Skip to main content
SpringerLink
Log in

Continuous dependence on the coefficients for double diffusive convection in Darcy flow with Magnetic field effect

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, the Darcy model is used to describe the double diffusive flow of a fluid containing a solute. An a priori result is established whereby we show the solution depends continuously on changes in the magnetic and the gravity vector coefficients.

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. Ames, K.A., Payne, L.E.: Continuous dependence results for solutions of the Navier–Stokes equations backward in time. Nonlinear Anal. Theor. Meth. Appl. 23, 103–113 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ames, K.A., Payne, L.E.: On stabilizing against modeling errors in a penetrative convection problem for a porous medium. Math. Models Meth. Appl. Sci. 4, 733–740 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ames, K.A., Payne, L.E.: Continuous dependence results for a problem in penetrative convection. Q. Appl. Math. 55, 769–790 (1997)

    Google Scholar 

  4. Ames, K.A., Payne, L.E.: Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation. Math. Models Meth. Appl. Sci. 8, 187–202 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ames, K.A., Straughan, B.: Non-standard and improperly posed problems. Mathematics in Science and Engineering Series, vol. 194. Academic Press, New York (1997)

  6. Bandle, C.: Isoperimetric Inequalities and Their Applications. Pitman Press, London (1980)

    Google Scholar 

  7. Bellomo, N., Preziosi, L.: Modelling mathematical methods and scientific computation. CRC, Boca Raton (1995)

    MATH  Google Scholar 

  8. Franchi, F., Straughan, B.: A continuous dependence on the body force for solutions to the Navier-Stokes equations and on the heat supply in a model for double-diffusive porous convection. J. Math. Anal. Appl. 172, 117–129 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Franchi, F., Straughan, B.: Continuous dependence on the relaxation time and modelling, and unbounded growth. J. Math. Anal. Appl. 185, 726–746 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Franchi, F., Straughan, B.: Spatial decay estimates and continuous dependence on modelling for an equation from dynamo theory. Proc. R. Soc. Lond. A 445, 437–451 (1994)

    Google Scholar 

  11. Franchi, F., Straughan, B.: Structural stability for the Brinkman equations of porous media. Math. Methods Appl. Sci. 19, 1335–1347 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galdi, G.P., Straughan, B.: Exchange of stabilities, symmetry and nonlinear stability. Arch. Rational Mech. Anal. 89, 211–228 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin, C., Payne, L.E.: On the spatial decay of ill-posed parabolic problems. Math. Meth. Appl. Sci. 3, 563–575 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, C., Payne, L.E.: The influence of domain and diffusivity perturbations on the decay of end effects in heat conduction. SIAM J. Math. Anal. 25, 1242–1258 (1994)

    MathSciNet  Google Scholar 

  15. Lin, C., Payne, L.E.: Structural stability for a Brinkman fluid. Math. Math. Methods Appl. Sci. 30, 567–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, C., Payne, L.E.: Structural stability for the Brinkman equations of flow in double diffusive convection. J. Math. Anal. Appl. 325, 1479–1490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, C., Payne, L.E.: Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow. J. Math. Anal. Appl. 342, 311–325 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Payne, L.E., Song, J.C.: Spatial decay estimates for the Brinkman and Darcy flows in a semi-infinite cylinder. Continuum Mech. Thermodyn. 9, 175–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Payne, L.E., Song, J.C.: Spatial decay for a model of double diffusive convection in Darcy and Brinkman flows. Z. Angew. Math. Phys. 51, 867–880 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Payne, L.E., Song, J.C.: Spatial decay bounds in double diffusive convection in Darcy flow. J. Math. Anal. Appl. 330, 864–875 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Payne, L.E., Song, J.C., Straughan, B.: Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity. Proc. R. Soc. Lond. A 455, 2173–2190 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Payne, L.E., Straughan, B.: Stability in the initial-time geometry problem for the Brinkman and Darcy equations of flow in porous media. J. Math. Pures Appl. 75, 225–271 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Payne, L.E., Straughan, B.: Analysis of the boundary condition at the interface between a viscous and a porous medium and related modelling questions. J. Math. Pures Appl. 77, 317–354 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Payne, L.E., Straughan, B.: Structural stability for the Darcy equations of flow in porous media. Proc. R. Soc. Lond. A 454, 1691–1698 (1998)

    Google Scholar 

  25. Payne, L.E., Weinberger, H.F.: New bounds for solutions of second order elliptic partial differential equations. Pacific J. Math. 8, 551–573 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  26. Straughan, B., Hutter, K.: A priori bounds and structural stability for double diffusive convection incorporating the Soret effect. Proc. R. Soc. Lond. A 455, 767–777 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Series in Applied Mathematical Sciences, vol. 91, 2nd edn. Springer, Berlin (2004)

  28. Straughan, B.: Stability and wave motion in porous media. Series in Applied Mathematical Sciences, vol. 165. Springer, Berlin (2008)

  29. Weatherburn, C.E.: Differential Geometry of Three Dimensions. Cambridge Univ Press, Cambridge (1980)

    Google Scholar 

Download references

Acknowledgments

This work was supported by a scholarship from the Iraqi ministry of higher education and scientific research. The author would like to thank Prof. B. Straughan for his guidance. Also, the author would like to thank anonymous referees for their comments that have led to improvements in the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

    A. J. Harfash

Authors
  1. A. J. Harfash
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. J. Harfash.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harfash, A.J. Continuous dependence on the coefficients for double diffusive convection in Darcy flow with Magnetic field effect. Anal.Math.Phys. 3, 163–181 (2013). https://doi.org/10.1007/s13324-013-0053-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-013-0053-x

Keywords

Search

Navigation