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Heat transfer and stability properties of convection rolls in an internally heated fluid layer

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Summary

A study has been made of stationary two-dimensional convection in an internally heated, infinite Prandtl number, horizontal fluid layer bounded above and below by rigid plates of unequal temperature. Sufficient conditions for instability of the quiescent state, as well as post-instability, two-dimensional finite amplitude convective solutions, and heat transfer are obtained as a function of the two Rayleigh number parameters and the wavenumber. Investigation of the stability of the finite amplitude convective motion reveals a region in the dual Rayleigh number domain wherein stationary, two-dimensional convection represents a stable solution of the equations of motion.

Zusammenfassung

Eine Studie der stationären zweidimensionalen Konvektion in einer homogen erhitzten horizontalen Flüssigkeitsschicht wird gemacht im Fall einer unendlichen Prandtl-Zahl, wenn oben und unten feste Platten mit verschiedenen Temperaturen als Randbedingungen vorgegeben sind. Hinreichende Bedingungen für die Instabilität der statischen Schichtung und Lösungen für die voll entwickelte Instabilität in Form zweidimensionaler Konvektion endlicher Amplitude werden in Abhängigkeit von den zwei Rayleighzahl-Parametern und der Wellenzahl berechnet. Die Untersuchung der Stabilität der Konvektionsströmung endlicher Amplitude ergibt ein Gebiet innerhalb des durch die beiden Rayleighzahlen aufgespannten Raumes, im dem zweidimensionale stationäre Konvektion eine physikalisch realisierbare, stabile Lösung der Bewegungsgleichungen darstellt.

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References

  1. F. H. Busse,On the Stability of Two-dimensional Convection in a Layer Heated from Below, J. Math. Phys.46, 140–150 (1967).

    Google Scholar 

  2. R. M. Clever andF. H. Busse,Transition to Time-dependent Convection, J. Fluid Mech.65, 625–645 (1974); also presented at 7th U.S. Nat. Cong. Appl. Mech., Univ. Colorado, Boulder (June, 1974).

    Google Scholar 

  3. M. M. Chen andJ. A. Whitehead,Evolution of Two-dimensional Periodic Rayleigh Convection Cells of Arbitrary Wave Numbers, J. Fluid Mech.31, 1–15 (1968).

    Google Scholar 

  4. R. Krishnamurti,On the Transition to Turbulent Convection, Part I: The Transition from Two- to Three-dimensional Flow, J. Fluid Mech.42, 295–307 (1970).

    Google Scholar 

  5. F. H. Busse andJ. A. Whitehead,Instabilities of Convection Rolls in a High Prandtl Number Fluid, J. Fluid Mech.47, 305–320 (1971).

    Google Scholar 

  6. F. H. Busse,The Stability of Finite Amplitude Convection and its Relation to an Extremum Principle, J. Fluid Mech.30, 625–649 (1967).

    Google Scholar 

  7. R. Krishnamurti,Finite Amplitude Convection with Changing Mean Temperature, Part I: Theory, J. Fluid Mech.33, 445–455 (1968).

    Google Scholar 

  8. R. Krishnamurti,Finite Amplitude Convection with Changing Mean Temperature, Part II: An Experimental Test of the Theory, J. Fluid Mech.33, 457–463 (1968).

    Google Scholar 

  9. P. H. Roberts,Convection in Horizontal Layers with Internal Heat Generation, J. Fluid Mech.30, 33–49 (1967).

    Google Scholar 

  10. M. Tveitereid andE. Palm,Convection Due to Internal Heat Sources, J. Fluid Mech.76, 481–499 (1976).

    Google Scholar 

  11. D. J. Tritton andM. N. Zarraga,Convection in Horizontal Layers with Internal Heat Generation. Experiments, J. Fluid Mech.30, 21–31 (1967).

    Google Scholar 

  12. E. W. Schwiderski andH. J. H. Schwab,Convection Experiments with Electrolytically Heated Fluid Layers, J. Fluid Mech.48, 703–719 (1971).

    Google Scholar 

  13. F. A. Kulacki andR. J. Goldstein,Thermal Convection in a Horizontal Layer with Uniform Volumetric Energy Sources, J. Fluid Mech.55, 271–287 (1972).

    Google Scholar 

  14. S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961).

    Google Scholar 

  15. E. M. Sparrow, R. J. Goldstein andV. K. Jonsson,Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Conditions and Non-linear Temperature Profile, J. Fluid Mech.18, 513–528 (1964).

    Google Scholar 

  16. W. V. R. Malkus andG. Veronis,Finite Amplitude Cellular Convection, J. Fluid Mech.4, 225–260 (1958).

    Google Scholar 

  17. W. Eckhaus,Studies in Nonlinear Stability Theory, Springer, Berlin/Heidelberg/New York (1965).

    Google Scholar 

  18. F. H. Busse,Stability Regions of Cellular Fluid Flow, IUTAM Symp. Herr. 1969, Springer, Berlin/Heidelberg/New York (1971).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Geophysics and Planetary Physics, University of California, Los Angeles, USA

    Richard M. Clever

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  1. Richard M. Clever
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Clever, R.M. Heat transfer and stability properties of convection rolls in an internally heated fluid layer. Journal of Applied Mathematics and Physics (ZAMP) 28, 585–597 (1977). https://doi.org/10.1007/BF01601337

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  • DOI: https://doi.org/10.1007/BF01601337

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