Skip to main content
SpringerLink
Log in

Global Exponential Nonlinear Stability for Double Diffusive Convection in Porous Medium

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

Nonlinear stability of the motionless double-diffusive solution of the problem of an infinite horizontal fluid layer saturated porous medium is studied. The layer is heated and salted from below. By introducing two balance fields and through defining new energy functionals it is proved that for CLe ≥ R, Le ≤ 1 the motionless double-diffusive solution is always stable and for CLe < R, Le < 1 the solution is globally exponentially and nonlinearly stable whenever R < 4π2+LeC, where Le, C and R are the Lewis number, Rayleigh number for solute and heat, respectively. Moreover, the nonlinear stability proved here is global and exponential, and the stabilizing effect of the concentration is also proved.

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. Gilman A, Bear J. The influence of free convection on soil salinization in arid regions. Transport Porous MED, 1996, 23: 275–301

    Article  Google Scholar 

  2. Pieters G J M, Van Duijn C J. Transient growth in linearly stable gravity-driven flow in porous media. Eur J Mech B/Fluids, 2006, 25: 83–94

    Article  MathSciNet  MATH  Google Scholar 

  3. Payne L E, Straughan B. A naturally efficient numerical technique for porous convection stability with non-trivial boundary conditions. Int J Num Anal Meth Geomech, 2000, 24(10): 815–836

    Article  MATH  Google Scholar 

  4. Nield D, Bejan A D. Convection in Porous Media. 5th ed. New York: Springer, 2017

    Book  MATH  Google Scholar 

  5. Capone F, Rionero S. Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient. Continuum Mech Thermodyn, 2003, 15: 529–538

    Article  MathSciNet  MATH  Google Scholar 

  6. Lombardo S, Mulone G. Necessary and sufficient conditions for global nonlinear stability for rotating double-diffusive convection in a porous medium. Continuum Mech Thermodyn, 2002, 14: 527–540

    Article  MathSciNet  MATH  Google Scholar 

  7. Lombardo S, Mulone G. Necessary and sufficient stability conditions via the eigenvalues-eigenvectors method: an application to the magnetic Bénard problem. Nonlinear Anal, 2005, 63(5/7): e2091–e2101

    Article  MATH  Google Scholar 

  8. Payne L E, Straughan B. Unconditional nonlinear stability in temperature-dependent viscosity flow in a porous medium. Stud Appl Math, 2000, 105: 59–81

    Article  MathSciNet  MATH  Google Scholar 

  9. Straughan B. The Energy Method, Stability, and Nonlinear Convection. 2nd ed. New York: Springer, 2004

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Joseph D D. Global stability of the conduction-diffusion solution. Arch Rational Mech Anal, 1970, 36: 285–292

    Article  MathSciNet  MATH  Google Scholar 

  12. Lombardo S, Mulone G, Straughan B. Stability in the Bénard problem for a double-diffusive mixture in a porous medium. Math Meth Appl Sci, 2001, 24: 1229–1246

    Article  MATH  Google Scholar 

  13. Galdi G P, Padula M. A new approach to energy theory in the stability of fluid motion. Arch Rational Mech Anal, 1990, 110: 187–286

    Article  MathSciNet  MATH  Google Scholar 

  14. Xu L X, Lan W L. On the nonlinear stability of parallel shear flow in the presence of a coplanar magnetic field. Nonlinear Anal, 2014, 95: 93–98

    Article  MathSciNet  MATH  Google Scholar 

  15. Mulone G, Straughan B. An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media. Z Angew Math Mech, 2006, 86(7): 507–520

    Article  MathSciNet  MATH  Google Scholar 

  16. Schmitt B J, Von Wahl W. Decomposition of solenoidal fields into poloidal fields, toroidal fields and mean flow. Applications to the Boussinesq equations// Heywood J G, Masuda K, Rautmann R, Solonnikov S A, eds. The Navier-Stokes Equations II-Theory and Numerical Methods. Lecture Notes in Mathematics 1530, Berlin, Heidelberg, New York: Springer, 1992: 291–305

    Chapter  Google Scholar 

  17. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Oxford: Oxford University Press, 1961

    MATH  Google Scholar 

  18. Yang Z X, Zhang G B. Global stability of traveling wavefronts for nonlocal rection-diffusion equations with time delay. Acta Math Sci, 2018, 38B(1): 291–304

    Google Scholar 

  19. He L, Tang S J, Wang T. Stability of viscous shock waves for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Acta Math Sci, 2016, 36B(1): 36–50

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing University of Chemical Technology, Beijing, 100029, China

    Lanxi Xu  (许兰喜) & Ziyi Li  (李紫奕)

Authors
  1. Lanxi Xu  (许兰喜)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ziyi Li  (李紫奕)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lanxi Xu  (许兰喜).

Additional information

This work was supported by National Natural Science Foundation Project (41671229).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, L., Li, Z. Global Exponential Nonlinear Stability for Double Diffusive Convection in Porous Medium. Acta Math Sci 39, 119–126 (2019). https://doi.org/10.1007/s10473-019-0109-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-019-0109-6

Key words

2010 MR Subject Classification

Search

Navigation