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Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile

Published online by Cambridge University Press:  28 March 2006

E. M. Sparrow ,
R. J. Goldstein  and
V. K. Jonsson
E. M. Sparrow
Affiliation:
Heat Transfer Laboratory, Department of Mechanical Engineering, University of Minnesota, Minneapolis 14, Minnesota
R. J. Goldstein
Affiliation:
Heat Transfer Laboratory, Department of Mechanical Engineering, University of Minnesota, Minneapolis 14, Minnesota
V. K. Jonsson
Affiliation:
Heat Transfer Laboratory, Department of Mechanical Engineering, University of Minnesota, Minneapolis 14, Minnesota
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Abstract

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 18 , Issue 4 , April 1964 , pp. 513 - 528
Copyright
© 1964 Cambridge University Press

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References

Bénard, H. 1900 Tourbillions cellulaires dan une nappe liquids. Rev. gén. Sci. pur. appl. 11, 1261–71, 1309–28.Google Scholar
Block, M. J. 1956 Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature, Lond., 178, 650–651.Google Scholar
Goldstein, A. W. 1959 Stability of a horizontal fluid layer with unsteady heating from below and time dependent body forces. NASA Tech. Rep. no. R-4.Google Scholar
Jeffreys, H. 1926 The stability of a layer of fluid heated below. Phil. Mag., 2, 833–44.Google Scholar
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. Roy. Soc. A, 118, 195–208.Google Scholar
Lin, C. C. 1958 The Theory of Hydrodynamic Stability, pp. 106–9. Cambridge University Press.
Low, A. R. 1929 On the criterion for stability of a layer of viscous fluid heated from below. Proc. Roy. Soc. A, 125, 180–95.Google Scholar
Ostrach, S. 1957 Convection phenomena in fluids heated from below. Trans. A.S.M.E., 79, 299–305.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech., 4, 489–500.Google Scholar
Pellew, A. R. & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. Roy. Soc. A, 176, 312–43.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529–46.Google Scholar
Sani, R. 1963 Convective instability. Ph.D. Thesis in Chemical Engineering, University of Minnesota.
Thomson, J. 1982 On a changing tessellated structure in certain liquids. Proc. Glasgow Phil. Soc. 13, 469.Google Scholar