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A stability analysis of non-time-periodic perturbations of buoyancy-induced flows in pure water near 4 °C

Published online by Cambridge University Press:  21 April 2006

I. M. El-Henawy ,
B. D. Hassard  and
N. D. Kazarinoff
I. M. El-Henawy
Affiliation:
Department of Mathematics, Mansoora, University, Mansoora, Egypt
B. D. Hassard
Affiliation:
Department of Mathematics, State University of New York at Buffalo, N.Y. 14214 U.S.A.
N. D. Kazarinoff
Affiliation:
Department of Mathematics, State University of New York at Buffalo, N.Y. 14214 U.S.A.
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Abstract

A new approach to determine stability of multiple steady-state similarity solutions corresponding to laminar flows is introduced and applied to laminar flows in cold, pure water at temperature T∞ °C (near 4 °C) adjacent to a vertical, isothermal, plane surface at temperature T0 °C when 0 < R ≡ (4−T∞)/(T0−T∞) < 0.5, the region of buoyancy-force reversals. The results show that the steady-state similarity solutions recently found in this region by El-Henawy et al. (1982) are unstable, and thus should not be observed experimentally; while those solutions found earlier by Carey, Gebhart & Mollendorf (1980) may be stable. No unstable modes corresponding to their solutions were found. Some flows for R in the range of strong buoyancy-force reversals, 0.14 < R < 0.32 at Prandtl number Pr = 11.6, have been observed, for example at R = 0.143,0.254 and 0.317 by Carey & Gebhart (1981) and Wilson & Vyas (1979). The latter found time-varying flows in this region of strongest flow reversals.

The advantages of the method introduced are reduction of mathematical shortcomings of the traditional approach and relative ease of numerical calculation of the real eigenvalues and eigenfunctions. The disadvantage is that information on downstream, selective frequency, exponential growth of amplitude is lost. The theory presented may be regarded as an asymptotic limit of the standard hydrodynamic theory as the frequency of perturbations approaches zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 163 , February 1986 , pp. 1 - 20
Copyright
© 1986 Cambridge University Press

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