Viscous flow in collapsible tubes of slowly varying elliptical cross-section

Abstract

This paper is concerned with steady flow in collapsible tubes, such as veins, at fairly low Reynolds number. Lubrication theory is used to calculate the velocity and pressure distribution in an elliptical tube whose cross-sectional area and eccentricity vary slowly and in a given way with longitudinal distance x. The transverse velocity field and the effect of inertia on the primary velocity and pressure distributions are calculated to first order in the relevant small parameter. The results of these calculations are combined with a relationship between transmural pressure and the cross-sectional area at any x which is close to that measured in (large) veins, and are used to predict the pressure and flow in a collapsible tube when a given distribution of external pressure is applied. Different relationships between the tube perimeter and cross-sectional area are examined. The theory is applied to an experiment in which a segment of collapsible tube is supported between two rigid segments, and squeezed; predictions of the relationship between the pressure drop and flow rate are made for various experimental conditions. In particular, when the resistance of the downstream rigid segment is held constant, a range of flow rates is found in which the pressure drop falls as the flow rate is raised; this agrees with experiment.