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The effect of geometry on the gravitational instability of a buoyant region of viscous fluid

Published online by Cambridge University Press:  26 April 2006

John R. Lister  and
Ross C. Kerr
John R. Lister
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Research School of Earth Sciences, The Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
Ross C. Kerr
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Research School of Earth Sciences, The Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
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Abstract

The low-Reynolds-number stability of a region of buoyant fluid surrounded by denser fluid is analysed in two situations. In the first study, the buoyant fluid lies in a thin layer sandwiched between two denser and much deeper layers. The growth rate and wavelength of the most unstable sinusoidal perturbation are calculated and the effects of the viscosity ratios and density differences between the fluids are investigated. It is found that if the buoyant fluid is much less viscous than the overlying fluid then, in quite general circumstances, both the most unstable wavelength and the corresponding growth rate are inversely proportional to the cube root of the viscosity of the buoyant fluid. A physical explanation of this result is given by scaling analysis of the total dissipation. In the second study, the buoyant fluid takes the form of a cylinder rising through a uniform environment. The eigenmodes of small perturbation about this state of motion are found for each axial wavenumber in terms of Fourier series of separable solutions to the Stokes equations. In contrast to the first study, it is found that the most unstable wavelength and growth rate are asymptotically independent of the viscosity of the buoyant fluid when this viscosity is small.

The difference between the results of the two studies is of importance, particularly for geophysical applications in which viscosity ratios are very large. Previous models of linear regions of volcanism at mid-ocean ridges and at island arcs have assumed that results obtained in simple two-layered systems can be generalized to other geometries. The conclusions of these models are discussed in the light of the stability results for a cylindrical (and hence linea.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 202 , May 1989 , pp. 577 - 594
Copyright
© 1989 Cambridge University Press

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