Zeitschrift für angewandte Mathematik und Physik

, Volume 62, Issue 1, pp 161–172 | Cite as

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

Article

Abstract

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.

Mathematics Subject Classification (2000)

76A05 

Keywords

Second grade fluid Shear stress Side walls Exact solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Rajagopal K.R.: On boundary conditions for fluids of the differential type. In: Sequira, A. (eds) Navier-Stokes Equations and Related Non-Linear Problems, pp. 273–278. Plenum Press, New York (1995)Google Scholar
  2. 2.
    Bird R.B.: Useful non-newtonian models. Ann. Rev. Fluid Mech. 8, 13–34 (1976)CrossRefGoogle Scholar
  3. 3.
    Erdogan M.E., Imrak C.E.: On unsteady unidirectional flows of a second grade fluid. Int. J. Non-Linear Mech. 40, 1238–1251 (2005)MATHCrossRefGoogle Scholar
  4. 4.
    Ting T.W.: Certain non-steady flows of second-order fluids. Arch. Rational Mech. Anal. 14, 1–26 (1963)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Rajagopal K.R.: A note on unsteady unidirectional flows of a non-newtonian fluid. Int. J. Non-Linear Mech. 17, 369–373 (1982)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bandelli R., Rajagopal K.R., Galdi G.P.: On some unsteady motions of fluids of second grade. Arch. Mech. 47, 661–676 (1995)MATHMathSciNetGoogle Scholar
  7. 7.
    Bandelli R., Rajagopal K.R.: Start-up flows of second grade fluids in domains with one finite dimension. Int. J. Non-Linear Mech. 30, 817–839 (1995)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hayat T., Asghar S., Siddiqui A.M.: Some unsteady unidirectional flow of a non-newtonian fluid. Int. J. Eng. Sci. 38, 337–346 (2000)CrossRefGoogle Scholar
  9. 9.
    Fetecau C., Zierep J.: On a class of exact solutions of the equations of motion of a second grade flud. Acta Mech. 150, 135–138 (2001)MATHCrossRefGoogle Scholar
  10. 10.
    Erdogan M.E.: On unsteady motions of a second-order fluid over a plane wall. Int. J. Non-Linear Mech. 38, 1045–1051 (2003)MATHCrossRefGoogle Scholar
  11. 11.
    Fetecau C., Corina Fetecau: Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int. J. Eng. Sci. 43, 781–789 (2005)CrossRefGoogle Scholar
  12. 12.
    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)MATHCrossRefGoogle Scholar
  13. 13.
    Fetecau C., Hayat T., Corina Fetecau, 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 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Khan M., Hyder Ali S., 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)MATHCrossRefGoogle Scholar
  15. 15.
    Fetecau C., Kannan K.: A note on an unsteady flow of an Oldroyd-B fluid. Int. J. Math. Math. Sci. 19, 3185–3194 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Corina Fetecau, Fetecau, C., Imran, M.: Axial couette flow of an oldroyd-B fluid due to a time-dependent shear stress. Math. Rep. Vol. 11(61), No.2, 145–154 (2009)Google Scholar
  17. 17.
    Corina Fetecau, Imran, M., Fetecau, C., Burdujan, L.: Helical flow of an oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses. Z. Angew. Math. Phys. (2009). doi:10.1007/s00033-009-0038-7
  18. 18.
    Dunn J.E., Fosdick R.L.: Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Ration. Mech. Anal. 56, 191–252 (1974)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rajagopal K.R.: Flow of viscoelastic fluids between rotating discs. Theor. Comput. Fluid. Dyn. 3, 185–206 (1992)MATHCrossRefGoogle Scholar
  20. 20.
    Dunn J.E., Rajagopal K.R.: Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci. 33, 689–729 (1995)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sneddon I.N.: Functional analysis. Encyclopedia of physics, vol. ll. Springer, Berlin (1955)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Theoretical MechanicsTechnical University of IasiIasiRomania
  2. 2.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan

Personalised recommendations