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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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