On accelerated flows of a viscoelastic fluid with the fractional Burgers’ model

Abstract

In this work, we consider the accelerated flows for a viscoelastic fluid governed by the fractional Burgers’ model. The velocity field of the flow is described by a fractional partial differential equation. By using the Fourier sine transform and the fractional Laplace transform, the exact solutions for the velocity distribution are obtained for the following two problems: (i) flow induced by constantly accelerating plate, and (ii) flow induced by variable accelerated plate. These solutions, presented under integral and series forms in terms of the generalized Mittag–Leffler function, are presented as the sum of two terms. The first terms represent the velocity field corresponding to a Newtonian fluid performing the same motion, and the second terms give the non-Newtonian contributions to the general solutions. The similar solutions for second grade, Maxwell and Oldroyd-B fluids with fractional derivatives as well as those for the ordinary models, are obtained as the limiting cases of our solutions. Moreover, in the special case when α = β = 1 , as it was to be expected, our solutions tend to the similar solutions for an ordinary Burgers’ fluid.