Skip to main content
SpringerLink
Log in

Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Taking into account the slip flow effects, Newtonian heating, and thermal radiation, two-dimensional magnetohydrodynamic (MHD) flows and heat transfer past a permeable stretching sheet are investigated numerically. We use one parameter group transformation to develop similarity transformation. By using the similarity transformation, we transform the governing boundary layer equations along with the boundary conditions into ordinary differential equations with relevant boundary conditions. The obtained ordinary differential equations are solved with the fourth-fifth order Runge-Kutta-Fehlberg method using MAPLE 13. The present paper is compared with a published one. Good agreement is obtained. Numerical results for dimensionless velocity, temperature distributions, skin friction factor, and heat transfer rates are discussed for various values of controlling parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part I: boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal, 7, 26–28 (1961)

    Article  Google Scholar 

  2. Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part II: the boundary-layer on a continuous flat surface. AIChE Journal, 7, 221–225 (1961)

    Article  Google Scholar 

  3. Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part III: the boundary-layer on a continuous cylindrical surface. AIChE Journal, 7, 467–472 (1961)

    Article  Google Scholar 

  4. Crane, L. J. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik, 21, 645–647 (1970)

    Article  Google Scholar 

  5. Chakrabarti, A. and Gupta, A. S. Hydromagnetic flow and heat transfer over a stretching sheet. Quarterly of Applied Mathematics, 37, 73–78 (1977)

    MATH  Google Scholar 

  6. Andersson, H. I. MHD flow of a viscous fluid past a stretching surface. Acta Mechanica, 95, 227–230 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chiam, T. C. Magnetohydrodynamic boundary layer flow due to a continuously moving flat plate. Computers & Mathematics with Applications, 26, 1–7 (1993)

    Article  MATH  Google Scholar 

  8. Vajravelu, K. and Hadjinicolaou, A. Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. International Journal of Engineering Science, 35, 1237–1244 (1997)

    Article  MATH  Google Scholar 

  9. Pop, I. and Na, T. Y. A note on MHD flow over a stretching permeable surface. Mechanics Research Communications, 25, 263–269 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kumari, M. and Nath, G. Flow and heat transfer in a stagnation point flow over a stretching sheet with a magnetic field. Mechanics Research Communications, 26, 469–478 (1999)

    Article  MATH  Google Scholar 

  11. Sajid, M. and Hayat, T. Influence of thermal radiation on the boundary layer flow due to exponentially stretching sheet. International Communications in Heat and Mass Transfer, 35, 347–356 (2008)

    Article  Google Scholar 

  12. Abo-Eldahab, E. M. and El Aziz, M. A. Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. International Journal of Thermal Sciences, 43, 709–719 (2004)

    Article  Google Scholar 

  13. Sahoo, B. Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip. Communications in Nonlinear Science and Numerical Simulation, 15, 602–615 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Datti, P. S., Prasad, K. V., Abel, M. S., and Joshi, A. MHD viscoelastic fluid flow over a nonisothermal stretching sheet. International Journal of Engineering Science, 42, 935–946 (2004)

    Article  MATH  Google Scholar 

  15. Elbashbeshy, E. M. A. Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31, 1951–1954 (1998)

    Article  Google Scholar 

  16. Seddeek, M. A. Heat and mass transfer on a stretching sheet with a magnetic field in a visco-elastic fluid flow through a porous medium with heat source or sink. Computational Materials Science, 38, 781–787 (2007)

    Article  Google Scholar 

  17. Shit, G. C. and Haldar, R. Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet. Appl. Math. Mech. -Engl. Ed., 32, 677–688 (2011) DOI 10.1007/s10483-011-1448-6

    Article  MathSciNet  MATH  Google Scholar 

  18. Vajravelu, K., Prasad, K. V., Sujatha A., and Ng, C. O. MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface. Appl. Math. Mech. -Engl. Ed., 33, 899–910 (2012) DOI 10.1007/s10483-012-1593-8

    Article  MathSciNet  MATH  Google Scholar 

  19. Merkin, J. H. Natural convection boundary-layer flow on a vertical surface with Newtonian heating. International Journal of Heat and Fluid Flow, 15, 392–398 (1994)

    Article  Google Scholar 

  20. Lesnic, D., Ingham, D. B., and Pop, I. Free convection boundary layer flow along a vertical surface in a porous medium with Newtonian heating. International Journal of Heat and Mass Transfer, 42, 2621–2627 (1999)

    Article  MATH  Google Scholar 

  21. Lesnic, D., Ingham, D. B., and Pop, I. Free convection from a horizontal surface in a porous medium with Newtonian heating. Journal of Porous Media, 3, 227–235 (2000)

    Article  MATH  Google Scholar 

  22. Lesnic, D., Ingham, D. B., Pop, I., and Storr, C. Free convection boundary layer flow above a nearly horizontal surface in a porous medium with Newtonian heating. Heat and Mass Transfer, 40, 665–672 (2004)

    Article  Google Scholar 

  23. Pop, I., Lesnic, D., and Ingham, D. B. Asymptotic solutions for the free convection boundarylayer flow along a vertical surface in a porous medium with Newtonian heating. Hybrid Methods in Engineering, 2, 31–40 (2000)

    Article  Google Scholar 

  24. Salleh, M. Z., Nazar, R., and Pop, I. Forced convection boundary layer flow at a forward stagnation point with Newtonian heating. Chemical Engineering Communications, 196, 987–996 (2009)

    Article  Google Scholar 

  25. Salleh, M. Z., Nazar, R., and Pop, I. Mixed convection boundary layer flow about a solid sphere with Newtonian heating. Archives of Mechanics, 62, 283–303 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Salleh, M. Z., Nazar, R., and Pop, I. Modeling of free convection boundary layer flow on a sphere with Newtonian heating. Acta Applicandae Mathematicae, 112, 263–274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Salleh, M. Z., Nazar, R., and Pop, I. Modelling of boundary layer flow and heat transfer over a stretching sheet with Newtonian heating. Journal of the Taiwan Institute of Chemical Engineers, 41, 651–655 (2010)

    Article  Google Scholar 

  28. Salleh, M. Z., Nazar, R., Arifin, N. M., Merkin, J. H., and Pop, I. Forced convection heat transfer over a horizontal circular cylinder with Newtonian heating. Journal of Engineering Mathematics, 69, 101–110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Merkin, J. H., Nazar, R., and Pop, I. The development of forced convection heat transfer near a forward stagnation point with Newtonian heating. Journal of Engineering Mathematics, 74, 53–60 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chaundary, R. C. and Preeti, J. Unsteady free convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating. Romanian Journal of Physics, 51, 911–925 (2006)

    Google Scholar 

  31. Chaundary, R. C. and Preeti, J. An exact solution to the unsteady free convection boundary layer flow past an impulsively started vertical surface with Newtonian heating. Journal of Engineering Physics and Thermophysics, 80, 954–960 (2007)

    Article  Google Scholar 

  32. Beavers, G. S. and Joseph, D. D. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30, 197–207 (1967)

    Google Scholar 

  33. Ariel, P. D. Two dimensional stagnation point flow of an elastico-viscous fluid with partial slip. Zeitschrift für angewandte Mathematik und Mechanik, 88, 320–324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fang, T., Zhang, J., and Yao, S. Slip MHD viscous flow over a stretching sheet—an exact solution. Communications in Nonlinear Science and Numerical Simulation, 14, 3731–3737 (2009)

    Article  Google Scholar 

  35. Aman, F., Ishak, A., and Pop, I. Mixed convection boundary layer flow near stagnation-point on vertical surface with slip. Appl. Math. Mech. -Engl. Ed., 32, 1599–1606 (2011) DOI 10.1007/s10483-011-1526-x

    Article  MathSciNet  MATH  Google Scholar 

  36. Yazdi, M. H., Abdullah, S., Hashim, I., and Sopian, K. Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction. International Journal of Heat and Mass transfer, 54, 3214–3225 (2011)

    Article  MATH  Google Scholar 

  37. Bhattacharyya, K., Mukhopadhyay, S., and Layek, G. C. Slip effects on boundary layer stagnationpoint flow and heat transfer towards a shrinking sheet. International Journal of Heat and Mass Transfer, 54, 308–313 (2011)

    Article  MATH  Google Scholar 

  38. Hansen, A.G. Similarity Analysis of Boundary Layer Problems in Engineering, Prentice Hall, Englewood Cliffs, U. S.A. (1964)

    Google Scholar 

  39. Olver, P. J. Application of Lie Groups to Differential Equations, Springer-Verlag, New York (1986)

    Book  MATH  Google Scholar 

  40. Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer, New York (1989)

    Book  MATH  Google Scholar 

  41. Fan, J. R., Shi, J. M., and Xu, X. Z. Similarity solution of free convective boundary-layer behavior at a stretching surface. Heat and Mass Transfer, 35, 191–196 (1999)

    Article  Google Scholar 

  42. Afify, A. A. Some new exact solutions for MHD aligned creeping flow and heat transfer in second grade fluids by using Lie group analysis. Nonlinear Analysis-Theory Methods & Applications, 70, 3298–3306 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Abdul-Kahar, R., Kandasamy, R., and Muhaimin, I. Scaling group transformation for boundary layer flow of a nanofluid past a porous vertical stretching surface in the presence of chemical reaction with heat radiation. Computers and Fluids, 52, 15–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Afify, A. A. and Elgazery, N. S. Lie group analysis for the effects of chemical reaction on MHD stagnation-point flow of heat and mass transfer towards a heated porous stretching sheet with suction or injection. Nonlinear Analysis: Modelling and Control, 17, 1–15 (2012)

    MathSciNet  MATH  Google Scholar 

  45. Hamad, M. A. A. and Ferdows, M. Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysis. Communication in onlinear Science and Numerical Simulation, 17, 132–140 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rosmila, A. B., Kandasamy, R., and Muhaimin, I. Lie symmetry group transformation for MHD natural convection flow of nanofluid over linearly porous stretching sheet in presence of thermal stratification. Appl. Math. Mech. -Engl. Ed., 33, 593–604 (2012) DOI 10.1007/s10483-012-1573-9

    Article  MathSciNet  Google Scholar 

  47. Uddin, M. J., Khan, W. A., and Ismail, A. I. M. Free convection boundary layer flow from a heated upward facing horizontal flat plate embedded in a porous medium filled by a nanofluid with convective boundary condition. Transport in Porous Media, 92, 867–881 (2012)

    Article  MathSciNet  Google Scholar 

  48. Ferdows, M., Uddin, M. J., and Afify, A. A. Scaling group transformation for MHD boundary layer free convective heat and mass transfer flow past a convectively heated nonlinear radiating stretching sheet. International Journal of Heat and Mass Transfer, 56, 181–187 (2013)

    Article  Google Scholar 

  49. Sparrow, E. M. and Cess, R. D. Radiation Heat Transfer, Hemisphere, Washington, D. C. (1978)

    MATH  Google Scholar 

  50. Helmy, K. A. MHD boundary layer equations for a power-law fluid with variable electric conductivity. Meccanica, 30, 187–200 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tapanidis, T., Tsagas, G., and Mazumdar, H. P. Application of scaling group of transformations to viscoelastic second grade fluid flow. Nonlinear Functional Analysis and Applications, 8, 345–350 (2003)

    MathSciNet  MATH  Google Scholar 

  52. Mukhopadhyay, S. and Layek, G. C. Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink. Meccanica, 47, 863–876 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Cortell, R. Heat and fluid flow due to non-linearly stretching surfaces. Applied Mathematics and Computation, 217, 7564–7572 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. White, R. E. and Subramanian, V. R. Computational Methods in Chemical Engineering with Maple, Springer, New York (2010)

    Book  Google Scholar 

  55. Uddin, M. J., Yusoff, N. H. M., Bég, O. A., and Ismail, A. I. M. Lie group analysis and numerical solutions of boundary layer flow of non-Newtonian nanofluids along a horizontal plate in porous medium with internal heat generation. Physica Scripta, 87, 025401 (2013)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Helwan University, Cairo, 11431, Egypt

    A. A. Afify

  2. American International University-Bangladesh, Dhaka, 1213, Bangladesh

    M. J. Uddin

  3. Department of Mathematics, University of Dhaka, Dhaka, 1000, Bangladesh

    M. Ferdows

Authors
  1. A. A. Afify
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. J. Uddin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Ferdows
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Afify.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Afify, A.A., Uddin, M.J. & Ferdows, M. Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects. Appl. Math. Mech.-Engl. Ed. 35, 1375–1386 (2014). https://doi.org/10.1007/s10483-014-1873-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-014-1873-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation