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Instability of the equilibrium of a liquid below its vapour between horizontal heated plates

Published online by Cambridge University Press:  26 April 2006

Adam Huang  and
Daniel D. Joseph
Adam Huang
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St S.E., Minneapolis MN 55455, USA
Daniel D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St S.E., Minneapolis MN 55455, USA
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Abstract

We study the stability of a motionless liquid below its vapour between heated horizontal plates. The temperature of the bottom plate is held below the vaporization temperature and the top plate is hotter than the vaporization temperature. A water film is on the cold plate and a vapour film on the hot plate. We find a basic solution depending only on the variable y normal to the plates, with steady distributions of temperature, a null velocity and no phase change. The linear stability of this basic state is studied in the frame of incompressible fluid dynamics, without convection, but allowing for phase change. An ambiguity in the choice of the conditions to be required of the temperature at the phase-change boundary is identified and discussed. It is shown that the basic state of equilibrium is overstable under conditions of large temperature gradient, when the other parameters have suitable values. An analysis of the energy of the most dangerous disturbance shows that the source of the instability is associated with change of phase.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 235 - 247
Copyright
© 1992 Cambridge University Press

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References

Aoki, K. & Cercignani, C. 1983 Evaporation and condensation on two parallel plates at finite Reynolds numbers. Phys. Fluids 26, 1163.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Blennerhassett, P. J. 1980 On the generation of waves by wind.. Proc. R. Soc. Lond. A 298, 451–494.Google Scholar
Busse, F. H. & Schubert, G. 1971 Convection in a fluid with two phases. J. Fluid Mech. 46, 801–812.Google Scholar
Cercignani, C., Fiszdon, W. & Frezzotti, A. 1985 The paradox of the inverted temperature profiles between an evaporating and a condensing surface. Phys. Fluids 28, 237.Google Scholar
Gebhart, B. 1971 Heat Transfer, 2nd edn. McGraw-Hill.
Hirschfelder, J. V., Curtis, C. F. & Bird, R. B. 1954 Molecular Theory of Gases and Liquids. John Wiley and Sons.
Hu, H. & Joseph, D. D. 1989 Lubricated pipelining: stability of core—annular flow. Part 2. J. Fluid Mech. 205, 359.Google Scholar
Ishii, M. 1975 Thermo-Fluid Dynamic Theory of Two-phase Flow. Eyrolles.
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.
Onishi, Y. 1984 14th Intl Symp. on Rarefied Gas Dynamics, Tsukuba, Japan.
Pao, Y. P. 1971 Application of kinetic theory to the problem of evaporation and condensation. Phys. Fluids 14, 306.Google Scholar
Plesset, M. S. & Zwick, S. A. 1954 The growth of vapor bubbles in superheated liquids. J. Appl. Phys. 25, 493.Google Scholar
Renardy, Y. 1987 Viscosity and density stratification in vertical Poiseuille flow. Phys. Fluids 30, 1658.Google Scholar
Schrage, R. 1953 A Theoretical Study of Interphase Mass Transfer. Columbia University Press.
Schubert, G. & Straus, J. 1980 Gravitational stability of water over steam in vapor-dominated geothermal systems. J. Geophys. Res. 85, 6505.Google Scholar
Shankar, P. N. & Deshpande, M. D. 1990 On the temperature distribution in liquid-vapor phase change between plane liquid surfaces.. Phys. Fluids A 2, 1030.Google Scholar
Sotin, C. & Parmentier, E. M. 1989 On the stability of a fluid layer containing a univariant phase transition: application to planetary interiors. Phys. Earth Planet. Inter. 55, 10–25.Google Scholar
Truskinovsky, L. 1991 Kinks versus shocks. In Shock Induced Transitions and Phase Structures in General Media (ed. R. Fosdick, E. Dunn, M. Slemrod). IMA vol. 36, Springer (to appear).
Unsal, M. & Thomas, W. C. 1978 Linear stability analysis of film condensation. Trans. ASME C: J. Heat Transfer 100, 629.Google Scholar
Unsal, M. & Thomas, W. C. 1980 Nonlinear stability analysis of film condensation. Trans. ASME C: J. Heat Transfer 102, 483.Google Scholar
Yih, C. S. 1955 Stability of parallel laminar flow with a free surface In Proc. Second US Natl Congr. of Applied Mechanics, pp. 623. ASME.
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321.Google Scholar