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This paper deals with the rotational flow of a fractional Maxwell fluid in an infinite circular cylinder, due to the torsional variable time-dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the constitutive relationship model of a Maxwell fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to ordinary Maxwell fluids as well as those for Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the relaxation time and the fractional parameter on the velocity of the fluid is analyzed by graphical illustrations.
PAPER REVISED: 2011-05-20
PAPER ACCEPTED: 2011-07-11
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THERMAL SCIENCE YEAR 2012, VOLUME 16, ISSUE Issue 2, PAGES [345 - 355]
  1. Dunn, J. E., Rajagopal, K. R., Fluids of differential type: Critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (1995), 5, pp. 689-729.
  2. Rajagopal, K. R., Mechanics of non-Newtonian fluids In: Recent developments in theoretical fluids mechanics. Pireman Research Notes in Mathematics, Vol. 291, Longman, New York, (1993) 129-162
  3. Han, S. F., Constitutive equation and computational analytical theory of non-Newtonian fluids. Science, Beijing 2000
  4. Maxwell, J. C., On the dynamical theory of gases, Philos. Trans. Roy. Soc. Lond. A, 157 (1866), ?, pp. 26-78
  5. Palade, L. I., Attane, P., Huilgol, R. R., Mena, B., Anomalous stability behavior of a properly invariant equation which generalizes fractional derivative models, Int. j. Eng. Sci. 37 (1999),?, pp. 315-329
  6. Rossihin, Y. A., Shitikova, M. V., A new method for solving dynamic problems of fractional derivative viscoelasticity, Int. J. Eng. Sci., 39 (2001), ?, pp. 149-176
  7. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999
  8. Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific Press, Singapore, 2000
  9. 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 (2005),?, pp. 485-495
  10. Tong, D., Liu, Y., Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe, Int. J. Eng. Sci., 43 (2005), ?, pp. 281-289
  11. 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, 52 (2009), 2, pp. 117-128
  12. Fetecau, C., Fetecau, C., Imran, M., Axial Couette flow of an Oldroyd-B fluid due to a time dependent shear stress, Math. Reports, 11 (2009), 2, pp. 145-154
  13. 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, Commu. Nonlinear Sci. Numer. Simu., 15 (2010), 12, pp. 3931-3938
  14. Nazar, M., Fetecau, C., Awan, A. U., A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress, Nonlinear Analysis: Real World Applications, 11 (2010), 4, 2207-2214
  15. Siddique, I., Vieru, D., Exact solution for the longitudinal flow of a generalized second grade fluid in a circular cylinder, Acta Mech. Sin., 25 (2009), ?, pp. 777-785
  16. Wang, S., Xu, M., Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Analysis: Real World Applications, 10 (2009), 2, pp. 1087-1096
  17. Qi, H., Jin, H., Unsteady helical flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Analysis: Real World Applications, 10 (2009), ?, pp. 2700-2708
  18. Athar, M., Kamran, M., Fetecau, C., Taylor-Couette flow of a generalized second grade fluid due to a constant couple, Nonlinear Analysis: Modeling and Control, 15 (2010),?, pp. 3-13
  19. Shah, S. H. A. M., Qi, H., Starting solutions for a viscoelastic fluid with fractional Burgers model in an annular pipe, Nonlinear Analysis: Real World Applications, 11 (2010), 1, pp. 547-554
  20. Heibig, A., Palade, L. I., On the rest state stability of an objective fractional derivative viscoelastic fluid model, J. Math. Phys., 49 (2008),?, pp. 043101-22
  21. Friedrich, C., Relaxation and retardation functions of the Maxwell models with fractional derivatives, Rheol. Acta, 30 (1991), 2, pp. 151-158
  22. Schiessel, H., Fiedrich, C., Blumen, A., Applications to problems in polymer physics and rheology. In: R. Hilfer (Ed), Applications of fractional calculus, NASA/TP-1999-209424/Rev 1, 1999
  23. Lorenzo, C. F., Hartley, T. T., Generalized functions for the fractional Calculus, NASA/TP-1999-209424/Rev1, 1999
  24. Debnath, L., Bhatta, D., Integral Transforms and their applications (second ed.), Chapman and Hall/CRC Press, Boca Raton London New York, 2007
  25. McLachlan, N. W., Bessel Functions for Engineers, Oxford University Press, London, 1995
  26. Abramowitz, M., Stegun, I. A., Handbook of Mathematical Functions, NBS, Appl. Math. Series 55, Washington, D.C, 1964
  27. Ditkin, V., Proudnicov, A., Transformation Integral et Calculus operational, Editions Mir-Moscou, 1987
  28. Fetecau, C., Imran, M., Fetecau, C., Burdujan, I., Helical flow of an Oldroyd-B fluid to a circular cylinder subject to time-dependent shear stress, ZAMP, 61 (2010), 5, pp. 959-969

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