Abstract
This paper concerned with the unsteady rotational flow of fractional Oldroyd-B fluid, between two infinite coaxial circular cylinders. To solve the problem we used the finite Hankel and Laplace transforms. The motion is produced by the inner cylinder that, at time t=0+, is subject to a time-dependent rotational shear. The solutions that have been obtained, presented under series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade, and Newtonian fluids, performing the same motion, are obtained as limiting cases of general solutions.
The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of a fractional second grade fluid as limiting cases of general solutions corresponding to the fractional Oldroyd-B fluid, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G functions, in contrast with (Imran and Zamra in Commun. Nonlinear Sci. Numer. Simul. 16:226–238, 2011) in which the expression for the velocity field involving the convolution product as well as the integral of the product of the generalized G functions.
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References
Imran S, Zamra S (2011) Exact solutions for the unsteady axial flow of non-Newtonian fluids through a circular cylinder. Commun Nonlinear Sci Numer Simul 16:226–238
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
Hayat T, Siddiqui AM, Asghar A (2001) Some simple flows of an Oldroyd-B fluid. Int J Eng Sci 39:135–147
Fetecau C, Fetecau C (2003) The first problem of Stokes for an Oldroyd-B fluid. Int J Non-Linear Mech 38:1539–1544
Hayat T, Khan M, Ayub M (2004) Exact solutions of flow problems of an Oldroyd-B fluid. Appl Math Comput 151:105–119
Fetecau C, Fetecau C (2005) Unsteady flows of Oldroyd-B fluids in a channel of rectangular cross-section. Int J Non-Linear Mech 40:1214–1219
Tan WC, Masuoka T (2005) Stokes first problem for an Oldroyd-B fluid in a porous half space. Phys Fluids 17:023101
Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge
Ting TW (1963) Certain non-steady flows of second-order fluids. Arch Ration Mech Anal 14:1–26
Srivastava PN (1966) Non-steady helical flow of a visco-elastic liquid. Arch Mech Stosow 18:145–150
Waters ND, King MJ (1971) The unsteady flow of an elasto-viscous liquid in a straight pipe of circular cross section. J Phys D, Appl Phys 4:204–211
Fetecau C, Fetecau C (1985) On the unsteady of some helical flows of a second grade fluid. Acta Mech 57:247–252
Nadeem S, Asghar S, Hayat T (2008) The Rayleigh Stokes problem for rectangular pipe in Maxwell and second grade fluid. Meccanica 43:495–504
Wood WP (2001) Transient viscoelastic helical flows in pipes of circular and annular cross-section. J Non-Newton Fluid Mech 100:115–126
Fetecau C, Fetecau C, Vieru D (2007) On some helical flows of Oldroyd-B fluids. Acta Mech 189:53–63
Tong D, Shan L (2009) Exact solutions for generalized Burgers fluid in an annular pipe. Meccanica 44:427–431
Friedrich CHR (1991) Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta 30(2):151–158
Shah SHAM (2010) Some helical flows of a Burgers fluid with fractional derivative. Meccanica 45:143–151
Kamran M, Imran M, Athar M (2010) Exact solutions for the unsteady rotational flow of a generalized second grade fluid through a circular cylinder. Nonlinear Anal Model Control 15(4):437–444
Athar M, Kamran M, Imran M (2010) On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder. Meccanica. doi:10.1007/s11012-010-9373-1
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(11):1387–1399
Song DY, Jiang TQ (1998) Study on the constitutive equation with fractional derivative for the viscoelastic fluids modified Jeffreys model and its application. Rheol Acta 27:512–517
Tan WC, Wenxiao P, Xu MY (2003) A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int J Non-Linear Mech 38(5):645–650
Tan WC, Xu MY (2004) Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates. Acta Mech Sin 20(5):471–476
Fetecau C, Imran M, Fetecau C, Burdujan I (2010) 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
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Lorenzo CF, Hartley TT (1999) Generalized functions for the fractional calculus. NASA/TP-1999-209424/REV1
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
Bandelli R, Rajagopal KR (1995) Start-up flows for second grade fluids in domains with one finite dimension. Int J Non-Linear Mech 30:817–839
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Kamran, M., Imran, M., Athar, M. et al. On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains. Meccanica 47, 573–584 (2012). https://doi.org/10.1007/s11012-011-9467-4
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DOI: https://doi.org/10.1007/s11012-011-9467-4