RETRACTED ARTICLE: Flow of fractional Maxwell fluid between coaxial cylinders
 C. Fetecau,
 Corina Fetecau,
 M. Jamil,
 A. Mahmood
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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 timedependent 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.
 Yu Z.S., Lin J.Z.: Numerical research on the coherent structure in the viscoelastic secondorder mixing layers. Appl. Math. Mech. 8, 717–723 (1998)
 Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)
 Drazin P.G., Reid W.H.: Hydromagnetic Stability. Cambridge University Press, Cambridge (1981)
 Shifang H.: Constitutive Equation and Computational Analytical Theory of NonNewtonian Fluids. Science Press, Beijing (2000)
 Ting T.W.: Certain nonsteady flows of secondorder fluids. Arch. Rational Mech. Anal. 14, 1–23 (1963) CrossRef
 Srivastava P.N.: Nonsteady helical flow of a viscoelastic liquid. Arch. Mech. 18, 145–150 (1966)
 Waters N.D., King M.J.: The Unsteady flow of an elasticoviscous liquid in a straight pipe of ciscular crosssection. J. Phys. D Appl. Phys. 4, 207–211 (1971) CrossRef
 Rajagopal K.R.: Longitudinal and torsional osillations of a rod in a nonNewtonian fluid. Acta Mech. 49, 281–285 (1983) CrossRef
 Bandelli R., Rajagopal K.R.: Startup flows of second grade fluids in domains with one finite dimension. Int. J. NonLinear Mech. 30, 817–839 (1995) CrossRef
 Rahman K.D., Ramkisson H.: Unsteady axial viscoelastic pipe flows. J. NonNewtonian Fluid Mech. 57, 27–38 (1995) CrossRef
 Rajagopal K.R., Bhatnagar R.K.: Exact solutions for some simple flows of an OldroydB fluid. Acta Mech. 113, 223–239 (1995) CrossRef
 Wood W.P.: Transient viscoelastic helical flows in pipes of circular and annular crosssection. J. NonNewtonian Fluid Mech. 100, 115–126 (2001) CrossRef
 Fetecau C.: Analytical solutions for nonNewtonian fluid flow in pipelike domains. Int. J. Nonlinear Mech. 39, 225–231 (2004) CrossRef
 Hayat T., Khan M., Wang T.: NonNewtonian flow between concentric cylinders. Commun. Nonlinear Sci. Numer. Simulat. 11, 297–305 (2006) CrossRef
 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)
 Fetecau C., Hayat T., Fetecau C.: Starting solutions for oscillating motions of OldroydB fluids in cylinderical domains. J. NonNewtonian Fluid Mech. 153, 191–201 (2008) CrossRef
 Bandelli R., Rajagopal K.R., Galdi G.P.: On some unsteady motions of fluids of second grade. Arch. Mech. 47, 661–676 (1995)
 Waters N.D., King M.J.: Unsteady flow of an elasticoviscous liquid. Rheol. Acta. 9, 345–355 (1970) CrossRef
 Erdogan M.E.: On unsteady motion of a second grade fluid over a plane wall. Int. J. Nonlinear Mech. 38, 1045–1051 (2003) CrossRef
 Fetecau C., Kannan K.: A note on an unsteady flow of an OldroydB fluid. Int. J. Math. Math. Sci. 19, 3185–3194 (2005) CrossRef
 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)
 Fetecau C., Fetecau C., Imran M.: Axial Couette flow of an OldroydB fluid due to a timedependent shear stress. Math. Reports 11, 145–154 (2009)
 Fetecau C., Awan A.U., Fetecau C.: Taylor–Couette flow of an OldroydB fluid in a circular cylinder subject to a timedependent rotation. Bull. Math. Soc. Sci. Math. Roumanie Tom 52, 117–128 (2009)
 Fetecau C., Imran M., Fetecau C., Burdujan I.: Helical flow of an OldroydB fluid due to a circular cylinder subject to timedependent shear stresses. Z. Angew. Math. Phys. 61, 959–969 (2010) CrossRef
 Tong D., Wang R., Yang H.: Exact solutions for the flow of nonNewtonian fluid with fractional derivative in an annular pipe. Sci. China Ser. G 48, 485–495 (2005) CrossRef
 Tong D., Liu Y.: Exact solutions for the unsteady rotational flow of nonNewtonian fluid in an annular pipe. Int. J. Eng. Sci. 43, 281–289 (2005) CrossRef
 Fetecau C., Mahmood A., Fetecau C., Vieru D.: Some exact solutions for the helical flow of a generalized OldroydB fluid in a circular cylinder. Comput. Math. Appl. 56, 3096–3108 (2008) CrossRef
 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) CrossRef
 Qi H., Jin H.: Unsteady helical flow of a generalized OldroydB fluid with fractional derivative. Nonlinear Anal. Real World Appl. 10, 2700–2708 (2009) CrossRef
 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) CrossRef
 Athar M., Kamran M., Fetecau C.: TaylorCouette flow of a generalized second grade fluid due to a constant couple. Nonlinear Anal. Model. Control 15, 3–13 (2010)
 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) CrossRef
 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) CrossRef
 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) CrossRef
 Friedrich C.: Relaxation and retardation functions of a Maxwell model with fractional derivatives. Rheol. Acta 30, 151–158 (1991) CrossRef
 Germant A.: On fractional differentials. Philosophical Magazine 25, 540–549 (1938)
 Bagley R.L., Torvik P.J.: A theoretical basis for the applications of fractional calculus to viscoelasticity. J. Reheol. 27, 201–210 (1983) CrossRef
 Makris M., Dargush G.F., Constantinou M.C.: Dynamic analysis of generalized viscoelastic fluids. J. Eng. Mech. 119, 1663–1679 (1993) CrossRef
 Fetecau C., Fetecau Corina, Vieru D.: On some helical flows of OldroydB fluids. Acta Mech. 189, 53–63 (2007) CrossRef
 Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)
 Podlubny I.: Fractional Differential Equations. Academic press, San Diego (1999)
 Lorenzo, C.F., Hartley, T.T.: Generalized Functions for the Fractional Calculus. NASA/TP1999209424 (1999)
 Debnath L., Bhatta D.: Integral Transforms and Their Applications. 2nd edn. Chapman & Hall/CRC, New York (2007)
 Title
 RETRACTED ARTICLE: Flow of fractional Maxwell fluid between coaxial cylinders
 Journal

Archive of Applied Mechanics
Volume 81, Issue 8 , pp 11531163
 Cover Date
 20110801
 DOI
 10.1007/s004190110536x
 Print ISSN
 09391533
 Online ISSN
 14320681
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Maxwell fluid
 Fractional calculus
 Coaxial cylinders
 Velocity field
 Timedependent shear stress
 Laplace and Hankel transforms
 Industry Sectors
 Authors

 C. Fetecau ^{(1)} ^{(2)}
 Corina Fetecau ^{(3)}
 M. Jamil ^{(1)} ^{(4)}
 A. Mahmood ^{(1)} ^{(5)}
 Author Affiliations

 1. Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
 2. Department of Mathematics, Technical University of Iasi, Iasi, Romania
 3. Department of Theoretical Mechanics, Technical University of Iasi, Iasi, Romania
 4. Department of Mathematics, NED University of Engineering and Technology, Karachi, 75270, Pakistan
 5. Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan