The Use of the Dimensionless Womersley Number to Characterize the Unsteady Nature of Internal Flow

Abstract

Dimensionless numbers are very useful in characterizing mechanical behavior because their magnitude can often be interpreted as the relative importance of competing forces that will influence mechanical behavior in different ways. One dimensionless number, the Womersley number (Wo), is sometimes used to describe the unsteady nature of fluid flow in response to an unsteady pressure gradient; i.e., whether the resulting fluid flow is quasi-steady or not. Fluids surround organisms which themselves contain fluid compartments; the behaviors exhibited by these biologically-important fluids (e.g. air, water, or blood) are physiologically significant because they will determine to a large extent the rates of mass and heat exchange and the force production between an organism and its environment or between different parts of an organism.In the biological literature, the use of the Womersley number is usually confined to a single geometry: the case of flow inside a circular cylinder. We summarize the evidence for a broader role of the Womersley number in characterizing unsteady flow than indicated by this geometrical restriction. For the specific category of internal flow, we show that the exact analytical solution for unsteady flow between two parallel walls predicts the same pattern of fluid behavior identified earlier for flow inside cylinders; i.e., a dichotomy in fluid behavior for values ofWo<1 andWo>1. WhenWo<1, the flow is predicted to faithfully track the oscillating pressure gradient, and the velocity profiles exhibit a parabolic shape such that the fluid oscillating with the greatest amplitude is farthest from the walls (“quasi-steady” behavior). WhenWo>1, the velocity profiles are no longer parabolic, and the flow is phase-shifted in time relative to the oscillating pressure gradient. The amplitude of the oscillating fluid may either increase or decrease asWo>1, as described in the text.