Skip to main content
SpringerLink
Log in

Shifted Legendre Collocation Method for the Flow and Heat Transfer due to a Stretching Sheet Embedded in a Porous Medium with Variable Thickness, Variable Thermal Conductivity and Thermal Radiation

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

This paper is devoted to introduce a numerical simulation with a theoretical study for flow of a Newtonian fluid over an impermeable stretching sheet which embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing PDEs are transformed into a system of coupled non-linear ODEs which are using appropriate boundary conditions for various physical parameters. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, thermal conductivity parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. Comparison of obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. The results attained in this paper confirm the idea that proposed method is powerful mathematical tool and it can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

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. Crane L.J.: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645–647 (1970)

    Article  Google Scholar 

  2. Gupta P.S., Gupta A.S.: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55(6), 744–746 (1979)

    Article  Google Scholar 

  3. Soundalgekar V.M., Ramana T.V.: Heat transfer past a continuous moving plate with variable temperature. Warme-Und Stoffuber Tragung. 14, 91–93 (1980)

    Article  Google Scholar 

  4. Grubka L.J., Bobba K.M.: Heat transfer characteristics of a continuous stretching surface with variable temperature. J. Heat Transf. 107, 248–250 (1985)

    Article  Google Scholar 

  5. Chen C.K., Char M.: Heat transfer on a continuous stretching surface with suction or blowing. J. Math. Anal. Appl. 35, 568–580 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hayat T., Abbas Z., Javed T.: Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Phys. Lett. A 372, 637–647 (2008)

    Article  MATH  Google Scholar 

  7. Cheng P., Minkowycz W.J.: Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res. 82, 2040–2044 (1977)

    Article  Google Scholar 

  8. Elbashbeshy E.M.A., Bazid M.A.A.: Heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection. Appl. Math. Comp. 158, 799–807 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cortell R.: Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn. Res. 37, 231–245 (2005)

    Article  MATH  Google Scholar 

  10. Hayat T., Abbas Z., Pop I., Asghar S.: Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium. Int. J. Heat Mass Transf. 53, 466–474 (2010)

    Article  MATH  Google Scholar 

  11. Hossain M.A., Alim M.A., Rees D.: The effect of radiation on free convection from a porous vertical plate. Int. J. Heat Mass Transf. 42, 181–191 (1999)

    Article  MATH  Google Scholar 

  12. Elbashbeshy E.M.A., Dimian M.F.: Effects of radiation on the flow and heat transfer over a wedge with variable viscosity. Appl. Math. Comput. 132, 445–454 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Abel M.S., Mahesha N.: Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl. Math. Model. 32, 1965–1983 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bataller R.C.: Radiation effects in the Blasius flow. Appl. Math. Comput. 198, 333–338 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Fang T., Zhang J., Zhong Y.: Boundary layer flow over a stretching sheet with variable thickness. Appl. Math. Comput. 218, 7241–7252 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Al-Housseiny T.T., Stone H.A.: On boundary-layer flows induced by the motion of stretching surfaces. J. Fluid Mech. 706, 597–606 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fang T., Tarek M.A.E.L.: Self-similar flow due to the stretching of a deformable fiber. Eur. Phys. J. Plus 129, 252 (2014)

    Article  Google Scholar 

  18. Khader M.M.: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 16, 2535–2542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khader M.M., Hendy A.S.: The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudo-spectral method. Int. J. Pure Appl. Math. 74(3), 287–297 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Khader M.M., Sweilam N.H., Mahdy A.M.S.: Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM. Appl. Math. Inf. Sci. 7(5), 2013–2020 (2013)

    Article  MathSciNet  Google Scholar 

  21. Bell, W.W.: Special Functions for Scientists and Engineers. Great Britain, Butler and Tanner Ltd, Frome and London (1968)

  22. Sweilam N.H., Khader M.M.: A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations. ANZIAM 51, 464–475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sweilam N.H., Khader M.M., Mahdy A.M.S.: Numerical studies for fractional-order Logistic differential equation with two different delays. J. Appl. Math. 2012, 1–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Khader M.M., Hendy A.S.: A new Legendre computational matrix method: An application for solving the high order fractional differential equations. Walailak J. Sci. Technol. 11(4), 289–305 (2014)

    Google Scholar 

  25. Raptis A.: Flow of a micropolar fluid past a continuously moving plate by the presence of radiation. Int. J. Heat Mass Transf. 41, 2865–2866 (1998)

    Article  MATH  Google Scholar 

  26. Raptis A.: Radiation and viscoelastic flow. Int. Commun. Heat Mass Transf. 26, 889–895 (1999)

    Article  Google Scholar 

  27. Chiam T.C.: Magnetohydrodynamic heat transfer over a non-isothermal stretching sheet. Acta Mech. 122, 169–179 (1997)

    Article  MATH  Google Scholar 

  28. Mahmoud M.A.A., Megahed A.M.: MHD flow and heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet with variable fluid properties. Can. J. Phys. 87, 1065–1071 (2009)

    Article  Google Scholar 

  29. Das K.: A mathematical model on magnetohydrodynamic slip flow and heat transfer over a non-linear stretching sheet. Therm. Sci. 18(2), S475–S488 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    M. M. Khader

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

    M. M. Khader

Authors
  1. M. M. Khader
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Khader.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khader, M.M. Shifted Legendre Collocation Method for the Flow and Heat Transfer due to a Stretching Sheet Embedded in a Porous Medium with Variable Thickness, Variable Thermal Conductivity and Thermal Radiation. Mediterr. J. Math. 13, 2319–2336 (2016). https://doi.org/10.1007/s00009-015-0594-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-015-0594-3

Mathematics Subject Classification

Keywords

Search

Navigation