Skip to main content
SpringerLink
Log in

Onset of Rayleigh-Benard Convection with Periodic Boundary Temperatures Using Weakly Nonlinear Theory

  • Original Article
  • Published:
Microgravity Science and Technology Aims and scope Submit manuscript

Abstract

This paper investigates the Rayleigh Benard instability of a viscous, Newtonian, Boussinesq fluid with time-periodic boundary temperature modulation using the framework of weakly nonlinear theory. Critical Rayleigh number is computed for asymptotic stability criterion using the energy method accompanied by variational algorithm. Subcritical instability is found to occur under two conditions: when modulation is in anti-phase and when modulation is imposed only on the lower boundary. Supercritical stability is witnessed during in-phase modulation. In all the three cases of the relative phase of two boundary temperatures, the effect of modulation is found to be weaker for infinitesimal disturbances. The findings of the present study could be referred in several applications where appropriate temperature modulation is of prime concern.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Arthurs, A.M.: Complementary Variational Principles. Clarendon Press, Oxford (1970)

    MATH  Google Scholar 

  • Bénard, H.: Les tourbillons cellulaires dans une nappe liquide propageant de la chaleur par convection en régime permanent, Ann. de Ch. et de Phys. (1901)

  • Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London (1961)

    MATH  Google Scholar 

  • Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Cambridge University press, Cambridge (2004)

    Book  Google Scholar 

  • Hancock MJ.: 18.303 linear partial differential equations, Fall (2005)

  • Homsy, G.M.: Global stability of time-dependent flows. Part 2. Modulated fluid layers. J. Fluid Mech. 62(02), 387–403 (1974)

    Article  MathSciNet  Google Scholar 

  • Joseph, D.D.: On the stability of the Boussinesq equations. Arch. Ration. Mech. Anal. 20(1), 59–71 (1965)

    Article  MathSciNet  Google Scholar 

  • Malkus, W.V., Veronis, G.: Finite amplitude cellular convection. J. Fluid Mech. 4(03), 225–260 (1958)

    Article  MathSciNet  Google Scholar 

  • Nayfeh, A.H.: Perturbation Methods. Wiley, Hoboken (2008)

    Google Scholar 

  • Raju, V.R.K., Bhattacharyya, S.N.: Onset of thermal instability in a horizontal layer of fluid with modulated boundary temperatures. J. Eng. Math. 66(4), 343–351 (2010)

    Article  MathSciNet  Google Scholar 

  • Rayleigh, L.: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. London Edinburgh Dublin Philosophic Mag J Sci. 32(192), 529–546 (1916)

    Article  Google Scholar 

  • Serrin, J.: On the stability of viscous fluid motions. Arch. Ration. Mech. Anal. 3(1), 1–13 (1959)

    Article  MathSciNet  Google Scholar 

  • Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer-Verlag, New York (1992)

    Book  Google Scholar 

  • Venezian, G.: Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35(02), 243–254 (1969)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vehicle CFD Department, Mercedes-Benz R&D, Bengaluru, Karnataka, 560066, India

    Arun Kumar

  2. Department of Mechanical Engineering, National Institute of Technology, Warangal, 506004, India

    Arun Kumar & V. R. K. Raju

  3. Department of Electrical Engineering, Indian Institute of Technology, Roorkee, 247667, India

    S. Das

Authors
  1. Arun Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. R. K. Raju
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. R. K. Raju.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, A., Raju, V.R.K. & Das, S. Onset of Rayleigh-Benard Convection with Periodic Boundary Temperatures Using Weakly Nonlinear Theory. Microgravity Sci. Technol. 32, 1237–1243 (2020). https://doi.org/10.1007/s12217-020-09844-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12217-020-09844-6

Keywords

Search

Navigation