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On the stability of vertical double-diffusive interfaces. Part 1. A single plane interface

Published online by Cambridge University Press:  26 April 2006

I. A. Eltayeb  and
D. E. Loper
I. A. Eltayeb
Affiliation:
Department of Mathematics and Computing, Sultan Qaboos University, Muscat, Sultanate of Oman
D. E. Loper
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
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Abstract

This is the first part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. Flow is forced by a prescribed jump of composition across the interfaces. Compositional diffusivity is ignored, while thermal diffusivity and viscosity are finite. In this first part, basic-state solutions are presented and discussed for three configurations: a single plane interface, two parallel interfaces and a circular cylindrical interface.

The stability of a single plane interface is then analysed. It is shown that the presence of the compositional jump gives rise to a new type of three-dimensional instability which occurs for any non-zero forcing. This is in contrast to the thermally driven flow adjacent to a rigid wall, which is unstable only for a finite value of the forcing and results in the growth of a two-dimensional perturbation. The timescale for growth of the new instability is given by \[ \tau = \left[\frac{\nu \rho}{(\Delta \rho)_c}\right]^2\frac{\alpha}{\kappa g}\frac{{\rm d}T}{{\rm d}z}, \] where (Δρ)C is the prescribed jump in composition and dT/dz is the imposed temperature gradient.

The influence of thermal diffusion is to enhance instability, while viscosity is stabilizing for nearly all wavenumbers. The interface is unstable for all finite wavenumbers if the Prandtl number is less than 1.472, while regions of stability in the wavenumber plane develop for small horizontal wavenumber and moderate vertical wavenumber for larger values of the Prandtl number. The neutral stability curves are investigated and the maximum growth rate of instability is identified for the whole range of values of the Prandtl number and its properties are elucidated by comparison with previous studies of flows near heated vertical walls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 149 - 181
Copyright
© 1991 Cambridge University Press

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