Skip to main content
SpringerLink
Log in

Convection in multi-component rotating fluid layers via the auxiliary system method

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

The onset of thermal convection in a uniformly rotating horizontal layer filled by a Navier–Stokes multi-component fluid mixture, heated from below and salted partly from above and partly from below, is investigated via the new approach named auxiliary system method Rionero (Rend Lincei Mat Appl 25:1–44, 2014). In the free–free case, via the generalization of the Rionero Linearization Principle: “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” [given in Rionero (Rend Lincei Mat Appl 25:1–44, 2014) in the absence of rotation], it is shown that conditions guaranteeing linear stability of thermal conduction solution guarantee also absence of subcritical instabilities and global exponential asymptotic nonlinear stability. The classical Bènard problem is investigated via a procedure different from the celebrated one given in Chandrasekhar (Hydrodynamic and hydromagnetic stability, 1981).

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

Similar content being viewed by others

References

  1. Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Dover, New York (1981)

  2. De Luca, R., Rionero, S.: Steady and oscillatory convection in rotating fluid layers heated and salted from below. Int. J. Nonlinear Mech. doi:10.1016/j.ijnonlinmec.2015.10.011

  3. Flavin, J., Rionero, S.: An Introduction Qualitative estimates for Partial Differential Equations. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  4. Joseph, D.D.: Stability of fluid motions I,II. vol 27–28 of Springer Tracts in Natural Philosophy. Springer, New York (1976)

    Google Scholar 

  5. Mulone, G., Rionero, S.: On the nonlinear stability of the rotating Bènard problem via the Lyapunov Direct Method. J. Math. Anal. Appl. 144, 109–127 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mulone, G., Rionero, S.: The rotating Bènard problem: new stability results for any Prandtl and Taylor numbers. Continuum Mech. Thermodyn. 9, 347–363 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nield, D.A., Bejan, A.: Convection in Porous Media, IV edn. Springer, Berlin Heidelberg New York (2012)

    MATH  Google Scholar 

  8. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection, Second Edition, Appl. Math. Sci., 91. Springer (2004)

  9. Mulone, G., Rionero, S.: Non-linear stability analysis of the magnetic Bènard problem through the Lyapunov direct method. Arch. Ration. Mech. Anal. 103, 347–368 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Capone, F., Rionero, S.: Inertia effect on the onset of convection in rotating porous layers via the “auxiliary system method”. Int. J. Nonlinear Mech. 52, 192–200 (2013)

    Article  Google Scholar 

  11. Rionero, S.: Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures. Phys. Fluids 24, 104101 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rionero, S.: Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global nonlinear stability for any number of salts. Phys. Fluids 25, 054104 (2013)

    Article  MATH  Google Scholar 

  13. Rionero, S.: Soret effects on the onset of convection in rotating porous layers via the “auxiliary system method”. Ric. Mat. 62(2), 183–208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rionero, S.: Onset of convection in rotating porous layers via a new approach. AIMS 19(7), 2279–2296 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Rionero, S.: Heat and mass transfer by convection in multicomponent Navier–Stokes mixture: absence of subcritical instabilities and global nonlinear stability via the auxiliary system method. Rend. Lincei Mat. Appl. 25, 1–44 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Merkin, D.R.: Introduction to the theory of stability. Texts in Applied Mathematics, 24, p. 319. Springer, New York (1997)

    Google Scholar 

Download references

Acknowledgments

This paper has been performed under the auspices of G.N.F.M. of INdAM. One of the authors (R. De Luca) acknowledges Progetto Giovani GNFM 2015 “Dinamica di sistemi complessi infinito dimensionali con applicazioni in Fluidodinamica, Economia e Biologia”.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applications “Renato Caccioppoli”, University of Naples Federico II, Via Cinzia, 80126, Naples, Italy

    Roberta De Luca & Salvatore Rionero

  2. Accademia Nazionale dei Lincei, Via della Lungara 10, 06165, Rome, Italy

    Salvatore Rionero

Authors
  1. Roberta De Luca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salvatore Rionero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberta De Luca.

Additional information

Communicated by Salvatore Rionero.

Some results contained in the present paper have been presented at XVIII International Conference on Waves and Stability in Continuous Media—WASCOM 2015—Cetraro (CS), June, 1–5, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Luca, R., Rionero, S. Convection in multi-component rotating fluid layers via the auxiliary system method. Ricerche mat 65, 363–379 (2016). https://doi.org/10.1007/s11587-015-0251-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-015-0251-y

Keywords

Mathematics Subject Classification

Search

Navigation