Advertisement

Nonlinear Radiative Heat Transfer in Blasius and Sakiadis Flows Over a Curved Surface

  • M. NaveedEmail author
  • Z. Abbas
  • M. Sajid
Article

Abstract

This study investigates the heat transfer characteristics for Blasius and Sakiadis flows over a curved surface coiled in a circle of radius R having constant curvature. Effects of thermal radiation are also analyzed for nonlinear Rosseland approximation which is valid for all values of the temperature difference between the fluid and the surface. The considered physical situation is represented by a mathematical model using curvilinear coordinates. Similar solutions of the developed partial differential equations are evaluated numerically using a shooting algorithm. Fluid velocity, skin-friction coefficient, temperature and local Nusselt number are the quantities of interest interpreted for the influence of pertinent parameters. A comparison of the present and the published data for a flat surface validates the obtained numerical solution for the curved geometry.

Keywords

Blasius/Sakiadis flows Curved surface Numerical solution Thermal radiation 

Notes

Acknowledgements

We are thankful to the honorable reviewers for their constructive suggestions.

References

  1. 1.
    T. Altan, S. Oh, H. Gegel, Metal Forming Fundamental and Applications (American Society of Metals, Metals Park, 1979)Google Scholar
  2. 2.
    E.G. Fisher, Extrusion of Plastics (Wiley, New York, 1976)Google Scholar
  3. 3.
    Z. Tadmor, I. Klein, Engineering Principles of Plasticating Extrusion. Polymer Science and Engineering Series (Van Nostrand Reinhold, New York, 1970)Google Scholar
  4. 4.
    R.M. Griffith, Velocity temperature and concentration distribution during the fiber spinning. Ind. Eng. Chem. Fund. 3, 245–250 (1964)CrossRefGoogle Scholar
  5. 5.
    M.V. Karwe, Y. Jaluria, Numerical simulation of thermal transport associated with a continuous moving flat sheet in materials processing. ASME Heat Transf. 113, 612–619 (1991)CrossRefGoogle Scholar
  6. 6.
    H. Blasius, Grenzschicten in Flussigkeitenmitkleinerreibung. Z. Math. Phys. 56, 1–37 (1908)Google Scholar
  7. 7.
    A.M.M. Abussita, A note on a certain boundary-layer equation. Appl. Math. Comput. 64, 73–77 (1994)MathSciNetGoogle Scholar
  8. 8.
    A. Asaithambi, A finite-difference method for the Falkner–Skan equation. Appl. Math. Comput. 92, 135–141 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Wang, A new algorithm for solving classical Blasius equation. Appl. Math. Comput. 157, 1–9 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    R.C. Battaler, Numerical solutions of the classical Blasius flat-plate problem. Appl. Math. Comput. 170, 706–710 (2005)MathSciNetGoogle Scholar
  11. 11.
    T. Fang, F. Guo, C.F. Lee, A note on the extended Blasius equation. Appl. Math. Lett. 19, 613–617 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Fang, C.F.F. Lee, A new solution branch for the Blasius equation: a shrinking sheet problem. Comput. Math. Appl. 56, 3088–3095 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J.H. He, A simple perturbation approach to Blasius equation. Appl. Math. Comput. 140, 217–222 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    J. Zhang, B. Chen, An iterative method for solving the Falkner–Skan equation. Appl. Math. Comput. 210, 215–222 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Naveed, Z. Abbas, M. Sajid, Thermophoresis and Brownian effects on the Blasius flow of a nanofluid due to curved surface with thermal radiation. Eur. Phy. J. Plus 131, 214 (2016)CrossRefGoogle Scholar
  16. 16.
    B.C. Sakiadis, Boundary layer behavior on continuous solid surface: I. The boundary layer on a continuous flat surface. AICHE J. 7, 26–28 (1961)CrossRefGoogle Scholar
  17. 17.
    K. Tsou, E. Sparrow, R.J. Goldstein, Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf. 10, 219–235 (1967)CrossRefGoogle Scholar
  18. 18.
    E. Magyari, The moving plate thermometer. Int. J. Therm. Sci. 47, 1436–1441 (2008)CrossRefGoogle Scholar
  19. 19.
    A. Pantokratoras, The Blasius and Sakiadis flow with variable fluid properties. Heat Mass Transf. 44, 1187–1198 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    A. Pantokratoras, Asymptotic profiles for the Blasius and Sakiadis flows in a Darcy-Brinckman isotropic porous medium either with uniform suction or with zero transverse velocity. Transp. Porous Med. (2008). doi: 10.1007/s11242-008-9255-3
  21. 21.
    R.C. Bataller, Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Appl. Maths. Comput. 206, 832–840 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    R.C. Bataller, Numerical comparison of Blasius and Sakiadis flows. Mathematika 26, 187–196 (2010)MathSciNetGoogle Scholar
  23. 23.
    P.O. Olanrewaju, J.A. Gbadeyan, O.O. Agboola, S.O. Abah, Radiation and viscous dissipation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Int. J. Adv. Sci. Technol. 2, 102–115 (2011)Google Scholar
  24. 24.
    A. Ishak, R. Nazar, I. Pop, Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. Non-linear Anal. Real World Appl. 10, 2909–2913 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, Unsteady MHD boundary layer flow with diffusion and first-order chemical reaction over a permeable stretching sheet with suction or blowing. Chem. Eng. Commun. 200, 379–397 (2013)CrossRefGoogle Scholar
  26. 26.
    A.J. Chamkha, I. Pop, H.S. Takhar, Marangoni mixed convection boundary layer flow. Meccanica 41, 219–232 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M.M. Rashidi, S.A. Mohimanian, S. Abbasbandy, Analytic approximation solutions for heat transfer of a micropolar fluid through a porous medium with radiation. Commun. Non-Linear Sci. Numer. Simul. 16, 1874–1889 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    O.D. Makinde, Similarity solution of hydromagnetic heat and mass transfer over a vertical plate with a convective surface boundary condition. Int. J. Phy. Sci. 5, 700–710 (2010)Google Scholar
  29. 29.
    M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. J. Mol. Liq. 213, 153–161 (2016)CrossRefGoogle Scholar
  30. 30.
    M. Sheikholeslami, M.M. Rashidi, D.D. Ganji, Numerical investigation of magnetic nanofluid forced convective heat transfer in existence of variable magnetic field using two phase method. J. Mol. Liq. 212, 117–126 (2015)CrossRefGoogle Scholar
  31. 31.
    M. Sheikholeslami, H.R. Ashorynejad, P. Rana, Lattice Boltzmann simulation of nanofluid heat transfer enhancement and entropy generation. J. Mol. Liq. 214, 86–95 (2016)CrossRefGoogle Scholar
  32. 32.
    M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of \(Al_2 O_3 \)- water nanofluid considering thermal radiation: A numerical study. Int. J. Heat Mass Transf. 96, 513–524 (2016)CrossRefGoogle Scholar
  33. 33.
    M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field. Int. J. Heat Mass Transf. 92, 339–348 (2016)CrossRefGoogle Scholar
  34. 34.
    A. Raptis, C. Perdikis, H.S. Takhar, Effect of thermal radiation on MHD flow. Appl. Math. Comput. 153, 645–649 (2004)MathSciNetzbMATHGoogle Scholar
  35. 35.
    R.C. Battaler, Radiation effects in the Blasius flow. Appl. Math. Comput. 198, 333–338 (2008)MathSciNetGoogle Scholar
  36. 36.
    R.C. Battaler, A numerical tackling on Sakiadis flow with thermal radiation. Chin. Phys. Lett. 25, 1340–1342 (2008)CrossRefGoogle Scholar
  37. 37.
    A. Pantokratoras, T. Fang, Sakiadis flow with nonlinear Rosseland thermal radiation. Phys. Scr. 87, 015703 (2013)ADSCrossRefGoogle Scholar
  38. 38.
    A. Pantokratoras, T. Fang, Blasius flow with nonlinear Rosseland thermal radiation. Meccanica 49, 1539–1545 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    A. Mushtaq, M. Mustafa, T. Hayat, A. Alsaedi, Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: a numerical study. J. Taiwan Inst. Chem. Eng. 45, 1176–1183 (2014)CrossRefGoogle Scholar
  40. 40.
    M. Naveed, Z. Abbas, M. Sajid, Flow and heat transfer in a semi porous curved channel with radiation and porosity effects. J. Porous Med. 19, 1–11 (2016)CrossRefGoogle Scholar
  41. 41.
    Z. Abbas, M. Naveed, M. Sajid, Nonlinear radiative heat transfer and Hall effects on a viscous fluid in a semi-porous curved channel. AIP Adv. 5, 107124 (2015)ADSCrossRefGoogle Scholar
  42. 42.
    M. Sajid, N. Ali, T. Javed, Z. Abbas, Stretching a curved surface in a viscous fluid. Chin. Phys. Lett. 27, 024703 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Z. Abbas, M. Naveed, M. Sajid, Heat transfer analysis for stretching flow over curved surface with magnetic field. J. Eng. Therm. Phys. 22, 337–345 (2013)CrossRefGoogle Scholar
  44. 44.
    M. Naveed, Z. Abbas, M. Sajid, MHD flow of micropolar fluid due to a curved stretching sheet with thermal radiation. J. Appl. Fluid Mech. 9, 131–138 (2016)CrossRefGoogle Scholar
  45. 45.
    N.C. Rosca, I. Pop, Unsteady boundary layer flow over a permeable curved stretching/shrinking surface. Euro. J. Mech. B/Fluids 51, 61–67 (2015)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    M. Naveed, Z. Abbas, M. Sajid, Hydromagnetic flow over an unsteady curved stretching surface. Eng. Sci. Technol. Int. J. 19, 841–845 (2016)CrossRefGoogle Scholar
  47. 47.
    Z. Abbas, M. Naveed, M. Sajid, Hydromagnetic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation. J. Mol. Liq. 215, 756–762 (2016)CrossRefGoogle Scholar
  48. 48.
    S. Rosseland, Astrophysik und atom-theoretischeGrundlagen (Springer, Berlin, 1931)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceComsats Institute of Information Technology VehariVehariPakistan
  2. 2.Department of MathematicsThe Islamia University of BahawalpurBahawalpurPakistan
  3. 3.Department of Mathematics and StatisticsInternational Islamic UniversityIslamabadPakistan

Personalised recommendations