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Homotopy Solutions for a Generalized Second-Grade Fluid Past a Porous Plate

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Abstract

The flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate has been studied. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method (HAM). Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results showed that the flow is appreciably influenced by the material and normal stress coefficient. Several results of interest are deduced as the particular cases of the presented analysis.

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References

  1. Bird, R. B., Armstrong, R. C., and Hassager, J., Dynamics of Polymeric Liquids., Vol. 1, Wiley, New York, 1977.

  2. Slattery, J. C., Advanced Transport Phenomena., Cambridge University Press, New York, 1999.

    Google Scholar 

  3. Dunn, J. E. and Fosdick, R. L., ‘Thermodynamics stability, and boundedness of fluids of complexity 2 and fluids of second grade,’ Arch. Rat. Mech. Anal.. 56., 1974, 191–252.

    Article  MathSciNet  Google Scholar 

  4. Dunn, J. E. and Rajagopal, K. R., ‘Fluids of differential type: Critical review and thermodynamic analysis,’ Int. J. Eng. Sci.. 33., 1995, 689–729.

    Article  MathSciNet  Google Scholar 

  5. Fosdick, R. L. and Rajagopal, K. R., ‘Anomalous features in the model of second order fluids,’ Arch. Rat. Mech. Anal.. 70., 1979, 145–152.

    Article  MathSciNet  Google Scholar 

  6. Rajagopal, K. R., ‘A note on unsteady unidirectional flows of a non-Newtonian fluid,’ Int. J. Non-Linear Mech.. 17., 1982, 369–373.

    MATH  MathSciNet  Google Scholar 

  7. Rajagopal, K. R., ‘On the creeping flow of the second grade fluid,’ J. Non-Newtonian Fluid Mech.. 15., 1984, 239–246.

    MATH  Google Scholar 

  8. Rajagopal, K. R., ‘Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid,’ Acta Mech.. 49., 1983, 282–285.

    Article  Google Scholar 

  9. Hayat, T., Asghar, S., and Siddiqui, A. M., ‘On the moment of a plane disk in a non-Newtonian fluid,’ Acta Mech.. 136., 1999, 125–131.

    Article  MathSciNet  Google Scholar 

  10. Hayat, T., Asghar, S., and Siddiqui, A. M., ‘Periodic unsteady flows of a non-Newtonian fluid,’ Acta Mech.. 131., 1998, 169–175.

    Article  MathSciNet  Google Scholar 

  11. Hayat, T., Asghar, S., and Siddiqui, A. M., ‘Some unsteady unidirectional flows of a non-Newtonian fluid,’ Int. J. Eng. Sci.. 38., 2000, 337–346.

    Article  Google Scholar 

  12. Hayat, T., Khan, M., Siddiqui, A. M., and Asghar, S., ‘Transient flows of a second grade fluid,’ Int. J. Non-Linear Mech.. 39., 2004, 1621–1633.

    MathSciNet  Google Scholar 

  13. Benharbit, A. M. and Siddiqui, A. M., ‘Certain solutions of the equations of the planar motion of a second grade fluid for steady and unsteady cases,’ Acta Mech.. 94., 1992, 85–96.

    Article  MathSciNet  Google Scholar 

  14. Rajagopal, K. R. and Gupta, A. S., ‘An exact solution for the flow of a non-Newtonian fluid past an infinite plate,’ Meccanica. 19., 1984, 158–160.

    Article  MathSciNet  Google Scholar 

  15. Bandelli, R. and Rajagopal, K. R., ‘On the falling of objects in non-Newtonian fluids,’ Ann. Ferrara. 39., 1996, 1–18.

    Google Scholar 

  16. Bandelli, R., ‘Unsteady unidirectional flows of second grade fluids in domains with heated boundaries,’ Int. J. Non-Linear Mech.. 30., 1995, 263–269.

    MATH  MathSciNet  Google Scholar 

  17. Fetecau, C. and Zierep, J., ‘On a class of exact solutions of the equations of motion of a second grade fluid,’ Acta Mech.. 150., 2001, 135–138.

    Google Scholar 

  18. Fetecau, C., Fetecau, C., and Zierep, J., ‘Decay of a potential vortex and propagation of a heat wave in second grade fluid,’ Int. J. Non-Linear Mech.. 37., 2002, 1051–1056.

    Google Scholar 

  19. Man, C. S., ‘Nonsteady channel flow of ice as a modified second grade fluid with power law viscosity,’ Arch. Rat. Mech. Anal.. 119., 1992, 35–57.

    Article  MATH  MathSciNet  Google Scholar 

  20. Massoudi, M. and Phuoc, T. X., ‘Fully developed flow of a modified second grade fluid with temperature dependent viscosity,’ Acta Mech.. 150., 2001, 23–37.

    Article  Google Scholar 

  21. Straughan, B., ‘Energy stability in the Benard problem for a fluid of second grade,’ J. Appl. Math. Phys. (ZAMP). 34., 1983, 502–508.

    MATH  MathSciNet  Google Scholar 

  22. Straughan, B., The Energy Method, Stability and Nonlinear Convection., Springer, New York, 1992.

    Google Scholar 

  23. Gupta, G. and Massoudi, M., ‘Flow of a generalized second grade fluid between heated plates,’ Acta Mech.. 99., 1993, 21–33.

    Article  Google Scholar 

  24. Franchi, H. and Straughan, B., ‘Stability and nonexistence results in the generalized theory of a fluid of second grade,’ J. Math. Anal. Appl.. 180., 1993, 122–137.

    Article  MathSciNet  Google Scholar 

  25. Liao, S. J., ‘The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,’ Ph.D. Thesis, Shanghai Jiao Tong University, 1992.

  26. Liao, S. J. and Campo, A., ‘Analytic solutions of the temperature distribution in Blasius viscous flow problems,’ J. Fluid Mech.. 453., 2002, 411–425.

    Article  MathSciNet  Google Scholar 

  27. Liao, S. J., ‘On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet,’ J. Fluid Mech.. 488., 2003, 189–212.

    Article  MATH  MathSciNet  Google Scholar 

  28. Liao, S. J., ‘An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude,’ Int. J. Non-Linear Mech.. 38., 2003, 1173–1183.

    Google Scholar 

  29. Liao, S. J. and Pop, I., ‘Explicit analytic solution for similarity boundary layer equations,’ Int. J. Heat Mass Trans.. 47., 2004, 75–85.

    Article  Google Scholar 

  30. Ayub, M., Rasheed, A., and Hayat, T., ‘Exact flow of a third grade fluid past a porous plate using homotopy analysis method,’ Int. J. Eng. Sci.. 41., 2003, 2091–2103.

    Article  MathSciNet  Google Scholar 

  31. Hayat, T., Khan, M., and Ayub, M., ‘On the explicit analytic solutions of an Oldroyd 6-constant fluid,’ Int. J. Eng. Sci.. 42., 2004, 123–135.

    MathSciNet  Google Scholar 

  32. Liao, S. J., Beyond Perturbation: Introduction to Homotopy Analysis Method., Chapman & Hall/CRC Press, Boca Raton, FL, 2003.

    Google Scholar 

  33. Liao, S. J., ‘On the homotopy analysis method for nonlinear problems,’ Appl. Math. Comput.. 147., 2004, 499–513.

    Article  MATH  MathSciNet  Google Scholar 

  34. Liao, S. J. and Cheung, K. F., ‘Homotopy analysis of nonlinear progressive waves in deep water,’ J. Eng. Math.. 45., 2003, 105–116.

    Article  MathSciNet  Google Scholar 

  35. Hayat, T., Khan, M., and Asghar, S., ‘Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid,’ Acta Mech.. 168., 2004, 213–232.

    Article  Google Scholar 

  36. Rivilin, R. S. and Ericksen, J. L., ‘Stress deformation relations for isotropic materials,’ J. Rat. Mech. Anal.. 4., 1955, 323–425.

    Google Scholar 

  37. Man, C. S., Shields, D. H., Kjartanson, B., and Sun, Q., ‘Creeping of ice as a fluid of complexity 2: The pressuremeter problem,’ in Proceedings 10th Canadian Congress on Applied Mechanics., Vol. 1, H. Rasmussen (ed.), Univ. W. Ontario, London, Ontario, 1985.

  38. Siddiqui, A. M., Hayat, T., and Asghar, S., ‘Periodic flows of a non-Newtonian fluid between two parallel plates,’ Int. J. Non-Linear Mech.. 34., 1999, 895–899.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad, Pakistan

    Tasawar Hayat & Masood Khan

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  1. Tasawar Hayat
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  2. Masood Khan
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Correspondence to Tasawar Hayat.

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Hayat, T., Khan, M. Homotopy Solutions for a Generalized Second-Grade Fluid Past a Porous Plate. Nonlinear Dyn 42, 395–405 (2005). https://doi.org/10.1007/s11071-005-7346-z

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  • DOI: https://doi.org/10.1007/s11071-005-7346-z

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