Journal of Applied Mathematics

Linear and Nonlinear Stability Analysis of Double Diffusive Convection in a Maxwell Fluid Saturated Porous Layer with Internal Heat Source

Moli Zhao, Qiangyong Zhang, and Shaowei Wang

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Abstract

The onset of double diffusive convection is investigated in a Maxwell fluid saturated porous layer with internal heat source. The modified Darcy law for the Maxwell fluid is used to model the momentum equation of the system, and the criterion for the onset of the convection is established through the linear and nonlinear stability analyses. The linear analysis is obtained using the normal mode technique, and the nonlinear analysis of the system is studied with the help of truncated representation of Fourier series. The effects of internal Rayleigh number, stress relaxation parameter, normalized porosity, Lewis number, Vadasz number and solute Rayleigh number on the stationary, and oscillatory and weak nonlinear convection of the system are shown numerically and graphically. The effects of various parameters on transient heat and mass transfer are also discussed and presented analytically and graphically.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 489279, 12 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305672

Digital Object Identifier
doi:10.1155/2014/489279

Mathematical Reviews number (MathSciNet)
MR3198383

Zentralblatt MATH identifier
07010652

Citation

Zhao, Moli; Zhang, Qiangyong; Wang, Shaowei. Linear and Nonlinear Stability Analysis of Double Diffusive Convection in a Maxwell Fluid Saturated Porous Layer with Internal Heat Source. J. Appl. Math. 2014 (2014), Article ID 489279, 12 pages. doi:10.1155/2014/489279. https://projecteuclid.org/euclid.jam/1425305672


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References

  • J. Azaiez and M. Sajjadi, “Stability of double-diffusive double-convective miscible displacements in porous media,” Physical Review E, vol. 85, no. 2, Article ID 026306, 2012.
  • N. Rudraiah and P. G. Siddheshwar, “A weak nonlinear stability analysis of double diffusive convection with cross-diffusion in a fluid-saturated porous medium,” Heat and Mass Transfer, vol. 33, no. 4, pp. 287–293, 1998.
  • I. S. Shivakumara, J. Lee, S. Suresh Kumar, and N. Devaraju, “Linear and nonlinear stability of double diffusive convection in a couple stress fluid-saturated porous layer,” Archive of Applied Mechanics, vol. 81, no. 11, pp. 1697–1715, 2011.
  • O. V. Trevisan and A. Bejan, “Combined heat and mass transfer by natural convection in a porous medium,” Advances in Heat Transfer, vol. 20, pp. 315–352, 1990.
  • C. Zhao, B. E. Hobbs, H. B. Mühlhaus, A. Ord, and G. Lin, “Numerical modelling of double diffusion driven reactive flow transport in deformable fluid-saturated porous media with particular consideration of temperature-dependent chemical reaction rates,” Engineering Computations, vol. 17, no. 4, pp. 367–385, 2000.
  • C. Zhao, B. E. Hobbs, A. Ord, S. Peng, H. B. Mühlhaus, and L. Liu, “Double diffusion-driven convective instability of three-dimensional fluid-saturated geological fault zones heated from below,” Mathematical Geology, vol. 37, no. 4, pp. 373–391, 2005.
  • C. Zhao, B. E. Hobbs, A. Ord, M. Kühn, H. B. Mühlhaus, and S. Peng, “Numerical simulation of double-diffusion driven convective flow and rock alteration in three-dimensional fluid-saturated geological fault zones,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 19–22, pp. 2816–2840, 2006.
  • C. Zhao, B. E. Hobbs, and A. Ord, Convective and Advective Heat Transfer in Geological Systems, Springer, Berlin, Germany, 2008.
  • C. Zhao, B. E. Hobbs, and A. Ord, Fundamentals of Computational Geoscience: Numerical Methods and Algorithms, Springer, Berlin, Germany, 2009.
  • D. A. Nield and A. Bejan, Convection in Porous Media, Springer, New York, NY, USA, 3rd edition, 2006.
  • D. B. Ingham and I. Pop, Transport Phenomena in Porous Media, vol. III, Elsevier, Oxford, UK, 2005.
  • K. Vafai, Handbook of Porous Media, Taylor and Francis, London, UK, CRC Press, Boca Raton, Fla, USA, 2005.
  • C. W. Horton and F. T. Rogers Jr., “Convection currents in a porous medium,” Journal of Applied Physics, vol. 16, pp. 367–370, 1945.
  • E. R. Lapwood, “Convection of a fluid in a porous medium,” Proceedings of the Cambridge Philosophical Society, vol. 44, pp. 508–521, 1948.
  • C. Zhao, H. B. Mühlhaus, and B. E. Hobbs, “Finite element analysis of steady-state natural convection problems in fluid-saturated porous media heated from below,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 21, no. 12, pp. 863–881, 1997.
  • C. Zhao, B. E. Hobbs, and H. B. Mühlhaus, “Finite element modelling of temperature gradient driven rock alteration and mineralization in porous rock masses,” Computer Methods in Applied Mechanics and Engineering, vol. 165, no. 1–4, pp. 175–187, 1998.
  • C. Zhao, H. B. Mühlhaus, and B. E. Hobbs, “Effects of geological inhomogeneity on high rayleigh number steady state heat and mass transfer in fluid-saturated porous media heated from below,” Numerical Heat Transfer A, vol. 33, no. 4, pp. 415–431, 1998.
  • C. Zhao, B. E. Hobbs, and H. B. Mühlhaus, “Theoretical and numerical analyses of convective instability in porous media with upward throughflow,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 23, no. 7, pp. 629–646, 1999.
  • C. Zhao, B. E. Hobbs, and H. B. Mühlhaus, “Effects of medium thermoelasticity on high Rayleigh number steady-state heat transfer and mineralization in deformable fluid-saturated porous media heated from below,” Computer Methods in Applied Mechanics and Engineering, vol. 173, no. 1-2, pp. 41–54, 1999.
  • C. Zhao, B. E. Hobbs, H. B. Mühlhaus, A. Ord, and G. Lin, “Finite element modelling of three-dimensional convection problems in pore-fluid saturated porous media heated from below,” Communications in Numerical Methods in Engineering, vol. 17, pp. 101–114, 2001.
  • C. Zhao, B. E. Hobbs, H. B. Mühlhaus, A. Ord, and G. Lin, “Convective instability of 3-D fluid-saturated geological fault zones heated from below,” Geophysical Journal International, vol. 155, no. 1, pp. 213–220, 2003.
  • C. Zhao, B. E. Hobbs, A. Ord, H. B. Mühlhaus, and G. Lin, “Effect of material anisotropy on the onset of convective flow in three-dimensional fluid-saturated faults,” Mathematical Geology, vol. 35, no. 2, pp. 141–154, 2003.
  • C. Zhao, B. E. Hobbs, A. Ord, S. Peng, H. B. Mühlhaus, and L. Liu, “Theoretical investigation of convective instability in inclined and fluid-saturated three-dimensional fault zones,” Tectonophysics, vol. 387, no. 1–4, pp. 47–64, 2004.
  • C. Parthiban and P. R. Patil, “Thermal instability in an anisotropic porous medium with internal heat source and inclined temperature gradient,” International Communications in Heat and Mass Transfer, vol. 24, no. 7, pp. 1049–1058, 1997.
  • B. Straughan and J. Tracey, “Multi-component convection-diffusion with internal heating or cooling,” Acta Mechanica, vol. 133, no. 1–4, pp. 219–238, 1999.
  • E. Magyari, I. Pop, and A. Postelnicu, “Effect of the source term on steady free convection boundary layer flows over a vertical plate in a porous medium. I,” Transport in Porous Media, vol. 67, pp. 49–67, 2007.
  • E. Magyari, I. Pop, and A. Postelnicu, “Effect of the source term on steady free convection boundary layer flows over a vertical plate in a porous medium. II,” Transport in Porous Media, vol. 67, no. 2, pp. 189–201, 2007.
  • A. A. Hill, “Double-diffusive convection in a porous medium with a concentration based internal heat source,” Proceedings of The Royal Society of London A, vol. 461, no. 2054, pp. 561–574, 2005.
  • B. S. Bhadauria, A. Kumar, J. Kumar, N. C. Sacheti, and P. Chandran, “Natural convection in a rotating anisotropic porous layer with internal heat generation,” Transport in Porous Media, vol. 90, no. 2, pp. 687–705, 2011.
  • B. S. Bhadauria, “Double-diffusive convection in a saturated anisotropic porous layer with internal heat source,” Transport in Porous Media, vol. 92, no. 2, pp. 299–320, 2012.
  • S. Wang and W. Tan, “Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below,” Physics Letters A, vol. 372, no. 17, pp. 3046–3050, 2008.
  • S. N. Gaikwad, M. S. Malashetty, and K. Rama Prasad, “An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect,” Applied Mathematical Modelling, vol. 33, no. 9, pp. 3617–3635, 2009.
  • S. N. Gaikwad, M. S. Malashetty, and K. Rama Prasad, “Linear and non-linear double diffusive convection in a fluid-saturated anisotropic porous layer with cross-diffusion effects,” Transport in Porous Media, vol. 80, no. 3, pp. 537–560, 2009.
  • M. C. Kim, S. B. Lee, S. Kim, and B. J. Chung, “Thermal instability of viscoelastic fluids in porous media,” International Journal of Heat and Mass Transfer, vol. 46, no. 26, pp. 5065–5072, 2003.
  • M. S. Malashetty and B. S. Biradar, “The onset of double diffusive convection in a binary Maxwell fluid saturated porous layer with cross-diffusion effects,” Physics of Fluids, vol. 23, no. 6, Article ID 064109, 2011.
  • M. S. Malashetty, W. Tan, and M. Swamy, “The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer,” Physics of Fluids, vol. 21, no. 8, Article ID 084101, 2009. \endinput