Steady Flow in Collapsible Tubes

Abstract

The one-dimensional theory of steady flow in a thin-walled tube, partially collapsed by a negative transmural pressure difference, is developed in a general way. The mechanics of the flow is closely coupled to the mechanics of the tube. The latter is characterized by a “tube law”: the relationship between cross-sectional area and transmural pressure difference. Features analogous to those in gas dynamics and free-surface flow may manifest themselves: a characteristic wave propagation speed; opposite phenomena at flow speeds, respectively, less than and greater than the wave speed; choking; and shocklike transitions. There are many practical examples of such flows, mainly in physiology and medicine. The one-dimensional, steady analysis includes the effects of friction, lengthwise variations in external pressure, variations in elevation, resting area, wall stiffness, and mechanical properties. The speed index S (ratio of flow speed to wave speed), analogous to the Mach and Froude numbers, appears naturally in the results as a controlling parameter of behavior. Various practical ways of passing continuously from subcritical (S < 1) to supercritical (S > 1)speed are suggested. A preliminary theory of shocklike, dissipative transitions is developed, the results of which depend sensitively on the tube law. Explicit working formulas are developed for several simple types of flow (friction alone; changes in rest area alone; changes in external pressure or elevation alone) for a simple, approximate tube law. Various modes of flow behavior for a flow affected by both friction and gravity are explored.