The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this Liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed.