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

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.