High Reynolds number flow in tubes of complex geometry with application to wall shear stress in arteries

The arterial systems of mammals contain many generations of branching, curved, elastic tubes, and the flow in them is unsteady. The flow details are important in determining the distribution of wall shear stress in arteries, a major factor in atherogenesis. The purpose of this paper is to point out that unsteady flow at realistic values of the Reynolds number (several hundred) and frequency parameter is much more complicated than a knowledge of steady flow in simpler geometries would suggest. Three-dimensional effects in curved, branched or indented tubes introduce secondary motions and horseshoe vortices, under which the wall shear can remain high for a considerable distance downstream. Unsteady flow in two-dimensional, non-uniform tubes can lead to regions of flow separation far from the non-uniformity, again associated with high wall shear, not low shear as for two-dimensional, steady, separated flow. Moreover, both two- and three-dimensional flows can be non-unique, so that the same driving pressures in the same geometry can lead to different flow patterns, depending on the initial conditions. There is great sensitivity to small geometric and temporal perturbations, making it very difficult to predict flow in one arterial bifurcation from a knowledge of another, or even at a particular site in one subject from a knowledge of another. Vessel elasticity is an additional complicating factor whose effects remain to be assessed in detail.