Abstract: The motion of a viscoelastic incompressible flow of the upper-convected Maxwell fluid, due to the motion of a solid circular cylinder, that is undergoing both a longitudinal and torsional oscillation, is investigated. Expressions for the velocity field, the tangential drag and the work done by the drag force have been obtained, and the corresponding Newtonian cases deduced. Thereafter, some numerical work is done and comparisons made with Newtonian fluids so as to get some insight into the results obtained.