Abstract. The tangential stress and velocity field corresponding to the flow of a generalized Oldroyd-B fluid in an infinite circular cylinder will be determined by mean of Laplace and finite Hankel transform. The motion is produced by the cylinder, that after t = 0 + , begins to rotate about its axis, under the action of oscillating shear stress ΩR sin(ωt) given on boundary. The solutions are based on an important remark regarding the governing equation for the non-trivial shear stress. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The obtained solution will be presented under series form in term of generalized G-function. The similar solutions for the ordinary Oldroyd-B fluid, Maxwell, ordinary Maxwell and Newtonian fluids performing the same motion will be obtained as special cases of our general solutions.