The fluid flow through a tube with an oscillating elliptical cross-section was analyzed in order to understand better the effects of lateral deformation and movement on flow patterns in large and medium-sized blood vessels. Time-dependent elliptical deformation may be caused by external forces, as is the case for the large vessels near the heart, or by lateral movement of the entire compliant vessel, as in the coronary arteries. An analytic, perturbation-type solution was found for the case of fully developed flow in a tube where the cross-section oscillated periodically from an ellipse to a circle. Analytic expressions were found for all three components of the velocity vector. The effects of the deformation on the axial velocity profile included near-wall fluctuations in velocity that depended on the local wall motion. At higher values of the Womersley parameter, these effects were more pronounced. Secondary flow patterns were established that swirled fluid from the center of the vessel to the walls, then back to the center. It was concluded that these phenomena could be important to flow in the largest vessels, but may not be so important in determining flow patterns in the coronary arteries.