Effects of Wall Stiffness on the Linear Stability of Flow in an Elastic Channel

The aim of this paper is to explore the key influences of the wall stiffness on the stability and self-excited oscillations of the flow in collapsible channels. To identify mechanism of the oscillations found in the full numerical solutions, a linear eigenvalue problem of the Orr-Sommerfield equations modified by the beam is solved. Excellent agreement is found between the onset of small amplitude oscillations in the full numerical simulations and the neutral stability curves from the eigenvalue problem. If the neutral stability curve is shown in the dimensionless pre-tension and wall stiffness space, there appears to be a "stability zone" inside an unstable region as either the pretension is reduced, with the wall stiffness fixed, or the tension fixed, and the wall stiffness is reduced. Further investigation of this "stability zone" over a wider range of parameters led to the conclusion that the pre-tension and the wall stiffness of the beam do not play independent roles in the linear stability of the system. Instead, the final tension, which is brought about by a combination of the pretension, wall stiffness and wall deformation is found to be the dominant factor in this system. A physical explanation of the existence of this stability zone is, however, still lacking.