholar