Longitudinal Tension Variation in Collapsible Channels: A New Mechanism for the Breakdown of Steady Flow

[+] Author and Article Information
T. J. Pedley

Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, U.K.

J Biomech Eng 114(1), 60-67 (Feb 01, 1992) (8 pages) doi:10.1115/1.2895451 History: Received October 25, 1990; Revised April 06, 1991; Online March 17, 2008


There are several mechanisms potentially involved in the breakdown of steady fluid flow in a collapsible tube under external pressure. Here we investigate one that has received little attention in the past: the fact that the longitudinal tension in the tube wall, T , decreases with distance downstream as a consequence of the viscous shear stress exerted by the fluid. If the tube is long enough, or the initial tension small enough, T may fall to zero before the end of the collapsible tube, and unsteady motion would presumably then ensue; this is what we mean by “breakdown.” We study the phenomenon theoretically, when the flow Reynolds number is of order one, using lubrication theory in a symmetric two-dimensional channel in which the collapsible tube is replaced by membranes occupying a segment of each wall. The resulting nonlinear ordinary differential equations are solved numerically for values of the dimensionless parameters that cover all the qualitatively different types of solution (e.g., in which the channel is distended over all its length, collapsed over all its length, or distended in the upstream part and collapsed downstream). Reducing the longitudinal tension has a marked effect on the shape of the collapsible segment, causing it to become much more deformed for the same flow rate and external pressure. Indeed, the wall slope is predicted to become very large when the downstream tension is very small, so the model is not self-consistent then. Nevertheless, the parameter values for which T becomes zero are mapped out and are expected to be qualitatively useful. The relationships between the values of T during flow and its value before the flow begins is also considered.

Copyright © 1992 by The American Society of Mechanical Engineers
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