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International Journal of Differential Equations
Volume 2011 (2011), Article ID 193813, 19 pages
doi:10.1155/2011/193813
Research Article
Slip Effects on Fractional Viscoelastic Fluids
1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan
3Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
Received 23 May 2011; Accepted 7 September 2011
Academic Editor: Wen Chen
Copyright © 2011 Muhammad Jamil and Najeeb Alam Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- C. Fetecau, M. Khan, C. Fetecau, and H. T. Qi, “Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a suddenly moved plate between two side walls perpendicular to the plate,” Proceeding of the Romanian academy, Series A, vol. 11, pp. 3–10, 2010.
- J. Zierep and C. Fetecau, “Energetic balance for the Rayleigh-Stokes problem of a Maxwell fluid,” International Journal of Engineering Science, vol. 45, no. 2–8, pp. 617–627, 2007. View at Publisher · View at Google Scholar · View at Scopus
- T. Hayat, A. H. Kara, and E. Momoniat, “Travelling wave solutions to Stokes' problem for a fourth grade fluid,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1613–1619, 2009. View at Publisher · View at Google Scholar · View at Scopus
- L. Zheng, F. Zhao, and X. Zhang, “Exact solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3744–3751, 2010. View at Publisher · View at Google Scholar · View at Scopus
- T. Hayat, S. Nadeem, and S. Asghar, “Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 153–161, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- R. S. Lakes, Viscoelastic Solids, CRC Press, Boca Raton, Fla, USA, 1999.
- R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, NewYork, NY, USA, 1999.
- D. Craiem, F. J. Rojo, J. M. Atienza, R. L. Armentano, and G. V. Guinea, “Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries,” Physics in Medicine and Biology, vol. 53, no. 17, pp. 4543–4554, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
- D. Craiem and R. L. Armentano, “A fractional derivative model to describe arterial viscoelasticity,” Biorheology, vol. 44, no. 4, pp. 251–263, 2007. View at Scopus
- M. Khan, S. Hyder Ali, C. Fetecau, and H. Qi, “Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2526–2533, 2009. View at Publisher · View at Google Scholar · View at Scopus
- A. Mahmood, S. Parveen, A. Ara, and N. A. Khan, “Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3309–3319, 2009. View at Publisher · View at Google Scholar
- S. Wang and M. Xu, “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1087–1096, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- C. Fetecau, M. Jamil, C. Fetecau, and D. Vieru, “The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 60, no. 5, pp. 921–933, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- C. Fetecau, A. Mahmood, and M. Jamil, “Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3931–3938, 2010. View at Publisher · View at Google Scholar
- L. C. Zheng, K. N. Wang, and Y. T. Gao, “Unsteady flow and heat transfer of a generalized Maxwell fluid due to a hyperbolic sine accelerating plate,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2209–2212, 2011. View at Publisher · View at Google Scholar
- C. Fetecau, M. Jamil, and A. Mahmood, “Flow of fractional Maxwell fluid betweencoaxial cylinders,” Archive of Applied Mechanics, vol. 81, pp. 1153–1163, 2011.
- M. Jamil, A. Rauf, A. A. Zafar, and N. A. Khan, “New exact analytical solutions for Stokes firstproblem of Maxwell fluid with fractional derivative approach,” Computers & Mathematics with Applications, vol. 62, pp. 1013–1023, 2011.
- M. Jamil, N. A. Khan, and A. A. Zafar, “Translational flows of an Oldroyd-B fluid with fractional derivatives,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1540–1553, 2011. View at Publisher · View at Google Scholar
- A. Heibig and L. I. Palade, “On the rest state stability of an objective fractional derivative viscoelastic fluid model,” Journal of Mathematical Physics, vol. 49, no. 4, Article ID 043101, 2008. View at Publisher · View at Google Scholar
- C. Friedrich, “Relaxation and retardation functions of the Maxwell model with fractional derivatives,” Rheologica Acta, vol. 30, no. 2, pp. 151–158, 1991. View at Publisher · View at Google Scholar · View at Scopus
- A. Germant, “On fractional differentials,” Philosophical Magazine, vol. 25, pp. 540–549, 1938.
- R. L. Bagley and P. J. Torvik, “A theoretical basis for the applications of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- N. Makris, G. F. Dargush, and M. C. Constantinou, “Dynamic analysis of generalized viscoelastic fluids,” Journal of Engineering Mechanics, vol. 119, no. 8, pp. 1663–1679, 1993. View at Scopus
- L. I. Palade, P. Attané, R. R. Huilgol, and B. Mena, “Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models,” International Journal of Engineering Science, vol. 37, no. 3, pp. 315–329, 1999. View at Scopus
- C. Derek, D. C. Tretheway, and C. D. Meinhart, “A generating mechanism for apparent fluid slip in hydrophobic microchannels,” Physics of Fluids, vol. 16, no. 5, article 1509, 7 pages, 2004. View at Publisher · View at Google Scholar
- G. V. Vinogradov and L. I. Ivanova, “Wall slippage and elastic turbulence of polymers in the rubbery state,” Rheologica Acta, vol. 7, no. 3, pp. 243–254, 1968. View at Publisher · View at Google Scholar · View at Scopus
- S. Luk, R. Mutharasan, and D. Apelian, “Experimental observations of wall slip: tube and packed bed flow,” Industrial and Engineering Chemistry Research, vol. 26, no. 8, pp. 1609–1616, 1987. View at Scopus
- K. B. Migler, H. Hervet, and L. Leger, “Slip transition of a polymer melt under shear stress,” Physical Review Letters, vol. 70, no. 3, pp. 287–290, 1993. View at Publisher · View at Google Scholar · View at Scopus
- S. G. Hatzikiriakos and J. M. Dealy, “Wall slip of molten high density polyethylenes. II. Capillary rheometer studies,” Journal of Rheology, vol. 36, pp. 703–741, 1992.
- A. R. A. Khaled and K. Vafai, “The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions,” International Journal of Non-Linear Mechanics, vol. 39, no. 5, pp. 795–809, 2004. View at Publisher · View at Google Scholar · View at Scopus
- T. Hayat, M. Khan, and M. Ayub, “The effect of the slip condition on flows of an Oldroyd 6-constant fluid,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 402–413, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- M. Khan, “Partial slip effects on the oscillatory flows of a fractional Jeffrey fluid in a porous medium,” Journal of Porous Media, vol. 10, no. 5, pp. 473–487, 2007. View at Publisher · View at Google Scholar · View at Scopus
- R. Ellahi, T. Hayat, and F. M. Mahomed, “Generalized couette flow of a third-grade fluid with slip: the exact solutions,” Zeitschrift fur Naturforschung—Section A, vol. 65, no. 12, pp. 1071–1076, 2010. View at Scopus
- T. Hayat, S. Najam, M. Sajid, M. Ayub, and S. Mesloub, “On exact solutions for oscillatory flows in a generalized Burgers fluid with slip condition,” Zeitschrift fur Naturforschung—Section A, vol. 65, no. 5, pp. 381–391, 2010. View at Scopus
- T. Hayat, S. Zaib, C. Fetecau, and C. Fetecau, “Flows in a fractional generalized Burgers' fluid,” Journal of Porous Media, vol. 13, no. 8, pp. 725–739, 2010. View at Publisher · View at Google Scholar · View at Scopus
- L. Zheng, Y. Liu, and X. Zhang, “Slip effects on MHD flow of a generalized Oldroyd-B fluid withfractional derivative,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 513–523, 2012.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- F. Mainardi, Frcational Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathemtical Models, Imperial College Press, London, UK, 2010.
- I. C. Christov, “Stokes' first problem for some non-Newtonian fluids: results and mistakes,” Mechanics Research Communications, vol. 37, no. 8, pp. 717–723, 2010. View at Publisher · View at Google Scholar · View at Scopus
- A. M. Mathai, R. K. Saxena, and H. J. Haubold, “Solutions of certain fractional kinetic equations and a fractional diffusion equation,” Journal of Mathematical Physics, vol. 51, no. 10, Article ID 103506, 2010. View at Publisher · View at Google Scholar
- L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman & Hall/CRC, 2nd edition, 2007.
- W. Tan and M. Xu, “Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model,” Acta Mechanica Sinica, vol. 18, no. 4, pp. 342–349, 2002. View at Scopus
- H. Qi and M. Xu, “Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model,” Acta Mechanica Sinica/Lixue Xuebao, vol. 23, no. 5, pp. 463–469, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- W. C. Tan and M. Y. Xu, “The impulsive motion of flat plate in a generalized second grade fluid,” Mechanics Research Communications, vol. 29, no. 1, pp. 3–9, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus