Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

  • M. Babaelahi


The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.


homotopy perturbation method second-grade fluid converging channel velocity equation 


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  1. 1.
    P. D. Ariel, “Axisymmetric Flow of a Second Grade Fluid Past a Stretching Sheet,” Int. J. Eng. Sci. 39, 529–553 (2001).CrossRefzbMATHGoogle Scholar
  2. 2.
    J. E. Dunn and R. L. Fosdick, “Thermodynamics, Stability and Boundedness of Fluids of Complexity-2 and Fluids of Second Grade,” Arch. Rational Mech. Anal. 56, 191–252 (1974).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. L. Fosdick and K. R. Rajagopal, “Anomalous Features in the Model of Second Order Fluids,” Arch. Rational Mech. Anal. 70, 145–152 (1979).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. E. Dunn and K. R. Rajagopal, “Fluids of Differential Type: Critical Review and Thermodynamic Analysis,” Int. J. Eng. Sci. 33, 689–729 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. S. Berman, “Laminar Flow in Channels with Porous Walls,” J. Appl. Phys. 24, 1232–1235 (1953).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. J. Choi, Z. Rusak, and J. A. Tichy, “Maxwell Fluid Suction Flow in a Channel,” J. Non-Newtonian Fluid Mech. 85, 165–187 (1999).CrossRefzbMATHGoogle Scholar
  7. 7.
    L. Rosenhead, Laminar Boundary Layers (Clarendon Press, Oxford, 1963), pp. 250–251.zbMATHGoogle Scholar
  8. 8.
    R. M. Terrill, “Slow Laminar Flow in a Converging or Diverging Channel with Suction at One Wall and Blowing at the Other Wall,” J. Appl. Math. Phys. 16, 306–308 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. S. Roy and P. Nayak, “Steady 2 Dimensional Incompressible Laminar Visco Elastic Flow in a Converging or Diverging Channel with Suction and Injection,” Acta Mech. 43, 129–136 (1982).CrossRefzbMATHGoogle Scholar
  10. 10.
    R. T. Balmer and J. J. Kauzlarich, “Similarity Solutions for Converging or Diverging Steady Flow of Non-Newtonian Elastic Power Law Fluids with Wall Suction or Injection,” AIChE J. 17, 1181–1188 (1971).CrossRefzbMATHGoogle Scholar
  11. 11.
    Y. Öztürk, A. Akyatan, and E. Senocak, “Slow Flow of the Reiner–Rivlin Fluid in a Converging or Diverging Channel with Suction and Injection,” Turk. J. Eng. Environ. Sci. 22, 179–183 (1998).Google Scholar
  12. 12.
    P. Denes and J. D. Finley, “Bäcklund Transformations for General PDE’s,” Physica D, Nonlinear Phenomena 9(1/2), 236–250 (1983).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bäcklund and Darboux Transformations, Ed. by A. Coely (Amer. Math. Soc., Providence, 2001).Google Scholar
  14. 14.
    Xing Biao Hu and Yong-Tang Wu, “Application of the Hirota Bilinear Formalism to a New Integrable Differential-Difference Equation,” Phys. Lett. A 246, 523–529 (1998).ADSCrossRefGoogle Scholar
  15. 15.
    Engui Fan, “Extended Tanh-Function Method and its Applications to Nonlinear Equations,” Phys. Lett. A 277, 212–218 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A.-M. Wazwaz, “A Sine-Cosine Method for Handling Nonlinear Wave Equations,” Math. Comput. Modeling 40, 499–508 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mingliang Wang, Yubin Zhou, and Zhibin Li, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics,” Phys. Lett. A 216, 67–75 (1996).ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Adomian, “Solution of Physical Problems by Decomposition,” Comput. Math. Appl. 27, 145–154 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Appl. Math. Comput. 114, 115–123 (2000).MathSciNetzbMATHGoogle Scholar
  20. 20.
    J. H. He, “Application of Homotopy Perturbation Method to Nonlinear Wave Equations,” Chaos Solitons Fractals 26, 695–700 (2005).ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    J. H. He, “Recent Development of the Homotopy Perturbation Method,” Topolog. Methods Nonlinear Anal. 31 (2), 205–209 (2008).ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    J. H. He, “Homotopy Perturbation Technique,” Comput. Math. Appl. Mech. Eng. 17 (8), 257–262 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. D. Ganji and A. Sadighi, “Application of He’s Homotopy-Perturbation Method to Nonlinear Coupled Systems of Reaction-Diffusion Equations,” Int. J. Nonlinear Sci. Numer. Simulat. 7 (4), 413–420 (2006).CrossRefGoogle Scholar
  24. 24.
    D. D. Ganji and A. Rajabi, “Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations,” Int. Comm. Heat Mass Transfer 33, 391–400 (2006).CrossRefGoogle Scholar
  25. 25.
    M. Rafei and D. D. Ganji, “Explicit Solutions of Helmholtz Equation and Fifth-Order KdV Equation using Homotopy Perturbation Method,” Int. J. Nonlinear Sci. Numer. Simulat. 7, 321–329 (2006).CrossRefGoogle Scholar
  26. 26.
    A. Rajabi, D. D. Ganji, and H. Taherian, “Application of Homotopy-Perturbation Method to Nonlinear Heat Conduction and Convection Equations,” Phys. Lett. A 360 (4/5), 570–573 (2007).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    R. S. Rivlin and J. L. Ericksen, “Stress Deformation Relations for Isotropic Materials,” J. Rational Mech. Anal. 4, 323–425 (1955).MathSciNetzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of QomQomIran

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