Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T03:09:25.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Structural stability for the resonant porous penetrative convection

Published online by Cambridge University Press:  10 August 2012

YAN LIU
YAN LIU*
Affiliation:
Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, P. R. China e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the structural stability of a problem in a porous medium when the density of saturating liquid is a nonlinear function of temperature and an internal heat source is present. We prove a convergence result for the Forchheimer coefficient. That is to say, when λ → 0, the solution of the non-isothermal flow in a porous medium of the Forchheimer type, see (1.1), can converge to the solution of the equivalent Darcy type.

Keywords

Structural stabilityForchheimer equationsThe Forchheimer coefficientDarcy equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 6 , December 2012 , pp. 761 - 775
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ames, K. A. & Payne, L. E. (1990) On stabilizing against modelling errors in a penetrative convection problem for a porous medium. Models Methods Appl. Sci. 4, 733–740.Google Scholar
[2]Ames, K. A. & Straughan, B. (1997) Non-Standard and Improperly Posed Problems (Mathematics in Science and Engineering series, Vol. 194), Academic Press, San Diego, CA.Google Scholar
[3]Celebi, A. O., Kalantarov, V. K. & Ugurlu, D. (2005) Continuous dependence for the convective Brinkman-Forchheimer equations. Appl. Anal. 84, 877–888.CrossRefGoogle Scholar
[4]Celebi, A. O., Kalantarov, V. K. & Ugurlu, D. (2006) On continuous dependence on coefficients of the Brinkman-Forchheimer equations. Appl. Math. Lett. 19, 801–807.CrossRefGoogle Scholar
[5]Chadam, J. & Qin, Y. (1997) Spatial decay estimates for flow in a porous medium, SIAM. J. Math. Anal. 28 (4), 808–830.Google Scholar
[6]Franchi, F. & Straughan, B. (2003) Continuous dependence and decay for the Forchheimer equations. Proc. R. Soc. Lond. A 459, 3195–3202.Google Scholar
[7]Kaloni, P. N. & Guo, J. (1996) Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman-Forchheimer Model. J. Math. Anal. Appl. 204, 138–155.Google Scholar
[8]Kaloni, P. N. & Qin, Y. (1996) Steady nonlinear double-diffusive convection in a porous medium base upon the Brinkman-Forchheimer model. J. Math. Anal. Appl. 204, 138–155.Google Scholar
[9]Kaloni, P. N. & Qin, Y. (1998) Spatial decay estimates for flow in the Brinkman-Forchheimer Model. Q. Appl. Math. 56, 71–87.Google Scholar
[10]Lin, C. & Payne, L. E. (2007a) Structural stability for a Brinkman fluid. Math. Meth. Appl. Sci. 30, 567–578.Google Scholar
[11]Lin, C. & Payne, L. E. (2007b) Structural stability for the Brinkman equations of flow in double diffusive convection. J. Math. Anal. Appl. 325, 1479–1490.Google Scholar
[12]Lin, C. & Payne, L. E. (2008) Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow. J. Math. Anal. Appl. 342, 311–325.CrossRefGoogle Scholar
[13]Liu, Y., DU, Y. & Lin, C. (2010) Convergence results for Forchheimers equations for fluid flow in porous media. J. Math. Fluid. Mech. 12, 576–593.CrossRefGoogle Scholar
[14]Nield, D. A. & Bejian, A., (1992) Convection in Porous Media, Springer, New York.CrossRefGoogle Scholar
[15]Payne, L. E., Rodrigues, J. F. & Straughan, B. (2001) Effect of anisotropic permeability on Darcy's law. Math. Methods Appl. Sci. 24, 427–438.Google Scholar
[16]Payne, L. E. & Song, J. C. (1997) Spatial decay estimates for the Brinkman and Darcy flows in a semi-infinite cylinder. Contin. Mech. Thermodyn. 9, 175–190.Google Scholar
[17]Payne, L. E. & Song, J. C. (2000) Spatial decay for a model of double diffusive convection in Darcy and Brinkman flows. Z. Angew. Math. Phys. 51, 867–880.Google Scholar
[18]Payne, L. E. & Song, J. C. (2002) Spatial decay bounds for the Forchheimer equations. Int. J. Eng. Sci. 40, 943–956.Google Scholar
[19]Payne, L. E. & Song, J. C. (2007) Spatial decay in a double diffusive convection problem in Darcy flow. J. Math. Anal. Appl. 330, 864–875.CrossRefGoogle Scholar
[20]Payne, L. E., Song, J. C. & Straughan, B. (1999) Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity. Proc. R. Soc. Lond. A 45S, 2173–2190.Google Scholar
[21]Payne, L. E. & Straughan, B. (1996) Stability in the initial-time geometry problem for the Brinkman and Darcy equations of flow in porous media. J. Math. Pures Appl. 75, 255–271.Google Scholar
[22]Payne, L. E. & Straughan, B. (1998) Structural stability for the Darcy equations of flow in porous media. Proc. R. Soc. Lond. A 454, 1691–1698.Google Scholar
[23]Payne, L. E. & Straughan, B. (1999) Convergence and continuous dependence for the Brinkman-Forchheimer equations. Stud. Appl. Math. 102, 419–439.Google Scholar
[24]Qin, Y. & Kaloni, P. N. (1998) Spatial decay estimates for plane flow in the Brinkman-Forchheimer model. Quart. Appl. Math. 56, 71–87.Google Scholar
[25]Song, J. C. (2002) Spatial decay estimates in time dependent double-diffusive Darcy plane flow. J. Math. Anal. Appl. 207, 76–88.Google Scholar
[26]Straughan, B. (2004) The Energy Method, Stability and Nonlinear Convection, 2nd ed. (Appl. Math. Sci. Ser., Vol. 91), Springer, New York.Google Scholar
[27]Straughan, B. (2008) Stability and Wave Motion in Porous Media (Appl. Math. Sci. Ser. Vol. 165), Springer, New York.Google Scholar
[28]Straughan, B. (2011) Continuous dependence on the heat source in resonant porous penetrative convection. Stud. Appl. Math. 127, 302–314.Google Scholar
[29]Weatherburn, C. E. (1980) Differential Geometry of Three Dimensions, Cambrige University Press, Cambridge, UK.Google Scholar