Summary
The stability problem for penetrative convection in a fluid layer that is heated internally is analyzed using the methods of linear instability theory and unconditional nonlinear energy theory. Critical Rayleigh numbers in the case of a constant heat source are determined numerically from both theories. Although it is not known whether exchange of stabilities holds for this problem, a comparison between the linear and nonlinear results suggests that for top temperatures close to 4°C stationary convection is predominant when the heat source is not too large. The nonlinear results delimit a band of Rayleigh numbers where possible subcritical instabilities could arise.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheney, W., Kincaid, D.: Numerical mathematics and computing. Monterey, California: Brooks/Cole 1985.
Davis, S. H.: On the principle of exchange of stabilities. Proc. Roy. Soc. LondonA 310, 341–358 (1969).
Drazin, P. G., Reid, W. H.: Hydrodynamic stability. Cambridge: University Press 1981.
Galdi, G. P., Payne, L. E., Proctor, M. R. E., Straughan, B.: Convection in thawing subsea permafrost. Proc. Roy. Soc. LondonA 414, 83–102 (1987).
Galdi, G. P., Straughan, B.: Exchange of stabilities, symmetry and nonlinear stability. Arch. Rational Mech. Anal.89, 211–228 (1985).
George, J. H., Gunn, R. D., Straughan, B.: Patterned ground formation and penetrative convection in porous media. Geophys. Astrophys. Fluid Dyn.46, 135–158 (1989).
Joseph, D. D.: Stability of fluid motions, vol. I. Berlin-Heidelberg-New York: Springer 1976.
Joseph, D. D.: Stability of fluid motions, vol. II. Berlin-Heidelberg-New York: Springer 1976.
Joseph, D. D., Shir, C. C.: Subcritical convective instability. Part 1. Fluid layers. J. Fluid Mech.26, 753–768 (1966).
Payne, L. E., Song, J. C., Straughan, B.: Double diffusive porous penetrative convection — thawing subsea permafrost. Int. J. Engng. Sci.26, 797–809 (1988).
Payne, L. E., Straughan, B.: Unconditional nonlinear stability in penetrative convection. Geophys. Astrophys. Fluid Dynamics39, 57–63 (1987).
Payne, L. E., Straughan, B.: Unconditional nonlinear stability in penetrative convection: corrected and extended numerical results. Geophys. Astrophys. Fluid Dynamics43, 307–309 (1988).
Payne, L. E., Straughan, B.: Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid. Int. J. Engng. Sci.27, 827–836 (1989).
Rionero, S., Straughan, B.: Convection in a porous medium with internal heat source and variable gravity effects. Int. J. Engng. Sci. (to appear).
Roberts, P. H.: Convection in horizontal layers with internal heat generation. Theory. J. Fluid Mechanics30, 33–49 (1967).
Straughan, B.: Finite amplitude instability thresholds in penetrative convection. Geophys. Astrophys. Fluid Dynamics34, 227–242 (1985).
Veronis, G.: Penetrative convection. Astrophysical J.137, 641–663 (1963).
Whitehead, J. A., Chen, M. M.: Thermal instability and convection of a thin layer bounded by a stably stratified region. J. Fluid Mechanics40, 549–576 (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ames, K.A., Straughan, B. Penetrative convection in fluid layers with internal heat sources. Acta Mechanica 85, 137–148 (1990). https://doi.org/10.1007/BF01181513
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01181513