Series solution of magneto-hydrodynamic boundary layer flow over bi-directional exponentially stretching surfaces

Technical Paper

Abstract

This study investigated theoretically the problem of three-dimensional, magnetohydrodynamic, boundary layer flow of a Jeffrey fluid with heat transfer in the presence of thermal radiation over an exponentially stretching surface. Highly nonlinear coupled partial differential equations are obtained using boundary layer approach. These equations are reduced to a set of ordinary differential equations using appropriate similarity transformations. The solution of the problem is found with the help of homotopy analysis method along with optimal homotopy analysis method to find optimal/best value for the convergence control parameter appearing in a series solution. The solution behaviors, for different emerging parameters, of velocity profiles (along \(x\) and \(y\) direction) as well as temperature profile are investigated and the effect of these parameters are explained through graphs. Moreover, for the present study, effective Prandtl number is used in the description of temperature profile. The skin friction coefficients along \(x\)-axis and \(y\)-axis are also discussed through graphs. The tabulated values of dimensionless heat transfer coefficient, Nusselt number, is presented.

Keywords

Jeffrey fluid model Three dimensional flow Heat transfer Thermal radiation MHD Similarity solution 

References

  1. 1.
    Sakiadis BC (1961) Boundary layer behavior on continuous solid surface: I boundary layer equations for two dimensional and axi-symmetric flow. AIChE J. 7(1):26–28CrossRefGoogle Scholar
  2. 2.
    Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647CrossRefGoogle Scholar
  3. 3.
    Chen CK, Char M-I (1988) Heat transfer of a continuous stretching surface with suction or blowing. Anal Appl 135(2):568–580MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Mastroberardino A (2014) Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet. Appl Math Mech Engl Ed 35(2):133–142MathSciNetCrossRefGoogle Scholar
  5. 5.
    Erickson LE, Fan LT, Fox VG (1966) Heat and Mass transfer in the laminar boundary layer flow of a moving surface with constant surface velocity and temperature focusing on the effects of suction/injection. Ind Eng Chem Fund 5:19–25CrossRefGoogle Scholar
  6. 6.
    Gurbaka LJ, Bobba KM (1985) Heat transfer characteristics of a continuous stretching surface with variable temperature. J Heat Transf 107(1):248–250CrossRefGoogle Scholar
  7. 7.
    Nadeem S, Hussain ST (2014) Heat transfer analysis of Williamson fluid over exponentially stretching surface. Appl Math Mech Engl Ed 35(4):489–502MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu IC (2005) A note on heat and mass transfer for a hydromagnetic flow over a stretching sheet. Int Commun Heat Mass Transf 32(8):1075–1084CrossRefGoogle Scholar
  9. 9.
    Biliana B, Roslinda N (2009) Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. Eur J Sci Res 33(4):710–717Google Scholar
  10. 10.
    Vajravelu K, Prasad KV, Lee J, Lee C, Pop I, Van Gorder RA (2011) Convective heat transfer in the flow of viscous Ag-water and Cu-water nanofluids over a stretching surface. Int J Therm Sci 50(5):843–851CrossRefGoogle Scholar
  11. 11.
    Jat RN, Chand G (2013) MHD flow and heat transfer over an exponentially stretching sheet with viscous dissipation and radiation effects. Appl Math Sci 7(4):167–180MathSciNetGoogle Scholar
  12. 12.
    Hamad MAA, Ferdows M (2012) Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet. Appl Math Mech Engl Ed 33(7):923–930MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Nadeem S, Lee C (2012) Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res Lett 7:94CrossRefGoogle Scholar
  14. 14.
    Gireesha BJ, Roopa GS, Bagewadi CS (2012) Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet. Appl Math Mech Engl Ed 33(8):1001–1014MathSciNetCrossRefGoogle Scholar
  15. 15.
    Qasim M, Khan I, Shafie S (2013) Heat transfer in a micropolar fluid over a stretching sheet with Newtonian heating. PLoS One 8(4)Google Scholar
  16. 16.
    Nadeem S, Hussain ST, Lee C (2013) Flow of a Williamson fluid over a stretching sheet. Braz J Chem Eng 30(3):619–625CrossRefGoogle Scholar
  17. 17.
    Shehzad SA, Alsaedi A, Hayat T (2013) Hydromagnetic steady flow of Maxwell fluid over a bidirectional stretching surface with prescribed surface temperature and prescribed surface heat flux. PLoS One 8(7):e68139. doi:10.1371/journal.pone.0068139 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nadeem S, Ul Haq R, Akbar NS, Lee C, Khan ZH (2013) Numerical study of boundary layer flow and heat transfer of oldroyd-B nanofluid towards a stretching sheet. PLoS One 8(8):e69811. doi:10.1371/journal.pone.0069811 CrossRefGoogle Scholar
  19. 19.
    Hayat T, Shafiq A, Alsaedi A (2014) Effect of Joule heating and thermal radiation in flow of third grade fluid over radiative surface. PLoS One 9(1):e83153. doi:10.1371/journal.pone.0083153 CrossRefGoogle Scholar
  20. 20.
    Gupta PS, Gupta AS (1977) Heat and mass transfer on a stretching sheet with suction or blowing. Can J Chem Eng 55(6):744–746CrossRefGoogle Scholar
  21. 21.
    Magyari E, Keller B (1999) Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J Phys D Appl Phys 32(5):577–585CrossRefGoogle Scholar
  22. 22.
    Nadeem S, Akram S (2010) Slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of induced magnetic field. Int J Numer Methods Fluids 63(3):374–394MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Nadeem S, Hussain A, Khan M (2010) Stagnation flow of a Jeffrey fluid over a shrinking sheet. M Z Naturforsch C 65(a):540–548Google Scholar
  24. 24.
    Turkyilmazoglu M, Pop I (2013) Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. Int J Heat Mass Trans 57(1):82–88CrossRefGoogle Scholar
  25. 25.
    Qasim M (2013) Heat and mass transfer in a Jeffrey fluid over a stretching sheet with heat source/sink. Alex Eng J 52(4):571–575CrossRefGoogle Scholar
  26. 26.
    Nadeem S, Akram S (2010) Peristaltic flow of a Jeffrey fluid in a rectangular duct. Nonlinear Anal Real World Appl 11(5):4238–4247MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Nadeem S, Abbasbandy S, Hussain M (2009) Series solutions of boundary layer flow of a micropolar fluid near the stagnation point towards a shrinking sheet. Z Naturforsch C 64(a):575–582Google Scholar
  28. 28.
    Nadeem S, Sadiq MA, Choi J, Lee C, Int C (2014) J Nonlinear Sci Numer Simul 15(3–4):171Google Scholar
  29. 29.
    Nadeem S, Zaheer S, Fang T (2011) Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. Numer Algorithms 57(2):187–205MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Liu IC, Wang HH, Peng YF (2012) Heat transfer for three dimensional flow over an exponentially stretching surface. Chem Eng Commun 200(2):253–268CrossRefGoogle Scholar
  31. 31.
    Magyari E, Pantokratoras A (2011) A Note on the effects of thermal radiationin linearized Rosseland approximation on the heat transfer characteristic of various boundary layer flow. Int Commun Heat Mass Trans 38(5):554–556CrossRefGoogle Scholar
  32. 32.
    Magyari E (2010) Comments on mixed convection boundary layer flow over a horizontal plate with thermal radiation by A. Ishak. Heat Mass Trans 46(8–9):809–810CrossRefGoogle Scholar
  33. 33.
    Liao S (2003) Beyond perturbation: introduction to the homotopy analysis method, vol 99. Chapman and Hall/CRCGoogle Scholar
  34. 34.
    Nadeem S, Haq R (2012) MHD boundary layer flow of nanofluid past a porous shrinking sheet with thermal radiation. J Aero Eng. doi:10.1061/(ASCE)AS.1943-5525.0000299
  35. 35.
    Abbasbandy S (2007) The application of homotopy analysis method to solve a generalized Hirota-Satsuma couple KdV equation. Phys Lett A 361(6):478–483MATHCrossRefGoogle Scholar
  36. 36.
    Ellahi R (2009) Effects of the slip boundary condition on non-Newtonian flows in a channel. Commun Nonlinear Sci Numer Simul 14(4):1377–1384CrossRefGoogle Scholar
  37. 37.
    Liao S (2012) Homotopy analysis method in nonlinear differential equations. Springer and Higher Education Press, HeidelbergMATHCrossRefGoogle Scholar
  38. 38.
    Nofal TA (2007) An approximation of the analytical solution of the Jeffery-Hamel flow by homotopy analysis method. Appl Math Sci 5(53):2603–2615MathSciNetGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2015

Authors and Affiliations

  • Sajjad ur-Rehman
    • 1
  • Sohail Nadeem
    • 2
  • Changhoon Lee
    • 1
    • 3
  1. 1.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea
  2. 2.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  3. 3.Department of Mechanical EngineeringYonsei UniversitySeoulKorea

Personalised recommendations