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In this investigation, the series solutions of mixed convection boundary layer flow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method (HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We offer various verifications of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding flow) but also for small values (opposing flow) of λ and the dual solutions do not obtain in both cases.
PAPER REVISED: 2012-04-26
PAPER ACCEPTED: 2012-04-27
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THERMAL SCIENCE YEAR 2014, VOLUME 18, ISSUE Issue 4, PAGES [1247 - 1258]
  1. Vafai, K., Handbook of Porous Media (Second Edition), Taylor & Francis, USA, 2005
  2. Vafai, K., Porous Media: Applications in Biological Systems and Biotechnology, Taylor & Francis, USA, 2010.
  3. Tan, W. C., Masuoka, T., Stokes' first problem for a second grade fluid in a porous half space with heated boundary, Int. J. Nonlinear Mech. 40 (2005), pp. 512-522
  4. Tan, W. C., Masuoka, T., Stability analysis of a Maxwell fluid in a porous medium heated from below. Phys. Lett. A, 360 (2007), pp. 454-460
  5. Fetecau, C., Hayat, T., MHD transient flows in a channel of rectangular cross-section with porous medium, Physics letters A, 369 (2007). pp. 44-54
  6. Hayat, T., Mambili-Mamboundou, H., Mahomed, F. M., Unsteady Solutions in a Third-Grade Fluid Filling the Porous Space, Mathematical Problems in Engineering, 2008 (2008), pp. 139560-13
  7. Pop, I., Ingham, D. B., Convective Heat Transfer, Pergamon, Amsterdam, 2001
  8. Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixed convection boundary-layer flow, J. Eng. Math. 22 (1988), pp. 73-92
  9. Ridha, A., Aiding flows non-unique similarity solutions of mixed convection boundary-layer equations, J. Appl. Math. Phys. 47 (1996), pp. 341-352
  10. Ishak, A., Nazar, R., Pop, I., The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder, Int. J. Non-Linear Mech. 42 (2007), 1010-1017
  11. Liao, S. J., Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press, 2003
  12. Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady boundary layer flow of a micropolar fluid near the forward stagnation point of a plane surface, Acta Mech. 184 (2006), pp. 87-101.
  13. Hayat, T., Ellahi, R., Asghar, S., Modelling of flow and heat transfer in a generalized second grade fluid, International Journal of Applied Mechan-ics. & Engineering, 13 (2008), pp. 101-121
  14. Ellahi, R., Ariel, P. D., Hayat, T., Asghar, S., Effect of heat transfer on a third grade fluid in a channel, International Journal of Numerical Method in Fluids, 63 (2010), pp. 847-859
  15. Ellahi, R., Riaz, A., Analytical solution for MHD flow in a third grade fluid with variable viscosity, Mathematical and Computer Modelling, 52 (2010), pp. 1783-1793
  16. Ellahi, R., Afzal, S., Effect of variable viscosity in a third grade fluid with porous medium. An analytical solution. Commun Nonlinear Sci Numer Simulation 14 (2009), 2056-2072
  17. Nadeem, S., Hayat, T., Abbasbandy, S., Ali, M., Effects of partial slip on a fourth-grade fluid with variable viscosity: An analytic solution, Nonlinear Analysis: Real World Applications, 11 (2010), pp. 856-868
  18. Abbasbandy, S., Yürüsoy, M., Pakdemirli, M., The analysis approach of boundary layer equations of power-law fluids of second grade, Z. Natur-forsch. A 63(a) (2008) 564-570.
  19. J. Cheng, S. J. Liao, R. N. Mohapatra, K. Vajravelu, Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. Math. Anal. Appl. 343 (2008), pp. 233-245
  20. Abbasbandy, S., Homotopy analysis method for generalized Benjamin-Bona-Mahony equation, Z. Angew. Math. Phys. (ZAMP) 59 (2008), pp. 51-62.
  21. Ellahi, R., Abbasbandy, S., Hayat, T., Zeeshan, A., On comparison of series and numerical solutions for second Painlevé equation, Numerical Methods for Partial Differential Equations, 26 (2010), pp. 1070-1078
  22. Abbasbandy, S., Hayat, T., M. Mahomed, F. M., Ellahi, R., On comparison of exact and series solutions for thin film flow of a third-grade fluid, International Journal for Numerical Methods in Fluids, 61, (2009), pp. 987-994
  23. Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady free convection flow in the stagnation-point region of a three-dimensional body, Int. J. Therm. Sci. 47 (2008), pp. 600-608
  24. Ellahi, R., Zeeshan, A., A study of pressure distribution of a slider bearing lubricated with second grade fluid, Numerical Methods for Partial Differ- ential Equations, 27 (2011), pp. 1231-1241
  25. Liao, S. J., A general approach to get series solution of non-similarity boundary-layer flows, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp 2144-2159
  26. Ellahi, R., Effects of the slip boundary condition on non-Newtonian flows in a channel, Communication in Nonlinear Science and Numerical Simulations, 14 (2009), pp. 1377-1384
  27. Gebhart, B., Jaluria, Y., Mahajan, R. L., Samakia, B., Buoyancy Induced Flows and Transport, Hemisphere, New York, 1988
  28. Rajagopal, K. R., Ruzicka, M., Srinivasa, A. R., On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996), pp. 1157-1167
  29. Van Gorder, R. A., Vajravelu, K., On the selection of auxiliary functions, operators, and con-vergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 4078-4089
  30. Liao, S. J., Notes on the homotopy analysis: Some de…nitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp. 983-997
  31. Ramachandran, N., Chen, T. S., Armaly, B. F., Mixed convection in stagnation flows adjacent to a vertical surface, ASME J. Heat Transfer 110 (1988), pp. 373-377
  32. Hassanien, I. A., Gorla, R. S., Combined forced and free convection in stagnation flows of micropolar fluids over vertical non-isothermal surface, Int. J. Eng. Sci. 28 (1990), pp. 783-792.
  33. Lok, Y. Y., Amin, N., Pop, I., Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface, Int. J. Therm. Sci. 45 (2006), pp. 1149-1157
  34. Ishak, A., Nazar, R., Pop, I., Dual solutions in mixed convection flow near a stagnation point on a vertical porous plate, Int. J. Thermal Sci. 47 (2008), pp. 417-422.
  35. Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixed-convection boundary-layer flow, J. Eng. Math. 22 (1988), pp. 73-92.

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