Stability of flow in a deformable channel with an unrestrained boundary

We report results from a linear stability analysis of Newtonian plane Poiseuille flow through a deformable linear elastic channel with an unrestrained boundary wherein the deformable wall is not rigidly bonded to a substrate and is free to undergo motion. The objective of this study is to address the experimental observations of instabilities for this configuration [S. S. Srinivas and V. Kumaran, Transitions to different kinds of turbulence in a channel with soft walls, J. Fluid Mech. 822, 267-306 (2017)]. We analyze the role of an unrestrained deformable boundary on the stability of channel flow using both asymptotic and numerical methods. Our results show that when the solid to fluid layer thickness ratio is O(1), both wall modes (whose critical Reynolds number Re-c proportional to G(3/4), with G being the shear modulus of the solid) and inviscid modes (whose Re-c proportional to G(1/2)) are significantly destabilized by the presence of an unrestrained boundary when compared to channels with completely bonded deformable boundaries. In agreement with experimental observations, the eigenfunctions corresponding to both these unstable modes exhibit a pronounced asymmetric behavior, thereby highlighting the influence of the unrestrained deformable boundary on the stability of the flow. The asymptotic predictions for the wall mode instability are shown to be in excellent agreement with our numerical results. However, for the solid to fluid thickness ratio similar to 7.7 (used in the aforementioned experiments), our results show that the reduction in the critical Reynolds number due to the unrestrained boundary is only moderate; we provide possible reasons for the same.