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On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder

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Abstract

Here the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. At time t=0 the fluid is at rest and the motion is produced by the rotation of the cylinder around its axis. The solutions that have been obtained are presented under series form in terms of the generalized G-functions. The similar solutions for ordinary second grade and Newtonian fluids are obtained from general solution for β→1, respectively, β→1 and α 1→0. Finally, the influences of the pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.

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References

  1. Rajagopal KR (1983) Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech 49:281–285

    Article  MATH  Google Scholar 

  2. Erdogan ME (1995) Plane surface suddenly set in motion in a non-Newtonian fluid. Acta Mech 108:179–187

    Article  MATH  Google Scholar 

  3. Erdogan ME, Imrak CE (2005) Effects of the side walls on starting flows in ducts. Int J Non-Linear Mech 40:107–111

    Article  Google Scholar 

  4. Nadeem S, Asghar S, Hayat T (2008) The Rayleigh Stokes problem for rectangular pipe in Maxwell and second grade fluid. Meccanica 43:495–504

    Article  MATH  MathSciNet  Google Scholar 

  5. Fetecau C, Vieru D, Fetecau C (2008) A note on the second problem of Stokes for Newtonian fluids. Int J Non-Linear Mech 43:451–457

    Article  MATH  Google Scholar 

  6. Rajagopal KR (1984) On the creeping flow of the second order fluid. J Non-Newton Fluid Mech 15:239–246

    Article  MATH  Google Scholar 

  7. Fetecau C, Fetecau C, Zierep J (2002) Decay of potential vortex and propagation of a heat wave in a second grade fluid. Int J Non-Linear Mech 37:1051–1056

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Hayat T, Hameed MI, Asghar S, Siddiqui AM (2004) Some steady MHD flows of the second order fluid. Meccanica 39:345–355

    Article  MATH  MathSciNet  Google Scholar 

  10. Fetecau C, Fetecau C (2005) Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int J Eng Sci 43:781–789

    Article  MATH  MathSciNet  Google Scholar 

  11. Vieru D, Akhtar W, Fetecau C, Fetecau C (2007) Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42:573–583

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Fetecau C, Fetecau C, Vieru D (2007) On some helical flows of Oldroyd-B fluids. Acta Mech 189:53–63

    Article  MATH  Google Scholar 

  14. Bagley RL (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201–210

    Article  MATH  ADS  Google Scholar 

  15. Friedrich C (1991) Relaxation and retardation function of the Maxwell model with fractional derivatives. Rheol Acta 30:151–158

    Article  Google Scholar 

  16. He G, Junqi H, Liu C (1995) General second order fluid flow in pipe. Appl Math Mech 16:767–773

    Google Scholar 

  17. Junqi H, He G, Liu C (1997) Analysis of general second order fluid flow in double cylinder rheometer. Sci China Ser A 40:183–190

    MATH  Google Scholar 

  18. Xu MY, Tan WC (2001) Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Sci China Ser A 44:1387–1399

    Article  MATH  Google Scholar 

  19. Xu MY, Tan WC (2002) The representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solution. Sci China Ser A 32:673–681

    Google Scholar 

  20. Tan WC, Xu MY (2002) Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell Model. Acta Mech Sin 18:342–349

    Article  MathSciNet  Google Scholar 

  21. Tan WC, Xian F, Wei L (2002) An exact solution of Couette flow of generalized second grade fluid. Chin Sci Bull 47:1783–1785

    Article  MathSciNet  Google Scholar 

  22. Tan WC, Pan WX, Xu MY (2003) A note on unsteady flows of a viscoelastic fluid with fractional Maxwell model between two parallel plates. Int J Non-Linear Mech 38:645–650

    Article  MATH  Google Scholar 

  23. Tan WC, Xu MY (2004) Unsteady flows of a generalized second grade fluid with fractional derivative model between two parallel plates. Acta Mech Sin 20:471–476

    Article  MathSciNet  Google Scholar 

  24. Bandelli R, Rajagopal KR (1995) Start-up flows of second grade fluids in domains with one finite dimension. Int J Non-Linear Mech 30:817–839

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  27. Tong D, Liu Y (2005) Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe. Int J Eng Sci 43:281–289

    Article  MATH  MathSciNet  Google Scholar 

  28. Lorenzo CF, Hartley TT (1999) Generalized functions for the fractional calculus. NASA/TP-1999-209424/REV1

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Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

    M. Athar, M. Kamran & M. Imran

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  1. M. Athar
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  2. M. Kamran
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  3. M. Imran
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Correspondence to M. Imran.

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Athar, M., Kamran, M. & Imran, M. On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder. Meccanica 47, 603–611 (2012). https://doi.org/10.1007/s11012-010-9373-1

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  • DOI: https://doi.org/10.1007/s11012-010-9373-1

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