Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders Authors A. Mahmood Department of Mathematics COMSATS Institute of Information Technology C. Fetecau Department of Mathematics Technical University of Iasi N. A. Khan Department of Mathematics University of Karachi M. Jamil Department of Mathematics COMSATS Institute of Information Technology Abdus Salam School of Mathematical Sciences GC University Research Paper

First Online: 26 June 2010 Received: 16 September 2008 Revised: 08 November 2009 Accepted: 11 January 2010 DOI :
10.1007/s10409-010-0353-4

Cite this article as: Mahmood, A., Fetecau, C., Khan, N.A. et al. Acta Mech Sin (2010) 26: 541. doi:10.1007/s10409-010-0353-4
Abstract The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0^{+} both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω_{1} and Ω_{2} . The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.

Keywords Generalized second grade fluid Velocity field Shear stress Longitudinal oscillatory flow Laplace and Hankel transforms

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