Arterial bends: the development and decay of helical flows

The macrocirculation is modelled by incompressible Newtonian flow through a rigid network of pipes for which possible simplifications are discussed. The common assumptions of two-dimensionality or axisymmetry can be generalised to helical symmetry, and in the first part of the paper, the three-dimensionality of arterial bends is considered by varying the curvature and torsion of a section of a helical pipe. The torsion is found to impart a preferential twist to the cross-sectional flow. This loss of symmetry ensures that flow separation is less severe for a helical bend than for a toroidal bend. The effects of variations in body size are examined using allometric scaling laws. In the second part of the paper, the approach to "fully developed" Dean or Womersley flow is considered in an attempt to quantify the regions of validity of idealised models. A perturbation approach, akin to hydrodynamic stability theory, is used. It is argued that often potential flows are more suitable for describing the rapid interactions between geometry and pulsatility rather than the eventual fully developed state so that, for example, the first 100 degrees of the aortic arch may be considered irrotational. Helical potential flows are found to develop faster than the corresponding toroidal flows, but slower than those in a straight pipe. The presence of vorticity in the core also retards the development of symmetric flows. It is concluded that while idealised flows can occur at some points in the body, in general experimental observation is needed to justify their use. Particular caution is recommended when interpreting calculations with Poiseuille input.