Skip to main content

The Rayleigh—Bénard Shallow Thermal Convection Problem

  • Chapter
  • 1717 Accesses

Part of the book series: Fluid Mechanics and its Applications ((FMIA,volume 90))

It is usual in the literature (see, for instance, the book by Drazin and Reid [1]) to denote as Rayleigh—Bénard (RB) shallow thermal convection, the instability problem produced mainly by buoyancy, possibly including the Marangoni and Biot effects in a non-deformable free surface.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P.G. Drazin and W.H. Reid, Hydrodynamic Stability. Cambridge University Press, Cambridge, 1981.

    MATH  Google Scholar 

  2. C. Ruyer-Quil, B. Scheid, S. Kalliadasis, M.G. Velarde and R.Kh. Zeytounian, Thermo-capillary long-waves in a liquid film flow; Part 1: Low-dimensional formulation. J. Fluid Mech. 538, 199–222, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Scheid, C. Ruyer-Quil, S. Kalliadasis, M.G. Velarde and R.Kh. Zeytounian, Thermo-capillary long-waves in a liquid film flow; Part 2: Linear stability and nonlinear waves. J. Fluid Mech. 538, 223–244, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  4. E.M. Sparrow, R.J. Goldstein and V.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.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961. See also Dover Publications, New York, 1981.

    MATH  Google Scholar 

  6. J.S. Turner, Buoyancy Effects in Fluids. Cambridge, Cambridge University Press, 1973.

    MATH  Google Scholar 

  7. M.G. Velarde and C. Normand, Sci. Amer. 243(1), 92, 1980.

    Google Scholar 

  8. C. Normand, Y. Pomeau and M.G. Velarde, Convective instability: A physicist's approach. Rev. Mod. Phys. 49(3), 581–624, 1977.

    Article  MathSciNet  Google Scholar 

  9. A.V. Getling, Rayleigh Bénard Convection: Structure and Dynamics. World Scientific, Singapore, 1998.

    MATH  Google Scholar 

  10. E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh—Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778, 2000.

    Article  MathSciNet  Google Scholar 

  11. P.C. Dauby and G. Lebon, Bénard—Marangoni instability in rigid rectangular containers. J. Fluid Mech. 329, 25–64, 1996.

    Article  MATH  Google Scholar 

  12. R.Kh. Zeytounian, Mécanique des Fluides Fondamentale. Springer-Verlag, Heidelberg 1991.

    MATH  Google Scholar 

  13. A.C. Newell and J. Whitehead, Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38(2), 279–303, 1966.

    Article  Google Scholar 

  14. P. Coullet and P. Huerre, Resonance and phase solitons in spatially-forced thermal convection. Physica D 23, 27–44, 1986.

    Article  MATH  Google Scholar 

  15. L.D. Landau, On the problem of turbulence. C.R. Acad. Sci. URSS 44, 311–314, 1944. See also Collected Papers, 387–391, Oxford, 1965.

    Google Scholar 

  16. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmospheric Sci. 20, 130–141, 1963.

    Article  Google Scholar 

  17. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer, 1982.

    Google Scholar 

  18. J.K. Platten and J.C. Legros, Convection in Liqids. Springer-Verlag, New York, 1984.

    Google Scholar 

  19. R.Kh. Zeytounian, Theory and Applications of Viscous Fluid Flows. Springer-Verlag, Heidelberg, 2004.

    MATH  Google Scholar 

  20. J.H. Curry et al., Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38, 1984.

    Article  MATH  Google Scholar 

  21. S. Orszag, Studies Appl. Math. L4, 293–327, 1971.

    MathSciNet  Google Scholar 

  22. J.P. Gollub and S.V. Benson, Many routes to turbulent convection. J. Fluid Mech. 100, 449–470, 1980.

    Article  Google Scholar 

  23. L.N. Howard and R. Krishnamurti, Large-scale flow in turbulent convection: A mathematical model. J. Fluid Mech. 170, 385–410, 1986.

    Article  MATH  Google Scholar 

  24. Z. Charki and R.Kh. Zeytounian, The Bénard problem for deep convection: Derivation of the Landau—Ginzburg equation. Int. J. Engng. Sci. 33(12), 1839–1847, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Siggia and A. Zippelius, Stability of finite-amplitude convection. Phys. Fluids 26, 2905, 1983.

    Article  MATH  Google Scholar 

  26. P. Coullet and S. Fauve, Propagative phase dynamics for systems with Galilean invari-ance. Phys. Rev. Lett. 55, 2857–2859, 1985.

    Article  Google Scholar 

  27. C. Doering, J. Gibbon, D. Holm and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg—Landau equation. Nonlinearity 1, 279–309, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  28. L.R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg—Landau equation. Stud. Appl. Math. 73, 91, 1985.

    MATH  MathSciNet  Google Scholar 

  29. A. Manela and I. Frankel, On the Rayleigh—Bénard problem in the continuum limit: Effects of temperature differences and model of interaction. Phys. Fluids 17, O36101-1–O36107, 2005.

    MathSciNet  Google Scholar 

  30. C. Cercignani, Rarefied Gas Dynamics. Cambridge, Cambridge University Press, 2000.

    MATH  Google Scholar 

  31. S. Stefanov, V. Roussinov and C. Cercignani, Rayleigh—Bénard flow of a rarefied gas and its attractors. I. Convection regime. Phys. Fluids 14, 2255, 2002.

    Article  MathSciNet  Google Scholar 

  32. R.Kh. Zeytounian, Topics in Hyposonic Flow Theory. Lecture Notes in Physics, Vol. 672. Springer-Verlag, Berlin/Heidelberg, 2006.

    Google Scholar 

  33. J. Frölich, P. Laure and R. Peyret, Phys. Fluids A 4, 1355, 1992.

    Article  Google Scholar 

  34. G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon, Oxford, 1994.

    Google Scholar 

  35. I. Boyd, In: Rarefied Gas Dynamics, A.D. Ketsdever and E.P. Muntz (Eds.), American Institute of Physics, New York, p. 899, 2003.

    Google Scholar 

  36. B. Straughan, The Energy Method, Stability, and Nonlinear Convection. Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  37. D.D. Joseph. Stability of Fluid Motions II. Springer-Verlag, Berlin, 1976.

    MATH  Google Scholar 

  38. M.G. Braunsfurth and G.M. Homsy, Combined thermocapillary-buoyancy convection in a cavity. Part II. An experimental study. Phys. Fluids 9(5), 1277–1287, 1997.

    Article  Google Scholar 

  39. T. Boeck, A. Nepomnyashchy, I. Simanovskii, A. Golovin, L. Braverman and A. Thess, Phys. Fluids 14(11), 3899–3911, 2002.

    Article  MathSciNet  Google Scholar 

  40. L. Ratke, H. Walter and B. Feuerbacher (Eds.), Materials and Fluids under Low Gravity. Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  41. H.C. Kuhlmann, Thermocapillary Convection in Models of Crystal Growth. Springer-Verlag, Berlin, 1999.

    Google Scholar 

  42. J.C. Legros, Acta Astron. 13, 697, 1986.

    Article  Google Scholar 

  43. I.B. Simanovskii and A.A. Nepomnyaschy. Convective Instabilities in Systems with Interface. Gordon and Breach, 1993.

    Google Scholar 

  44. A. Juel et al., Surface tension-driven convection patterns in two liquid layers. Physica D 143, 169–186, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  45. L.M. Bravermam et al., Convection in two-layer systems with an anomalous thermocap-illary effect. Phys. Rev. E 62, 3619–3631, 2000.

    Article  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, B.V.

About this chapter

Cite this chapter

(2009). The Rayleigh—Bénard Shallow Thermal Convection Problem. In: Convection in Fluids. Fluid Mechanics and its Applications, vol 90. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2433-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-90-481-2433-6_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-2432-9

  • Online ISBN: 978-90-481-2433-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics