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Experiments were conducted over a range of Reynolds numbers from 50 to 8000 to study the pressure-flow relationship for a single bifurcation in a multi-generation model during steady expiratory flow. Using the energy equation, the measured static pressure drop was decomposed into separate components due to fluid acceleration and viscous energy dissipation. The frictional pressure drop was found to closely approximate that for an equivalent length of curved tube with the same curvature ratio as in the model bifurcation. The sensitivity of these results to changes in airway cross-sectional shape, non-planar configuration, and flow regime (laminar-turbulent) was investigated. In separate experiments using dye visualization and hot-wire anemometry, a transition to turbulent flow was observed at Reynolds numbers between 1000 and 1500. Transition had very little effect on the pressure-flow relation.

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.

In this companion paper to “Part I: Numerical Simulations,” we report in vitro experimental studies performed on a simple model leg consisting of a “vein” of thin-walled latex tubing surrounded by “tissue” of open-pore foam rubber. Three modes of periodic external compression were investigated: i) uniform compression; (ii) graded compression, decreasing from ankle to knee; and (iii) sequential compression, progressing from ankle to knee. The modes are compared on the basis of three hemodynamic criteria: degree of vessel collapse, level of fluid velocity, and level of shear stress. In uniform compression these measures of merit are distributed very nonuniformly along the length of the leg: they are high near the proximal end of the cuff but low elsewhere, a result due to the formation proximally of a partially occlusive throat. The latter does not form in either graded or sequential compression, with the consequence that favorable values of the three measures of merit occur more uniformly along the length of the pressurized region. It is concluded that either the graded or sequential mode of compression, or perhaps a combination of the two, would be more effective than uniform compression as a prophylaxis against deep vein thrombosis.

A simple, continuous, one-dimensional model for the geometry and structure of the bronchial airways is used for the analysis of fluid flow patterns which have been observed in forced expiration maneuvers. Various phenomena within the conducting system associated with flow limitation are investigated: (a) the conditions in which a “choke” (flow limitation) can occur in a compliant system; (b) theoretical flows that are physically impossible; (c) the possibility of having elastic jumps downstream of the choke point; (d) perturbations in the physical parameters of the conducting system.

The mechanics of the lung parenchyma is studied using models comprised of line members interconnected to form 3-D cellular structures. The mechanical properties are represented as elastic constants of a continuum. These are determined by perturbing each individual cell from a reference state by an increment in stress which is superimposed upon the uniform stretching forces initially present in the members due to the transpulmonary pressure. A force balance on the distorted structure, together with a force-deformation law for the members, leads to a calculation of the strain increments of the members. Predictions based on the analysis of the 3-D isotropic dodecahedron are in good agreement with experimental values for the Young’s, shear, and bulk moduli reported in the literature. The model provides an explanation for the dependence of the elastic moduli on transpulmonary pressure, the geometrical details of the structure, and the stress-strain law of the tissue.

A computational model is presented for unsteady flow through a collapsible tube with variable wall stiffness. The one-dimensional flow equations are solved for inlet, outlet and external conditions that vary with time and for a tube with time-dependent, spatially-distributed local properties. In particular, the effects of nonuniformities and local perturbations in stiffness distribution in the tube are studied. By allowing the flow to evolve in time, asymptotically steady flows are calculated. When simulating a quasi-steady reduction in downstream pressure, the model demonstrates critical transitions, the phenomena of wave-speed limitation and the sites of flow limitation. It also exhibits conditions for which viscous flow limitation occurs. Computations of rapid, unsteady changes of the exit pressure illustrate the phenomena occurring at the onset of a cough, and the generation and propagation of elastic jumps.

Filling of a thin-walled, highly compliant tube in a partially collapsed condition is studied. The theory, based on one-dimensional flow, takes account of friction, longitudinal tension, and the highly nonlinear pressure-area law for the tube. Various aspects of filling behavior are revealed by alternative calculations using: (i) the method of characteristics; (ii) numerical integration of the continuity, momentum, and tube-law equations; and (iii) a crude but simple lumped-element capacitance-inertance-resistance model. Varied phenomena appear. At high Reynolds number, these include dispersive wave trains associated with circumferential bending stiffness and longitudinal tension, nonlinear changes of wave form, development of highly asymmetrical wave reflections, and sloshing. At low Reynolds number, the area changes with time in a diffusivelike manner. The experiments exhibited the dispersive phenomena predicted by the theory.