Computation of Stokes flow in a channel with a collapsible segment

Lowe, T. W. and Pedley, T. J. (1995). Computation of Stokes flow in a channel with a collapsible segment. Journal of Fluids and Structures, 9(8) pp. 885–905.

DOI: https://doi.org/10.1006/jfls.1995.1050

Abstract

The numerical solution of Stokes flow in a two dimensional channel in which a segment of one wall is formed by an elastic membrane under longitudinal tension and the remaining channel boundary is rigid is considered. This model problem is being used to gain an understanding of the complex interactions that occur between the fluid flow and the wall mechanics when fluid flows through a collapsible tube, examples of which are widespread in physiology. Previous work (Pedley 1992) considered a similar system using lubrication theory in which the wall slopes are assumed small. The results showed that as the longitudinal wall tension is reduced, the downstream end of the collapsible segment becomes ever steeper, thus violating the assumptions. Here, lubrication theory is abandoned and a numerical solution of the full governing equations, including the complete expression for wall curvature, is sought using an iterative scheme. The effect of the variation in wall tension due to the fluid shear stresses at the compliant boundary is also included.

Results are presented for a range of transmural (internal minus external) pressures and wall tensions. It is found, however, that as the wall tension is reduced, the iterative scheme considered fails to converge. This is similar behaviour to that seen in viscous free surface flows (Silliman \& Scriven 1980).
Possible reasons for this breakdown together with alternative solution strategies are discussed.

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