We now turn to the second main feature of the thoracic aorta: its curvature. The aim is to describe flow near the entrance of a curved tube in the same way that the previous section described flow near the entrance of a straight tube. However, we immediately come up against the major difficulty that the fully developed flow to which the entry flow tends, and which in a straight tube is Poiseuille flow (the mean flow) plus an easily calculated oscillatory component, is very complicated, and even the steady component is not yet completely understood. In the next three sections, therefore, we concentrate on fully developed flow in curved tubes, leaving a discussion of entry flow to §§ 4.4 and 4.5.

The reason why the flow in a curved tube is difficult to calculate lies in the fact that the motion cannot be everywhere parallel to the curved axis of the tube, but transverse (or secondary) components of velocity must be present. This follows because in order for a fluid particle to travel in a curved path of radius R with speed w it must be acted on by a lateral force (provided by the pressure gradients in the fluid) to give it a lateral acceleration w2/R. Now the pressure gradient acting on all particles will be approximately uniform, but the velocity of those particles near the wall will be much lower than that of particles in the core, as a result of the no-slip condition.