Skip to main content
SpringerLink
Log in

Poiseuille Flow of a Third Grade Fluid in a Porous Medium

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy’s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.

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

  • Abbasbandy S.: Approximate solution of the nonlinear model of diffusion and reaction catalysts by means of the homotopy analysis method. Chem. Eng. J. 136, 144–150 (2008)

    Article  Google Scholar 

  • Abbasbandy S., Hayat T.: Solution of the MHD Falkner-Skan flow by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 3591–3598 (2009)

    Article  Google Scholar 

  • Abbasbandy S., Zakaria F.S.: Soliton solution for the fifth-order Kdv equation with the homotopy analysis method. Nonlinear Dyn. 51, 83–87 (2008)

    Article  Google Scholar 

  • Abbasbandy S., Hayat T., Ellahi R., Asghar S.: Numerical results of a flow in a third grade fluid between two porous walls. Z. Naturforsch 64, 59–64 (2009a)

    Google Scholar 

  • Abbasbandy S., Hayat T., Mahomed F.M., Ellahi R.: On comparison of exact and series solutions for thin film flow of a third-grade fluid. Int. J. Numer. Meth. Fluids 67, 987–994 (2009b)

    Article  Google Scholar 

  • Akgul M.B., Pakdemirli M.: Analytical and numerical solutions of electro-osmotically driven flow of a third grade fluid between micro-parallel plates. Int. J. Non-Linear Mech. 43, 985–992 (2008)

    Article  Google Scholar 

  • Akyildiz F.T.: On the dispersion of a solute in Poiseuille flow of a third grade fluid in a channel. Int. J. Non-Linear Mech. 38, 1553–1559 (2003)

    Article  Google Scholar 

  • Ariel P. D.: On exact solutions of flow problems of a second grade fluid through two parallel porous walls. Int. J. Eng. Sci. 40, 913–941 (2002)

    Article  Google Scholar 

  • Ariel P.D.: Flow of a third grade fluid through a porous flat channel. Int. J. Eng. Sci. 41, 1267–1285 (2003)

    Article  Google Scholar 

  • Bataineh A.S., Noorani M.S.M., Hashim I.: Homotopy analysis method for singular IVPs of Emdan-Flower type. Commun. Nonlinear Sci. Numer. Simul. 14, 1121–1131 (2009a)

    Article  Google Scholar 

  • Bataineh A.S., Noorani M.S.M., Hashim I.: Modified homotopy analysis method for solving systems of second-order BVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 430–442 (2009b)

    Article  Google Scholar 

  • Cheng J., Liao S.J., Mohapatra R.N., Vajravelu K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method. J. Math. Anal. Appl. 343, 233–245 (2008)

    Article  Google Scholar 

  • Erdogan M. E., Imrak C. E.: On the comparison of two different solutions in the form of series of the governing equation of an unsteady flow of a second grade fluid. Int. J. Non-Linear Mech. 40, 545–550 (2005)

    Article  Google Scholar 

  • Erdogan M. E., Imrak C. E.: Effects of the side walls on unsteady flow of a second grade fluid over a plane wall. Int. J. Non-Linear Mech. 43, 779–782 (2008)

    Article  Google Scholar 

  • Feiz-Dizaji A., Salimpour M.R., Jam F.: Flow field of a third-grade non-Newtonian fluid in the annulus of rotating concentric cylinders in the presence of magnetic field. J. Math. Anal. Appl. 337, 632–645 (2008)

    Article  Google Scholar 

  • Fetecau C., Fetecau C.: The Rayleigh-Stokes problem for heated second grade fluids. Int. J. Non-Linear Mech. 37, 1011–1015 (2002)

    Article  Google Scholar 

  • Fetecau C., Fetecau C.: Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Int. J. Eng. Sci. 44, 788–796 (2006)

    Article  Google Scholar 

  • Fetecau C., Hayat T., Fetecau C., Ali N.: Unsteady flow of a second grade fluid between two side walls perpendicular to a plate. Nonlinear Anal. Real World Appl. 9, 1236–1252 (2007)

    Article  Google Scholar 

  • Fosdick R.L., Rajagopal K.R.: Thermodynamics and stability of fluids of third grade. Proc. R. Soc. Lond. A 339, 351–377 (1980)

    Google Scholar 

  • Hayat T., Ellahi R., Asghar S., Siddiqui A.M.: Flow induced by non-coaxial rotation of a porous disk executing non-torsional oscillations and a second grade fluid rotating at infinity. Appl. Math. Model. 28, 591–605 (2004)

    Article  Google Scholar 

  • Hayat T., Khan M., Ayub M.: Some analytical solutions for second grade fluid flows for cylindrical geometries. Math. Comp. Model. 43, 16–29 (2006a)

    Article  Google Scholar 

  • Hayat T., Ellahi R., Ariel P.D., Asghar S.: Homotopy solution for the channel flow of a third grade fluid. Nonlinear Dyn. 45, 55–64 (2006b)

    Article  Google Scholar 

  • Hayat T., Asghar S., Khalique C.M., Ellahi R.: Influence of partial slip on flows of a second grade fluid in a porous medium. J. Porous Media 10, 797–805 (2007)

    Article  Google Scholar 

  • Hayat T., Abbas Z., Asghar S.: Effects of Hall current and heat transfer on rotating flow of a second grade fluid through a porous medium. Commun. Nonlinear Sci. Numer. Simul. 13, 2177–2192 (2008)

    Article  Google Scholar 

  • Hayat T., Mustafa M., Pop I.: Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Commun. Nonlinear Sci. Numer. Simul. 15, 1183–1196 (2009a)

    Article  Google Scholar 

  • Hayat T., Maqbool K., Asghar S.: Hall and heat transfer effects on the steady flow of a Sisko fluid. Z. Naturforsch 64, 769–782 (2009b)

    Google Scholar 

  • Hayat T., Nadeem S., Ellahi R., Asghar S.: The influence of Hall current in a circular duct. Nonlinear Anal. Real World Appl. 11, 184–189 (2010a)

    Article  Google Scholar 

  • Hayat T., Naz R., Sajid M.: On the homotopy solution for Poiseuille flow of a fourth grade fluid. Commun. Nonlinear Sci. Numer. Simul. 15, 581–589 (2010b)

    Article  Google Scholar 

  • 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 (2010c)

    Article  Google Scholar 

  • Khan M., Ali S.H., Hayat T., Fetecau C.: MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium. Int. J. Non-Linear Mech. 43, 302–319 (2008)

    Article  Google Scholar 

  • Liao S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall, CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  • Makinde O.D.: On thermal stability of a reactive third-grade fluid in a channel with convective cooling the walls. Appl. Math. Comp. 213, 170–176 (2009)

    Article  Google Scholar 

  • Malik M.Y., Hussain A., Nadeem S., Hayat T.: Flow of a third grade fluid between coaxial cylinders with variable viscosity. Z. Naturforsch 64, 588–596 (2009)

    Google Scholar 

  • Nadeem S., Hussain A., Malik M. Y., Hayat T.: Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet. Appl. Math. Mech. -Engl. Ed. 30(10), 1255–1262 (2009)

    Article  Google Scholar 

  • Nadeem S., Hayat T., Abbasbandy S., Ali M.: Effects of partial slip on a fourth grade fluid with variable viscosity: an analytic solution. Nonlinear Anal. Real World Appl. 11, 856–868 (2010)

    Article  Google Scholar 

  • Okoya S.S.: On the transition for a generalized Couette flow of a reactive third-grade fluid with viscous dissipation. Int. Commun. Heat Mass Transf. 35, 188–196 (2008)

    Article  Google Scholar 

  • Pakdemirli M., Yilbas B.S.: Entropy generation for pipe flow of a third grade fluid with Vogel model viscosity. Int. J. Non-Linear Mech. 41, 432–437 (2006)

    Article  Google Scholar 

  • Sahoo B.: Hiemenz flow and heat transfer of a third grade fluid. Commun. Nonlinear Sci. Numer. Sim. 14, 811–826 (2009)

    Article  Google Scholar 

  • Sahoo B.: Effects of slip viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet. Appl. Math. Mech. 31, 159–173 (2010a)

    Article  Google Scholar 

  • Sahoo B.: Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip. Commun. Nonlinear Sci. Numer. Simul. 15, 602–615 (2010b)

    Article  Google Scholar 

  • Siddiqui A. M., Haroon T., Irum S.: Torsional flow of third grade fluid using modified homotopy perturbation method. Comp. Math. Appl. 58, 2274–2285 (2009)

    Article  Google Scholar 

  • Tan Y., Liao S.J.: Series solution of three-dimensional unsteady laminar viscous flow due to a stretching surface in a rotating fluid. J. Appl. Mech. 74, 1011–1018 (2007)

    Article  Google Scholar 

  • Tan W. C., Masuoka T.: Stokes’ first problem for a second grade fluid in a porous half space with heated boundary. Int. J. Non-Linear Mech. 40, 515–522 (2005)

    Article  Google Scholar 

  • Vajravelu K., Cannon J.R., Rollins D., Leto J.: On solutions of some non-linear differential equations arising in third grade fluid flows. Int. J. Eng. Sci. 40, 1791–1805 (2002)

    Article  Google Scholar 

  • Wang Y., Wu W.: Time-dependent magnetohydrodynamic flow induced by non-coaxial rotations of a non-torsionally oscillating porous plate and a third-order fluid at infinity. Math. Comput. Model. 46, 1277–1293 (2007)

    Article  Google Scholar 

  • Xu H., Liao S.J., Pop I.: Series solutions of unsteady three dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate. Eur. J. Mech.-B\Fluid. 26, 15–27 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan

    T. Hayat & Rahila Naz

  2. Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    T. Hayat

  3. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, 14515/775, Iran

    S. Abbasbandy

Authors
  1. T. Hayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahila Naz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Abbasbandy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Abbasbandy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hayat, T., Naz, R. & Abbasbandy, S. Poiseuille Flow of a Third Grade Fluid in a Porous Medium. Transp Porous Med 87, 355–366 (2011). https://doi.org/10.1007/s11242-010-9688-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-010-9688-3

Keywords

Search

Navigation