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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 908386, 23 pages
http://dx.doi.org/10.5402/2012/908386
Research Article

Oscillating Flows of Fractionalized Second Grade Fluid

1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Department of Mathematics, NED University of Engineering & Technology, Karachi 75270, Pakistan
3Department of Mathematics, University of Karachi, Karachi 75270, Pakistan

Received 22 October 2011; Accepted 14 November 2011

Academic Editors: F. Ardalan and M. Rasetti

Copyright © 2012 Muhammad Jamil et al. 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.

Abstract

New exact solutions for the motion of a fractionalized (this word is suitable when fractional derivative is used in constitutive or governing equations) second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of Laplace and finite Hankel transforms. These solutions are presented in series form in term of generalized 𝐺 π‘Ž , 𝑏 , 𝑐 ( β‹… , 𝑑 ) functions and satisfy all imposed initial and boundary conditions. In special cases, solutions for ordinary second grade and Newtonian fluids are obtained. Furthermore, other equivalent forms of solutions for ordinary second grade and Newtonian fluids are presented and written as sum of steady-state and transient solutions. The solutions for Newtonian fluid coincide with the well-known classical solutions. Finally, by means of graphical illustrations, the influence of pertinent parameters on fluid motion as well as comparison among different models is discussed.