The influence of geometry on inviscid decay rates in haemodynamic flows

Abstract

Fully developed flows are often used to describe fluid motion in complex geometrical systems, including the human macrocirculation. In fact they may frequently be quite inappropriate even for geometrically simple pipes, owing to the unfeasibly large viscous entry lengths required. Inviscid adjustment to changes in geometry, however, occurs on the lengthscale of the pipe diameter. Inviscid idealizations are therefore more likely to apply in relatively short arterial sections. We aim to quantify the distances involved by calculating the rates of spatial decay for a general disturbance superimposed on an idealized base flow. Both irrotational and rotational base flows are examined, although in the latter case there can exist non-decaying inertial waves, so that an arbitrary inflow need not attain an inviscid state independent of the downstream coordinate. In the rotational case, we therefore restrict attention to those flows which settle down to perturbations of such a state, whereas the potential flows can be regarded as developing from an arbitrary input.We focus on the last surviving mode of decay in simple uniform pipe geometries, in particular a straight pipe, part of a torus, and a helical pipe. In this way we are able to assess the effects of curvature and torsion on the inviscid entry lengths.Principally, it is shown that the rate of decay is fastest in a straight pipe and slowest in a toroidal pipe, with that in a helical pipe somewhere in between. Core vorticity tends to reduce the decay rate. If an idealized flow occurs in a geometrically simple arterial portion, our results determine its domain of validity.