A similarity transform is used to analyse the flow of an upper-convected Maxwell fluid in an infinitely long cylinder whose surface has a velocity that increases in magnitude linearly with axial coordinate. Two types of problem are considered, the accelerated surface flow - when the surface velocity is outward towards the tube ends, and the decelerated surface flow - when it is inward. For the accelerated surface flow, the introduction of elasticity prevents the loss of similarity solution that occurs without elasticity at a Reynolds number (Re) of 10.25: with elasticity, solutions up to a Reynolds number of 95 were computed. As elasticity is introduced, normal stress gradients in an elastic boundary layer near the accelerated surface help offset inertially generated negative axial pressure gradients; with sufficient elasticity the turning point in the non-elastic solution family at Re = 10.25 disappears. For the decelerated surface flow, solutions could not be computed beyond a critical Re that depends on the level of elasticity considered, because at this critical Re, the axial velocity profile at the centreline becomes infinitely blunt.