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The onset of transient convective instability

Published online by Cambridge University Press:  29 March 2006

D. A. Nield
D. A. Nield
Affiliation:
Department of Mathematics, University of Auckland, New Zealand
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Abstract

The stability of a horizontal fluid layer when the thermal (or concentration) gradient is not uniform is examined by means of linear stability analysis. Both buoyancy and surface-tension effects are considered, and the analogous problem for a porous medium is also treated. Attention is focused on the situation where the critical Rayleigh number (or Marangoni number) is less than that for a linear thermal gradient, and the convection is not (in general) maintained. The case of constant-flux boundary conditions is examined because then a simple application of the Galerkin method gives useful results and general basic temperature profiles are readily treated. Numerical results are obtained for special cases, and some general conclusions about the destabilizing effects, with respect to disturbances of infinitely long wavelength, of various basic temperature profiles are presented. If the basic temperature gradient (considered positive, for a fluid which expands on heating, if the temperature decreases upwards) is nowhere negative, then the profile which leads to the smallest critical Rayleigh (or Marangoni) number is one in which the temperature changes stepwise (at the level at which the velocity, if motion were to occur, would be vertical) but is otherwise uniform. If, as well as being non-negative, the temperature gradient is a monotonic function of the depth, then the most unstable temperature profile is one for which the temperature gradient is a step function of the depth.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 71 , Issue 3 , 14 October 1975 , pp. 441 - 454
Copyright
© 1975 Cambridge University Press

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References

Bannister, T. C., Grodzka, P. G., Spradley, L. W., Bourgeois, S. V., Heddon, R. O. & Facemire, B. R. 1973 Apollo 17 heat flow and convection experiments. Final data analysis results. N.A.S.A. Tech. Memo. X-64772.
Berg, J. C., Acrivos, A. & Boudart, M. 1966 Evaporation convection. Adv. in Chem. Engng, 6, 61–123.Google Scholar
Brindley, J. 1967 Thermal convection in horizontal fluid layers. J. Inst. Math. Appl. 3, 313–343.Google Scholar
Burde, G. I. 1971 Numerical investigation of convection arising with fluctuations of the temperature at horizontal boundaries. Fluid Dyn. 6, 125–131. (English trans. of Izv. Akad. Nauk S.S.S.R. Mekh. Zhid. Gaza, 6, 144–150.)Google Scholar
Chandra, K. 1938 Instability in fluids heated from below. Proc. Roy. Soc. A 164, 231–242.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.Google Scholar
Davis, S. H. 1972 On the stability of flows started from rest. Quart. J. Mech. Appl. Math. 24, 459–465.Google Scholar
Debler, W. R. & Wolf, L. F. 1970 The effects of gravity and surface tension gradients on cellular convection in fluid layers with parabolic temperature profiles. Trans. A.S.M.E., J. Heat Transfer, C 92, 351–358.Google Scholar
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles. Academic.
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids, 8, 1249–1257.Google Scholar
Gershuni, G. Z. & Zhukhovitskii, E. M. 1963 On parametric excitation of convective instability. J. Appl. Math. Mech. 27, 1197–1204. (English trans. of Prikl. Mat. Mekh. 27, 784–793.)Google Scholar
Goldstein, A. W. 1959 Stability of a horizontal fluid layer with unsteady heating from below and time-dependent body force. N.A.S.A. Tech. Rep. R-4.Google Scholar
Graaf, J. G. A. De & Van der held, E. F. M. 1953 The relation between heat transfer and the convection phenomena in enclosed plane air layers. Appl. Sci. Res. A 3, 393–409.Google Scholar
Graham, A. 1933 Shear patterns in an unstable layer of air. Phil. Trans. A 232, 285–296.Google Scholar
Gresho, P. M. & Sani, R. L. 1971 The stability of a fluid layer subjected to a step change in temperature: transient vs. frozen time analyses. Int. J. Heat Mass Transfer, 14, 207–221.Google Scholar
Grodzka, P. G. & Bannister, T. C. 1972 Heat flow and convection demonstration experiments aboard Apollo 14. Science, 176, 506–508.Google Scholar
Homsy, G. M. 1973 Global stability of time-dependent flows: impulsively heated or cooled fluid layers. J. Fluid Mech. 60, 127–129.Google Scholar
Kondo, H., Kimura, R. & Nishimoto, S. 1972 Effects of curved temperature profiles on the Bénard-Rayleigh convective instability. J. Met. Soc. Japan, 50, 65–69.Google Scholar
Koschmieder, E. L. 1974 Bénard convection. Adv. in Chem. Phys. 26, 177–212.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature. Part 1. Theory. J. Fluid Mech. 33, 445–455.Google Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating. J. Fluid Mech. 21, 265–276.Google Scholar
List, E. J. 1965 The convective stability of a confined saturated porous medium containing uniformly distributed heat sources. W.M.Keck Lab. Hydraul. & Water Resources, Div. Engng Appl. Sci., Calif. Inst. Tech., Memo. no. 65.10.Google Scholar
Malurkar, S. L. 1937 Instability in fluid layers when the lower surface is heated. Gerlands Beitr. z. Geophys. 51, 270–277.Google Scholar
Morton, B. R. 1957 On the equilibrium of a stratified layer of fluid. Quart. J. Mech. Appl. Math. 10, 433–447.Google Scholar
Nield, D. A. 1968 Onset of thermohaline convection in a porous medium. Water Resources Res. 4, 553–660.Google Scholar
Ogura, Y. & Kondo, H. 1970 A linear stability of convective motion in a thermally unstable layer below a stable region. J. Met. Soc. Japan, 48, 204–216.Google Scholar
Roberts, P. H. 1967 Convection in horizontal layers with internal heat generation Theory. J. Fluid Mech. 30, 33–49.Google Scholar
Robinson, J. L. 1967 A note on the stability of an infinite fluid heated from below. J. Fluid Mech. 29, 461–464.Google Scholar
Rogers, F. T. 1953 Convection currents in porous media. V. Variational form of the theory. J. Appl. Phys. 24, 877–880.Google Scholar
Rogers, F. T. & Morrison, H. L. 1950 Convection currents in porous media. III. Extended theory of the critical gradient. J. Appl. Phys. 21, 1177–1180.Google Scholar
Rogers, F. T., Schilberg, L. E. & Morrison, H. L. 1951 Convection currents in porous media. IV. Remarks on the theory. J. Appl. Phys. 22, 1476–1479.Google Scholar
Rosenblat, S. & Herbert, D. M. 1970 Low frequency modulation of thermal instability. J. Fluid Mech. 43, 385–389.Google Scholar
Rosenblat, S. & Tanaka, G. A. 1971 Modulation of thermal convection instability. Phys. Fluids, 14, 1319–1322.Google Scholar
Sasaki, Y. 1970 Influences of thermal boundary layer on atmospheric cellular convection. J. Met. Soc. Japan, 48, 492–502.Google Scholar
Schechter, R. S. & Velarde, M. 1974 The two-component Bénard problem. Adv. in Chem. Phys. 26, 265–301.Google Scholar
Segel, L. A. 1966 Nonlinear hydrodynamic stability theory and its application to thermal convection and curved flows. In Non-equilibrium Thermodynamics, Variational Techniques, and Stability (ed. R. J. Donnelly, R. Herman & I. Prigogine). pp. 165–197. University of Chicago Press.
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18, 513–528.CrossRefGoogle Scholar
Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352.Google Scholar
Spiegel, E. A. 1972 Convection in stars. II. Special effects. Ann. Rev. Astron. Astrophys. 10, 261–303.Google Scholar
Sutton, O. G. 1950 On the stability of a fluid heated from below. Proc. Roy. Soc. A 204, 297–309.Google Scholar
Takaki, R. & Yoshizawa, A. 1972 Thermal convective instability of fluid heated suddenly at the lower boundary. J. Phys. Soc. Japan, 32, 1394–1397.Google Scholar
Venezian, G. 1969 Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35, 243–254.Google Scholar
Vidal, A. & Acrivos, A. 1968 Effect of nonlinear temperature profiles on the onset of convection driven by surface tension gradients. Indust. Engng Chem. Fund. 7, 53–56.Google Scholar
Yih, C.-S. & Li, C.-H. 1972 Instability of unsteady flows or configurations. Part 2. Convective instability. J. Fluid Mech. 54, 143–152.Google Scholar