Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder

The helical flow of an Oldroyd-B fluid with fractional derivatives, also named generalized Oldroyd-B fluid, in an infinite circular cylinder is studied using Hankel and Laplace transforms. The motion is due to the cylinder that, at time t=0⁺ begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Vt. The components of the velocity field and the resulting shear stresses are presented under integral and series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions, and are presented as sums of two terms, one of them being a similar solution for a Newtonian fluid. Similar solutions for generalized Maxwell fluids, as well as those for ordinary Oldroyd-B and Maxwell fluids are obtained as limiting cases of our general solutions. Furthermore, the solutions for Newtonian fluids performing the same motion, are also obtained as special cases of our solutions for α=β=1 and λ_r→λ.