THE USE OF THE DIMENSIONLESS WOMERSLEY NUMBER TO CHARACTERIZE THE UNSTEADY NATURE OF INTERNAL FLOW

Dimensionless numbers are very useful in characterizing mechanical beh avior because their magnitude can often be interpreted as the relative importance of competing forces that will influence mechanical behavio r 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 th emselves contain fluid compartments; the behaviors exhibited by these biologically-important fluids (e.g. air, water, or blood) are physiolo gically significant because they will determine to a large extent the rates of mass and heat exchange and the force production between an or ganism and its environment or between different parts of an organism. In the biological literature, the use of the Womersley number is usual ly confined to a single geometry: the case of how inside a circular cy linder. We summarize the evidence for a broader role of the Womersley number in characterizing unsteady how than indicated by this geometric al restriction. For the specific category of internal flow, we show th at the exact analytical solution for unsteady flow between two paralle l walls predicts the same pattern of fluid behavior identified earlier for flow inside cylinders; i.e., a dichotomy in fluid behavior for va lues of Wo < 1 and Wo > 1. When Wo < 1, the flow is predicted to faith fully track the oscillating pressure gradient, and the velocity profil es exhibit a parabolic shape such that the fluid oscillating with the greatest amplitude is farthest from the walls (''quasi-steady'' behavi or). When Wo > 1, the velocity profiles are no longer parabolic, and t he flow is phase-shifted in time relative to the oscillating pressure gradient. The amplitude of the oscillating fluid may either increase o r decrease as Wo > 1, as described in the text. (C) 1998 Academic Pres s Limited.