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Buoyant instability of a viscous film over a passive fluid

Published online by Cambridge University Press:  26 April 2006

D. Canright  and
S. Morris
D. Canright
Affiliation:
Mathematics, Code MA/Ca, Naval Postgraduate School, Monterey, CA 93943, USA
S. Morris
Affiliation:
Mechanical Engineering, University of California, Berkeley, CA 94720, USA
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Abstract

In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d1) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8μ/Δρgd1. In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 255 , October 1993 , pp. 349 - 372
Copyright
© 1993 Cambridge University Press

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References

Amaranath, T. & Rajappa, N. R. 1976 A study of Taylor instability of superposed fluids. Acta Mech. 24, 87–97.Google Scholar
Aref, H. & Tryggvason, G. 1989 Model of Rayleigh–Taylor instability. Phys. Rev. Lett. 62, 749–752.Google Scholar
Baker, G. R. & Meiron, D. I. 1984 Boundary integral methods for axisymmetric and three-dimensional Rayleigh–Taylor instability problems. Physica 12D, 19–31.Google Scholar
Baker, L. & Freeman, J. R. 1981 Heuristic model of the nonlinear Rayleigh–Taylor instability. J. Appl. Phys. 52, 655–663.Google Scholar
Bellman, R. & Pennington, R. H. 1954 Effects of surface tension and viscosity on Taylor instability. Q. Appl. Maths 12, 151–162.Google Scholar
Canright, D. R. 1987 A finite-amplitude analysis of the buoyant instability of a highly viscous film over a less viscous half-space. PhD thesis, University of California, Berkeley.
Canright, D. R. & Morris, S. 1993 Long-wave buoyant instability of a thermo-viscous layer beneath a suddenly cooled plane. (In preparation.)
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Instability. Oxford University Press.
Davies, G. F. 1990 Heat and mass transport in the early earth. In Origin of the Earth (ed. H. E. Newsom & J. H. Jones), pp. 175–194. Oxford University Press.
Dienes, J. K. 1978 Method of generalized coordinates and an application to Rayleigh–Taylor instability. Phys. Fluids 21, 736–744.Google Scholar
Drazin, P. G. 1969 Non-linear internal gravity waves in a slightly stratified atmosphere. J. Fluid Mech. 36, 433–446.Google Scholar
Dussan, V., E. B. 1975 Hydrodynamic stability and instability of fluid systems with interfaces. Arch. Rat. Mech. Anal. 57, 363–379.Google Scholar
Emmons, H. W., Chang, C. T. & Watson, B. C. 1960 Taylor instability of finite surface waves. J. Fluid Mech. 7, 177–193.Google Scholar
Harlow, F. H. & Welch, J. E. 1966 Numerical study of large-amplitude free-surface motions. Phys. Fluids 9, 842–851.Google Scholar
Kull, H. J. 1986 Nonlinear free-surface Rayleigh–Taylor instability. Phys. Rev. A 33, 1957–1967.Google Scholar
Kull, H. J. 1991 Theory of the Rayleigh–Taylor instability. Phys. Rep. 206, 197–325.Google Scholar
Lewis, D. J. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, II. Proc. R. Soc. Lond. A 202, 81–96.Google Scholar
Menikoff, R., Mjolsness, R. C., Sharp, D. H. & Zemach, C. 1978 Initial value problem for Rayleigh–Taylor instability of viscous fluids. Phys. Fluids 21, 1674–1687.Google Scholar
Nayfeh, A. H. 1969 On the non-linear Lamb–Taylor instability. J. Fluid Mech. 38, 619–631.Google Scholar
Newhouse, L. A. & Pozrikidis, C. 1990 The Rayleigh–Taylor instability of a viscous liquid layer resting on a plane wall. J. Fluid Mech. 217, 615–638.Google Scholar
Ott, E. 1972 Nonlinear evolution of the Rayleigh–Taylor instability of a thin layer. Phys. Rev. Lett. 29, 1429–1432.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–197.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica 12 D, 3–18.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Tryggvason, G. 1988 Numerical simulations of the Rayleigh–Taylor instability. J. Comput. Phys. 75, 253–282.Google Scholar
Walcott, R. I. 1973 Structure of the earth from glacio-isostatic rebound. Ann. Rev. Earth Planet. Sci. 1, 15–37.Google Scholar
Whitehead, J. A. & Luther, D. S. 1975 Dynamics of laboratory diapir and plume models. J. Geophys. Res. 80, 705–717.Google Scholar
Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561–570.Google Scholar