Curvature forms an important feature of thoracic aorta and this paper deals with the flow of an idealized elastico-viscous liquid in a curved pipe of circular cross-section and slowly varying curvature, under a pressure gradient. The flow is assumed to be steady and at low Reynolds numbers. By using the series expansion method of Dean (Phil Mag 4 (1927) 208-223; Phil Mag 5 (1928) 673-693) in powers of a parameter L, which can be considered as the square of ratio of the centrifugal force induced by the circular motion of the fluid to the viscous force, it is shown that in a tube of increasing curvature, there will be delay in setting up of the secondary motion. The wall shear stress, an important parameter in physiological flows, is calculated. The flow of Newtonian fluid in a tube of circular cross section is discussed, as a particular case.