Exact solutions for some oscillating motions of a fractional Burgers' fluid

Abstract

This article presents some starting solutions corresponding to the oscillating flows of a Burgers' fluid with fractional derivatives. The fractional calculus approach in the constitutive relationship model is introduced and a fractional Burgers' model is built. The exact solutions for the oscillating motions of a fractional Burgers' fluid due to cosine and sine oscillations of an infinite flat plate as well as those corresponding to an oscillating pressure gradient are established with the help of integral transforms (Fourier sine and Laplace transforms). In order to avoid lengthy calculations of residues and contour integrals, the discrete Laplace transform method has been used. The expressions for the velocity field and the resulting shear stress that have been obtained, presented under integral and series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. Similar solutions for generalized Oldroyd-B, Maxwell and second grade fluids as well as those for ordinary Oldroyd-B, Maxwell, second grade and Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the obtained solutions are graphically analyzed for variations of interesting flow parameters. © 2009 Elsevier Ltd. All rights reserved.