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Structural stability for convection models in a reacting porous medium with magnetic field effect

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Abstract

This paper deals with two fundamental models for convection in a reacting porous medium with magnetic field effect. We demonstrate that the solution depends continuously on changes in the chemical reaction and the electrical conductivity coefficients.

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References

  1. 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  MATH  MathSciNet  Google Scholar 

  2. Amili, P., Yortsos, Y.C.: Stability of heat pipes in vapor-dominated system. Int. J. Heat Mass Transfer 47, 1223–1246 (2004)

    Article  Google Scholar 

  3. Capone, F., Gentile, M., Hill, A.A.: Penetrative convection in a fluid layer with throughflow. Ricerche mat. 57, 251–260 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Capone, F., Luca, R.D.: Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law. Int. J. non-linear mech. 57, 799–805 (2012)

    Article  Google Scholar 

  5. Capone, F., Luca, R.D., Torcicollo, I.: Longtime behavior of vertical throughflows for binary mixtures in porous layers. Int. J. non-linear mech. 57, 1–7 (2013)

    Article  Google Scholar 

  6. Cimatti, G.: A class of explicit solutions for the Soret-Dufour boundary value problem in arbitrary domains. Ricerche mat. 59, 199–205 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eltayeb, I.A., Hamza, E.A., Jervase, J.A., Krishnan, E.V., Loper, D.E.: Compositional convection in the presence of a magnetic field. I. A single interface. Proc. Roy. Soc. London A, 460, 3505–3528 (2004)

    Google Scholar 

  8. Eltayeb, I.A., Hamza, E.A., Jervase, J.A., Krishnan, E.V., Loper. D.E.: Compositional convection in the presence of a magnetic field. II. Cartesian plume. Proc. Roy. Soc. London A, 461, 2605–2633 (2005)

    Google Scholar 

  9. Fabrizio, M., Morro, A.: Electromagnetism of continuous media. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  10. Franchi, F., Straughan, B.: A comparison of the Graffi and Kazhikhov-Smagulov models for top heavy pollution instability. Adv Water Resour 24, 585–594 (2001)

    Article  Google Scholar 

  11. Franchi, F., Straughan, B.: 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Galdi, G.P., Joseph, D.D., Preziosi, L., Rionero, S.: Mathematical problems for miscible, incompressible fluids with Korteweg stresses. European J. Mech. B/fluid 10, 269–294 (1991)

    Google Scholar 

  15. Guo, J., Kaloni, P.N.: Double-diffusive convection in porous medium, nonlinear stability. Stud. App. Math., 94, 341–358 (2005)

    Google Scholar 

  16. Harfash, A.J., Straughan, B.: Magnetic effect on instability and nonlinear stability in a reacting fluid. Meccanica. 47, 1849–1857 (2012)

    Article  MathSciNet  Google Scholar 

  17. Harfash, A.J.: Magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid. Continuum Mech. Thermodyn. 25, 89–106 (2013)

    Article  MathSciNet  Google Scholar 

  18. Harfash, A.J.: Continuous dependence on the coefficients for double diffusive convection in Darcy flow with Magnetic field effect. Anal. Math. Phys. doi:10.1007/s13324-013-0053-x

  19. Hayat, T., Nawaz, M.: Unsteady stagnation point flow of viscous fluid caused by an impusively rotating disk. J. Taiwan Inst. Chem. Eng. 42, 41–49 (2011)

    Article  Google Scholar 

  20. Hirsch, M.W., Smale, S.: Differential equation, dynamical systems, and linear algebra. Academic Press, New York (1974)

    Google Scholar 

  21. Kaloni, P.N., Mahajan, A.: Stability of magnetic fluid motions in a saturated porous medium. ZAMP 62, 529–538 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. 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  MATH  MathSciNet  Google Scholar 

  24. 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  MATH  MathSciNet  Google Scholar 

  25. Maehlmann, S., Papageorgiou, D.T.: Interfacial instability in electrified plane Couette flow. J. Fluid Mech. 666, 155–188 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Malashetty, M.S., Biradar, B.S.: The onset of double diffusive reaction–convection in an anisotropic porous layer. Phys. Fluids 23, 064102 (2011)

    Article  Google Scholar 

  27. Mahidjiba, A., Vasseur, P., Bilgen, E.: Multiple solution for double-diffusive convection in porous enclosures. Int. J. Heat Mass Transfer 38, 623–630 (1995)

    Google Scholar 

  28. Mahidjiba, A., Mamou, A., Vasseur, P.: Onest of double-diffusive convection in in rectangular porous cavity subject to mixed boundary conditions. Int. J. Heat Mass Transfer 43, 1505–1522 (2000)

    Article  MATH  Google Scholar 

  29. Nanjundappa, C.E., Ravisha, M., Lee, J., Shivakumara, I.S.: Penetrative ferroconvection in a porous layer. Acta Mechanica 216, 243–257 (2011)

    Article  MATH  Google Scholar 

  30. Nield, D.A., Bejan, A.: Convection in porous media, 2nd edn. Springer, New York (1999)

    Book  MATH  Google Scholar 

  31. 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  MATH  MathSciNet  Google Scholar 

  32. 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  MATH  MathSciNet  Google Scholar 

  33. 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)

    MATH  MathSciNet  Google Scholar 

  34. 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)

    Article  MATH  MathSciNet  Google Scholar 

  35. 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 

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

    Article  MathSciNet  Google Scholar 

  37. Rahman, M.M., Al-Lawatia, M.: Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium. Can. J. Chem. Eng. 88, 23–32 (2010)

    Article  Google Scholar 

  38. Reddy, P.D.S., Bandyopadhyay, D., Joo, S.W., Sharma, A., Qian, S.: Parametric study on instabilities in a two-layer electromagnetohydrodynamic channel flow confined between two parallel electrodes. Phys. Rev. E 83, 036313 (2011)

    Article  Google Scholar 

  39. Rionero, S.: Sulla stabilità asintotica in media in magnetoidrodinamic. Ann. Matem. Pura Appl. 76, 75–92 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  40. Rionero, S.: Sulla stabilità asintotica in media in magnetoidrodinamic non isoterma. Ricerche mat. 16, 250–263 (1967)

    MathSciNet  Google Scholar 

  41. Rionero, S.: Metodi variazionali stabilità asintotica in media in magnetoidrodinamic. Ann. Matem. Pura Appl. 78, 339–364 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  42. Rionero, S.: Sulla stabilità magnetoidrodinamic in media con vari tipi di condizioni al contorno. Ricerche mat. 17, 64–78 (1968)

    MATH  MathSciNet  Google Scholar 

  43. Rionero, S.: Sulla stabilità nonlinear asintotica in media in magnetoidrodinamic. Ricerche mat. 19, 269–285 (1970)

    MATH  MathSciNet  Google Scholar 

  44. Rionero, S.: Sulla stabilità magnetoidrodinamic nonlinear asintotica in media in presenza di effetto hall. Ricerche mat. 20, 285–296 (1971)

    MATH  MathSciNet  Google Scholar 

  45. Rionero, S.: Onset of convection in porous materials with vertically stratified porosity. Acta Mech. 222, 261–272 (2011)

    Article  MATH  Google Scholar 

  46. Rionero, S.: Global non-linear stability in double diffusive convection via hidden symmetries. Int. J. non-linear mech. 47, 61–66 (2012)

    Google Scholar 

  47. Rionero, S.: Global nonlinear stability for a triply diffusive convection in a porous layer. Continuum Mech. Thermodyn. 24, 629–641 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  49. Rionero, S.: Triple diffusive convection in porous media. Acta Mech. 224, 447–458 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  50. Rionero, S., Mulone, G.: Nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method. Arch. Rational Mech. Anal. 103, 347–368 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  51. Rionero, S., Vergori, L.: Long-time behaviour of fluid motions in porous media according to the Brinkman model. Acta Mech. 210, 221–240 (2010)

    Article  MATH  Google Scholar 

  52. Roberts, P.H.: An introduction to magnetohydrodynamics. Longman, London (1967)

    Google Scholar 

  53. Shivakumara, I.S., Ng, C.O., Nagashree, M.S.: The onset of electrothermoconvection in a rotating Brinkman porous layer. Int. J. Eng. Sci. 49, 663–6464 (2011)

    Article  MathSciNet  Google Scholar 

  54. Shivakumara, I.S., Lee, J., Ravisha, M., Raddy, R.G.: Effects of MFD viscosity and LTNE on the onset of ferromagnetic convection in a porous medium. Int. J. Heat Mass Transfer 54, 2630–2641 (2011)

    Article  MATH  Google Scholar 

  55. 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  Google Scholar 

  56. Straughan, B.: The energy method, stability, and nonlinear convection, 2nd edn. Series in Applied Mathematical Sciences, vol. 91. Springer, Berlin (2004)

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

  58. Sharma, S.P. and Mahajan, A.: Onset of Darcy-Brinkman double-diffusive convection in a magnetized ferrofluid layer using a thermal non-equilibrium model: a nonlinear stability analysis. J. Geophys. Eng. 7, 417–430 (2010)

    Google Scholar 

  59. Trevian, O.V., Bejan, A.: Combined Heat and mass transfer by natural convection in a porous midium. Adv. Heat Transfer 28, 1587–1611 (1990)

    Google Scholar 

  60. Xu, L., Yang, S.: Stability analysis of thermosolutal second-order fluid in porous Bénard layer. Ricerche mat. 56, 149–160 (2007)

    Article  MATH  Google Scholar 

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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. The author acknowledges the comments and suggestions of an anonymous referee, which led to deep improvements in the manuscript.

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

    A. J. Harfash

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Harfash, A.J. Structural stability for convection models in a reacting porous medium with magnetic field effect. Ricerche mat. 63, 1–13 (2014). https://doi.org/10.1007/s11587-013-0152-x

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