Stability of the flow in a soft tube deformed due to an applied pressure gradient

Abstract

A linear stability analysis is carried out for the flow through a tube with a soft wall in order to resolve the discrepancy of a factor of 10 for the transition Reynolds number between theoretical predictions in a cylindrical tube and the experiments of Verma and Kumaran [J. Fluid Mech. 705, 322 (2012)]. Here the effect of tube deformation (due to the applied pressure difference) on the mean velocity profile and pressure gradient is incorporated in the stability analysis. The tube geometry and dimensions are reconstructed from experimental images, where it is found that there is an expansion and then a contraction of the tube in the streamwise direction. The mean velocity profiles at different downstream locations and the pressure gradient, determined using computational fluid dynamics, are found to be substantially modified by the tube deformation. The velocity profiles are then used in a linear stability analysis, where the growth rates of perturbations are calculated for the flow through a tube with the wall modeled as a neo-Hookean elastic solid. The linear stability analysis is carried out for the mean velocity profiles at different downstream locations using the parallel flow approximation. The analysis indicates that the flow first becomes unstable in the downstream converging section of the tube where the flow profile is more pluglike when compared to the parabolic flow in a cylindrical tube. The flow is stable in the upstream diverging section where the deformation is maximum. The prediction for the transition Reynolds number is in good agreement with experiments, indicating that the downstream tube convergence and the consequent modification in the mean velocity profile and pressure gradient could reduce the transition Reynolds number by an order of magnitude.

Stability and the transition to turbulence in the flow through conduits with compliant walls

“…Thus, a multi-fold reduction in the transition Reynolds number has been demonstrated for the flow conduits with compliant walls. The transition at high Reynolds number seems to be captured by an approximate local stability analysis of Verma & Kumaran (2013, 2015) (§ 8.2.2), in contrast to the flow in rigid tubes/channels where the linear stability analysis does not accurately predict transition.…”

“…Therefore, a stability analysis which considers a steady and fully developed base state may not accurately predict the experimental results. Stability analyses which incorporate the deformation due to the pressure gradient have been carried out (Verma & Kumaran 2012, 2015), as indicated in § 8.2.2. Here, the deformed shape is determined from experimental measurements, and the stability analysis is carried out at different downstream locations assuming the flow is unidirectional.…”

“…A global stability analysis for the flow in a deformed tube has not been carried out so far. An approximate analysis for a quasi-fully developed flow has been carried out by Verma & Kumaran (2015). This is valid when the slope of the wall is small, so that the flow can be considered locally parallel.…”

“…It was also reported that there is agreement only if the effect of tube deformation on the velocity profile and pressure gradient is considered in the linear stability analysis; for a parabolic flow and a constant pressure gradient, the transition Reynolds number is higher than the experimental value by a factor of 10. An analysis of the eigenfunctions of the most unstable mode in Verma & Kumaran (2015) indicated that the perturbations are confined to thin regions both in the fluid and in the gel, suggesting that transition is due to the wall-mode instability.…”

Experimental studies on the flow through soft tubes and channels

“…The change in the scaling was explained on the basis of the flow modification due to the tube convergence, and the consequent increase in the strain rate at the wall. A more detailed calculation (Verma & Kumaran 2015) indicated that the quantitative decrease in the Reynolds number is due to the modification in the velocity and pressure profiles caused by the deformation of the wall of the tube. The detailed shape of the tube wall was measured in the experiments.…”