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Structural Stability in Resonant Penetrative Convection in a Brinkman–Forchheimer Fluid Interfacing with a Darcy Fluid

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Abstract

The resonant penetrative convection in a Brinkman–Forchheimer fluid interfacing with a Darcy fluid is considered. It is our main purpose to study the continuous dependence of the solution on changes in the heat source and the continuous dependence of the solution on the Forchheimer coefficient.

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References

  1. Straughan, B.: Resonant porous penetrative convection. Proc. R. Soc. Lond. A 460, 2913–2927 (2004)

    Article  MathSciNet  Google Scholar 

  2. Straughan, B.: Continuous dependence on the heat source and non-linear stability in penetrative convection. Int. J. Non-Linear Mech. 26(2), 221–231 (1991)

    Article  MathSciNet  Google Scholar 

  3. Morro, A., Straughan, B.: Continuous dependence on the source parameters for convective motion in porous media. Nonlinear Anal. Theory Methods Appl. 18(4), 307–315 (1992)

    Article  MathSciNet  Google Scholar 

  4. Straughan, B.: Continuous dependence on the heat source in resonant porous penetrative convection. Stud. Appl. Math. 127(3), 302–314 (2011)

    Article  MathSciNet  Google Scholar 

  5. Gentile, M., Straughan, B.: Structural stability in resonant penetrative convection in a Forchheimer porous material. Nonlinear Anal. Real World Appl. 14(1), 397–401 (2013)

    Article  MathSciNet  Google Scholar 

  6. Ames, K.A., Straughan, B.: Stability and Newton’s law of cooling in double diffusive flow. J. Math. Anal. Appl. 230, 57–69 (1999)

    Article  MathSciNet  Google Scholar 

  7. Ames, K.A., Payne, L.E.: Continuous dependence results for solutions of the Navier-Stokes equations backward in time. Nonlinear Anal. Theory Methods Appl. 23(1), 103–113 (1994)

    Article  MathSciNet  Google Scholar 

  8. Franchi, F., Straughan, B.: Continuous dependence and decay for the Forchheimer equations. Proc. R. Soc. Lond. A 459, 3195–3202 (2003)

    Article  MathSciNet  Google Scholar 

  9. Horgan, L., Ibragimov, A.: Structural stability of generalized Forchheimer equations for compressible fluids in porous media. Nonlinearity 24, 1–41 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. 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  Google Scholar 

  12. 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  Google Scholar 

  13. Liu, Y.: Convergence and continuous dependence for the Brinkman-Forchheimer equations. Math. Comput. Model 49, 1401–1415 (2009)

    Article  MathSciNet  Google Scholar 

  14. Liu, Y.: Continuous dependence for a thermal convection model with temperature-dependent solubility. Appl. Math. Comput. 308, 18–30 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Liu, Y., Xiao, Sh, Lin, Y.W.: Continuous dependence for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded domain. Math. Comput. Simul. 150, 66–82 (2018)

    Article  MathSciNet  Google Scholar 

  16. Liu, Y., Du, Y., Lin, C.H.: Convergence results for Forchheimer’s equations for fluid flow in porous media. J. Math. Fluid Mech. 12, 576–593 (2010)

    Article  MathSciNet  Google Scholar 

  17. Liu, Y., Du, Y., Lin, C.H.: Convergence and continuous dependence results for the Brinkman equations. Appl. Math. Comput. 215, 4443–4455 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Scott, N.L.: Continuous dependence on boundary reaction terms in a porous mediu od Darcy type. J. Math. Anal. Appl. 399, 667–675 (2013)

    Article  MathSciNet  Google Scholar 

  19. Scott, N.L., Straughan, B.: Continuous dependence on the reaction terms in porous convection with surface reactions. Q. Appl. Math. 71, 501–508 (2013)

    Article  MathSciNet  Google Scholar 

  20. Li, Y., Lin, C.: Continuous dependence for the nonhomogeneous Brinkman-Forchheimer equations in a semi-infinite pipe. Appl. Math. Comput. 244, 201–208 (2014)

    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. Roy.Soc.London, A 455, 2173–2190 (1999)

    Article  MathSciNet  Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Payne, L.E., Straughan, B.: Convergence and continuous dependence for the Brinkman-Forchheimer equations. Stud. Appl. Math. 102, 419–439 (1999)

    Article  MathSciNet  Google Scholar 

  25. Straughan, B.: Continuous dependence on the heat source in resonant porous penetrative convection. Stud. Appl. Math. 127, 302–314 (2011)

    Article  MathSciNet  Google Scholar 

  26. McKay, G., Straughan, B.: A nonlinear analysis of convection near the density maximum. Acta Mech. 95, 9–28 (1992)

    Article  MathSciNet  Google Scholar 

  27. Gentile, M., Straughan, B.: Structural stability in resonant penetrative convection in a Forchheimer porous material. Nonlinear Anal. Real World Appl. 14(1), 397–401 (2013)

    Article  MathSciNet  Google Scholar 

  28. Chen, W., Dao, T.A.: On the Cauchy problem for semilinear regularity-loss-type \(\sigma \)-evolution models with memory term. Nonlinear Anal. 59, 103265 (2021)

    Article  MathSciNet  Google Scholar 

  29. Chen, W.: Cauchy problem for thermoelastic plate equations with different damping mechanisms. Commun. Math. Sci. 18(2), 429–457 (2020)

    Article  MathSciNet  Google Scholar 

  30. Li, Y.F., Xiao, S.Z., Zeng, P.: The applications of some basic mathematical inequalities on the convergence of the primitive equations of moist atmosphere. J. Math. Inequal. 15(1), 293–304 (2021)

    Article  Google Scholar 

  31. Li, Y.F., Zhang, S.H., Lin, C.L.: Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain. Boundary Value Probl. 2021, 27 (2021)

    Article  MathSciNet  Google Scholar 

  32. Néel, M.C.: Convection forcée en milieux poreux: écartà la loi de Darcy. C.R. Acad. Sci. Paris, Srie IIb 326, 615–620 (1998)

    Article  Google Scholar 

  33. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection, second ed., in: Applied Mathematical Sciences. 91, Springer-Verlag, NewYork, (2004)

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

    Article  MathSciNet  Google Scholar 

  35. Nield, D.A., Bejan, A.: Convection in Porous Media. Springer-Verlag, NewYork (1992)

    Book  Google Scholar 

  36. Payne, L.E., Rodrigues, J.F., Straughan, B.: Effect of anisotropic permeability on Darcy’s law. Math. Methods Appl. Sci. 24, 427–438 (2001)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

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Authors and Affiliations

  1. Huashang College Guangdong University of Finance & Economics, Guangzhou, 511300, People’s Republic of China

    Yuanfei Li, Xuejiao Chen & Jincheng Shi

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  1. Yuanfei Li
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  2. Xuejiao Chen
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  3. Jincheng Shi
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Correspondence to Yuanfei Li.

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Y. Li: This work is supported by the Key projects of universities in Guangdong Province (NATURAL SCIENCE) (2019KZDXM042)

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Li, Y., Chen, X. & Shi, J. Structural Stability in Resonant Penetrative Convection in a Brinkman–Forchheimer Fluid Interfacing with a Darcy Fluid. Appl Math Optim 84 (Suppl 1), 979–999 (2021). https://doi.org/10.1007/s00245-021-09791-7

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  • DOI: https://doi.org/10.1007/s00245-021-09791-7

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