Unsteady Helical Flow of a Generalized Oldroyd-B Fluid with Fractional Derivative

 International Journal of Mathematical Trends and Technology (IJMTT) © 2014 by IJMTT Journal Volume-5 Year of Publication : 2014 Authors : Yaqing Liu , Fenglei Zong , Jinbin Dai 10.14445/22315373/IJMTT-V5P524

Yaqing Liu , Fenglei Zong , Jinbin Dai Article:Unsteady Helical Flow of a Generalized Oldroyd-B Fluid with Fractional Derivative,International Journal of Mathematical Trends and Technology (IJMTT),V5:67-77: January 2014. Published by Seventh Sense Research Group.

Abstract
This paper considered some unsteady helical flows of a generalized Oldroyd-B fluid between two infinite concentrice cylinders and an infinite circular cylinder. The flow is due to the cylinders oscillate around their common axis and accelerating slide in the direction of the same axis with prescribed velocities. Exact solutions of some unsteady helical flows are obtained by using Laplace transform coupled with Hankel transform for fractional calculus. The corresponding solutions for generalized second grade fluid, Maxwell fluid, ordinary Oldroyd-B fluid or Newtonian fluid are obtained as limiting cases of general solutions. Finally, the influence of the fractional parameters and on the fluid motion is underlined by graphical illustrations.

References

[1] Podlubny I, Fractional differential equations, New York: Academic Press, 1999.
[2] T.Wenchang, X.Feng, W. Lan, “An exact solution of unsteady Couette flow of generalized second grade fluid, ” Chin.Sci.Bull. vol. 47, pp.1226-1228, Nov. 2002.
[3] X.Mingyu, T.Wenchang, “Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion,” Sci.Chin.(Ser.A), vol. 44, pp. 1387-1399, Nov.2001
[4] D.Y.Song, T.Q.Jiang, “Study on the constitutive equation with fractional derivative for the vicoelastic fluid modified Jeffreys model and its application,” Rheol.Acta, vol.27, pp.512-517, Nov. 1998.
[5] C. Fetecau, Corina Fetecau, “Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder,” International Journal of Engineering science, vol.44, pp.788-796, Aug. 2006.
[6] M.Nzazr, Corina Fetecau, A.U.Awan, “A note on the unsteady flow of a generalized second-grade fluid througt a circular cylinder subject to a time dependent shear stress, Nonlinear Analysis,” Real Word Applications , vol. 11, pp. 2207-2214,Aug. 2010.
[7] Shaowei Wang, Mingyu Xu, “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, ” Nonlinear analysis: real world applications, vol.10, pp.1087-1096, Apr. 2009.
[8] Corina Fetecau, T. Hayat, Constantin Fetecau, “Starting solutions for oscillating motions of Oldroyd-B fluids in cylinderical domians,” Journal of non-Newtonian fluid mechanics, vol. 153, pp. 191-201, Aug. 2008. [9] C. Fetecau, A. Mahmood, Corina Fetecau, D. Vieru, “Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder,” Computers and mathematics with applications, vol.56, pp.3096-3108, Dec. 2008.
[10] M. Jamil, C. Fetecau, “Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary,” Nonlinear Analysis: Real World Applications, vol. 11, pp.4302-4311, Oct. 2010.
[11] M. Jamil, C. Fetecau, M. Imran, “Unsteady helical flows of Oldroyd-B fluids, ” Commun Nonlinear Sci Numer Simulat, vol.16, pp.1378-1386, Mar. 2011.
[12] M. Khan, S. Hyder Ali, Haitao Qi, “Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders,” Nonlinear analysis: real world applications, vol.10, pp. 1775-1783, Jun. 2009.
[13] A. Mahmood, S. Parveen, A. Ara, N.A. Khan, “Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model,” Commun nonlinear sci numer simulate, vol.14, pp.3309-3319, Aug. 2009.
[14] Dengke Tong, Yusong Liu, “Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe,” International Journal of Engineering science, vol. 43, pp.281-289, Feb. 2005.
[15] Dengke Tong, Xianmin Zhang, Xinhong Zhang, “Unsteady helical flows of a generalized Oldroyd-B fluid,” Journal of non-newtonian fluid mechanics, vol. 156, pp. 75-83, Jan.2009.
[16] M.Khan, S.Hyder Ali, Haitao Qi, “Some accelerated flows for a generalized Oldroyd-B fluid,” Nonlinear analysis: real world applications, vol. 10, pp.980-991, Apr.2009.
[17] Qi Haitao, Xu Mingyu, “Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative,” Appl.Math. Modelling, vol. 33, pp. 4184-4191, Nov. 2009.
[18] Haitao Qi, Hui Jin, “Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative,” Nonlinear analysis: real world applications, vol. 10, pp.2700-2708, Oct.2009.
[19] I.N.Sneddon, Fourier transforms, New York, Toronto, London, McGraw Hill Book Company, Inc., 1951.

Keywords
Helical fluid, Oldroyd-B fluid, Laplace transform, Hankel transform.