A dynamical instability due to fluid–wall coupling lowers the transition Reynolds number in the flow through a flexible tube

Abstract

A flow-induced instability in a tube with flexible walls is studied experimentally. Tubes of diameter 0.8 and 1.2 mm are cast in polydimethylsiloxane (PDMS) polymer gels, and the catalyst concentration in these gels is varied to obtain shear modulus in the range 17-550 kPa. A pressure drop between the inlet and outlet of the tube is used to drive fluid flow, and the friction factor f is measured as a function of the Reynolds number Re. From these measurements, it is found that the laminar flow becomes unstable, and there is a transition to a more complicated flow profile, for Reynolds numbers as low as 500 for the softest gels used here. The nature of the f -Re curves is also qualitatively different from that in the flow past rigid tubes; in contrast to the discontinuous increase in the friction factor at transition in a rigid tube, it is found that there is a continuous increase in the friction factor from the laminar value of 16/Re in a flexible tube. The onset of transition is also detected by a dye-stream method, where a stream of dye is injected into the centre of the tube. It is found that there is a continuous increase of the amplitude of perturbations at the onset of transition in a flexible tube, in contrast to the abrupt disruption of the dye stream at transition in a rigid tube. There are oscillations in the wall of the tube at the onset of transition, which is detected from the laser scattering off the walls of the tube. This indicates that the coupling between the fluid stresses and the elastic stresses in the wall results in an instability of the laminar flow.

Experimental studies on the flow through soft tubes and channels

“…In the experiments of Verma & Kumaran (2012), the transition Reynolds number was found to be lower than theoretical predictions by a factor of 10. Moreover the scaling of the transition Reynolds number was found to be Re ∝ 5/8 , in between those for the inviscid and wall modes.…”

“…Due to this, it was considered infeasible to practically realise the high Reynolds number instability at a Reynolds number lower than that for transition in a rigid tube/channel. Due to this, the results of Verma & Kumaran (2012) for the flow through a soft tube, and Verma & Kumaran (2013) for a channel flow discussed a little later, were surprising.…”

“…Experiments on gel-walled tubes of diameter 800 μm and 1200 μm and length about 20 cm were carried out by Verma & Kumaran (2012). Polydimethyl siloxane (PDMS) gels were used in the experiments, and the catalyst concentration in the gel was varied between 10% and 1.75%, in order to vary the gel shear modulus between a maximum of 0.5 MPa and a minimum of about 18 kPa.…”

Stability of fluid flow through deformable tubes and channels: An overview

“…The predictions for low Re seem to be quite sensitive to the nature of the constitutive models used, while at high Re, predictions are relatively insensitive to the models. Experimental observations (Verma & Kumaran 2012 in deformable circular tubes and rectangular channels do show instabilities at Re ∼ 500, but they also show that there is a change in the tube cross-section due to applied pressure gradient. If this change is accounted for, then there is reasonably good agreement between the wall mode predictions and experimental observations, as argued by Verma & Kumaran (2013).…”

“…Later, Chokshi (2007) carried out a weakly nonlinear analysis for wall modes as well, and showed that the instability is always supercritical at higher Re. Although experimental studies in flow through deformable tubes exhibit an instability (Verma & Kumaran 2012), it is not yet clear whether there is evidence on the nature of the bifurcation.…”