Archive of Applied Mechanics

, Volume 81, Issue 8, pp 1153–1163 | Cite as

RETRACTED ARTICLE: Flow of fractional Maxwell fluid between coaxial cylinders

  • C. Fetecau
  • Corina Fetecau
  • M. Jamil
  • A. Mahmood


This paper deals with the study of unsteady flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t = 0+, is subject to a time-dependent longitudinal shear stress. Velocity field and the adequate shear stress are presented under series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between the three models is underlined by graphical illustrations.


Maxwell fluid Fractional calculus Coaxial cylinders Velocity field Time-dependent shear stress Laplace and Hankel transforms 

List of symbols

V(r, t)

Velocity field

S(r, t)

Extra-stress tensor

v(r, t)

Axial component of velocity field

τ(r, t)

Non-trivial shear stress


The dynamic viscosity


The relaxation time


Kinematic viscosity


Density of the fluid


The fractional parameter


Fractional differential operator

r, t, q


R1, R2

Radii of inner and outer cylinders

a, b, c, d

Real/complex numbers



Ra, b(c, t), Ga, b, c(d, t)

Generalized functions

\({\overline{v}(r,q), \overline{v}_{\rm H}(r_{n}, q)}\)

Laplace and finite Hankel transforms of v(r, t)


Laplace transform of τ(r, t)

vM(r, t), τM(r, t)

Velocity component and shear stress for classical Maxwell fluid

vN(r, t), τN(r, t)

Velocity component and shear stress for classical Newtonian fluid


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Yu Z.S., Lin J.Z.: Numerical research on the coherent structure in the viscoelastic second-order mixing layers. Appl. Math. Mech. 8, 717–723 (1998)Google Scholar
  2. 2.
    Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)MATHGoogle Scholar
  3. 3.
    Drazin P.G., Reid W.H.: Hydromagnetic Stability. Cambridge University Press, Cambridge (1981)Google Scholar
  4. 4.
    Shifang H.: Constitutive Equation and Computational Analytical Theory of Non-Newtonian Fluids. Science Press, Beijing (2000)Google Scholar
  5. 5.
    Ting T.W.: Certain non-steady flows of second-order fluids. Arch. Rational Mech. Anal. 14, 1–23 (1963)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Srivastava P.N.: Non-steady helical flow of a visco-elastic liquid. Arch. Mech. 18, 145–150 (1966)Google Scholar
  7. 7.
    Waters N.D., King M.J.: The Unsteady flow of an elastico-viscous liquid in a straight pipe of ciscular cross-section. J. Phys. D Appl. Phys. 4, 207–211 (1971)CrossRefGoogle Scholar
  8. 8.
    Rajagopal K.R.: Longitudinal and torsional osillations of a rod in a non-Newtonian fluid. Acta Mech. 49, 281–285 (1983)MATHCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Rahman K.D., Ramkisson H.: Unsteady axial viscoelastic pipe flows. J. Non-Newtonian Fluid Mech. 57, 27–38 (1995)CrossRefGoogle Scholar
  11. 11.
    Rajagopal K.R., Bhatnagar R.K.: Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech. 113, 223–239 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wood W.P.: Transient viscoelastic helical flows in pipes of circular and annular cross-section. J. Non-Newtonian Fluid Mech. 100, 115–126 (2001)MATHCrossRefGoogle Scholar
  13. 13.
    Fetecau C.: Analytical solutions for non-Newtonian fluid flow in pipe-like domains. Int. J. Non-linear Mech. 39, 225–231 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hayat T., Khan M., Wang T.: Non-Newtonian flow between concentric cylinders. Commun. Nonlinear Sci. Numer. Simulat. 11, 297–305 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fetecau C., Fetecau C.: Unsteady motion of a Maxwell fluid due to longitudinal and torsional oscillations of an infinite circular cylinder. Proc. Roy. Acad. Ser. A 8, 77–84 (2007)MathSciNetMATHGoogle Scholar
  16. 16.
    Fetecau C., Hayat T., Fetecau C.: Starting solutions for oscillating motions of Oldroyd-B fluids in cylinderical domains. J. Non-Newtonian Fluid Mech. 153, 191–201 (2008)CrossRefGoogle Scholar
  17. 17.
    Bandelli R., Rajagopal K.R., Galdi G.P.: On some unsteady motions of fluids of second grade. Arch. Mech. 47, 661–676 (1995)MathSciNetMATHGoogle Scholar
  18. 18.
    Waters N.D., King M.J.: Unsteady flow of an elastico-viscous liquid. Rheol. Acta. 9, 345–355 (1970)MATHCrossRefGoogle Scholar
  19. 19.
    Erdogan M.E.: On unsteady motion of a second grade fluid over a plane wall. Int. J. Non-linear Mech. 38, 1045–1051 (2003)MATHCrossRefGoogle Scholar
  20. 20.
    Fetecau C., Kannan K.: A note on an unsteady flow of an Oldroyd-B fluid. Int. J. Math. Math. Sci. 19, 3185–3194 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Akhtar W., Jamil M.: On the axial Couette flow of a Maxwell fluid due to longitudinal time dependent shear stress. Bull. Math. Soc. Sci. Roumanie Tome 51, 93–101 (2008)Google Scholar
  22. 22.
    Fetecau C., Fetecau C., Imran M.: Axial Couette flow of an Oldroyd-B fluid due to a time-dependent shear stress. Math. Reports 11, 145–154 (2009)MathSciNetGoogle Scholar
  23. 23.
    Fetecau C., Awan A.U., Fetecau C.: Taylor–Couette flow of an Oldroyd-B fluid in a circular cylinder subject to a time-dependent rotation. Bull. Math. Soc. Sci. Math. Roumanie Tom 52, 117–128 (2009)MathSciNetGoogle Scholar
  24. 24.
    Fetecau C., Imran M., Fetecau C., Burdujan I.: Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses. Z. Angew. Math. Phys. 61, 959–969 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Tong D., Wang R., Yang H.: Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe. Sci. China Ser. G 48, 485–495 (2005)CrossRefGoogle Scholar
  26. 26.
    Tong D., Liu Y.: Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe. Int. J. Eng. Sci. 43, 281–289 (2005)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Fetecau C., Mahmood A., Fetecau C., Vieru D.: Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder. Comput. Math. Appl. 56, 3096–3108 (2008)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Wang S., Xu M.: Axial Coutte flow of two kinds of fractional viscoelastic fluids in an annulus. Nonlinear Anal. Real World Appl. 10(2), 1087–1096 (2009)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Qi H., Jin H.: Unsteady helical flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlinear Anal. Real World Appl. 10, 2700–2708 (2009)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Khan M., Hyder A.S., Qi H.: Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders. Nonlinear Anal. Real World Appl. c 10(3), 1775–1783 (2009)MATHCrossRefGoogle Scholar
  31. 31.
    Athar M., Kamran M., Fetecau C.: Taylor-Couette flow of a generalized second grade fluid due to a constant couple. Nonlinear Anal. Model. Control 15, 3–13 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Fetecau C., Mahmood A., Jamil M.: Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress. Commun. Nonlinear Sci. Numer. Simulat. 15, 3931–3938 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Shah S.H.A.M., Qi H.T.: Starting solutions for a viscoelastic fluid with fractional Burgers’ model in an annular pipe. Nonlinear Anal. Real World Appl. 11, 547–554 (2010)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Heibig A., Palade L.I.: On the rest state stability of an objective fractional derivative viscoelastic fluid model. J. Math. Phys. 49, 043101–043122 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Friedrich C.: Relaxation and retardation functions of a Maxwell model with fractional derivatives. Rheol. Acta 30, 151–158 (1991)CrossRefGoogle Scholar
  36. 36.
    Germant A.: On fractional differentials. Philosophical Magazine 25, 540–549 (1938)Google Scholar
  37. 37.
    Bagley R.L., Torvik P.J.: A theoretical basis for the applications of fractional calculus to viscoelasticity. J. Reheol. 27, 201–210 (1983)MATHCrossRefGoogle Scholar
  38. 38.
    Makris M., Dargush G.F., Constantinou M.C.: Dynamic analysis of generalized viscoelastic fluids. J. Eng. Mech. 119, 1663–1679 (1993)CrossRefGoogle Scholar
  39. 39.
    Fetecau C., Fetecau Corina, Vieru D.: On some helical flows of Oldroyd-B fluids. Acta Mech. 189, 53–63 (2007)MATHCrossRefGoogle Scholar
  40. 40.
    Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)MATHGoogle Scholar
  41. 41.
    Podlubny I.: Fractional Differential Equations. Academic press, San Diego (1999)MATHGoogle Scholar
  42. 42.
    Lorenzo, C.F., Hartley, T.T.: Generalized Functions for the Fractional Calculus. NASA/TP-1999-209424 (1999)Google Scholar
  43. 43.
    Debnath L., Bhatta D.: Integral Transforms and Their Applications. 2nd edn. Chapman & Hall/CRC, New York (2007)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • C. Fetecau
    • 1
    • 2
  • Corina Fetecau
    • 3
  • M. Jamil
    • 1
    • 4
  • A. Mahmood
    • 1
    • 5
  1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  2. 2.Department of MathematicsTechnical University of IasiIasiRomania
  3. 3.Department of Theoretical MechanicsTechnical University of IasiIasiRomania
  4. 4.Department of MathematicsNED University of Engineering and TechnologyKarachiPakistan
  5. 5.Department of MathematicsCOMSATS Institute of Information TechnologyLahorePakistan

Personalised recommendations