Pressure-Flow Relationships in Collapsible Tubes

Abstract

The pressure-flow characteristic of a short collapsible tube held open at its ends and inserted between two constrictions was investigated as a function of flow for different external pressures and downstream constrictions. Taking the upstream and downstream pressure and the flow through the tube as coordinates, a surface can be determined from which the results of any steady flow investigation at constant external pressure can be calculated. The surface has three regions corresponding to the open, partially collapsed, and almost completely collapsed tube. Steady flow through the tube was a multiple-valued function of the pressure drop across it. A collapsible tube is thus a flow-controlled nonlinear resistance, (QNLR) whose shape depends on external pressure and downstream circuit. The magnitude of the negative dynamic resistance (slope of the pressure drop-flow curve at the operating point) of the QNLR was approximately equal to the dynamic resistance of the downstream circuit. Negative dynamic resistance appeared when the tube collapsed at a critical transmural pressure independent of flow. The tube oscillated and amplified when the dynamic resistance was negative, but would not switch. An equivalent circuit for the tube was proposed with a variable compliance in shunt across a QNLR and variable inertance in series. Van der Polls equation qualitatively described oscillations in the tube. The collapsible tube model should apply to fluid conductors whose lumen significantly varies, for example, veins, coronary arteries, muscle pumps, and cardiac assist devices.