Instability induced by wall deformability in sliding Couette flow

Abstract

The linear stability of “sliding Couette flow” of a Newtonian fluid through the annular gap formed by two concentric cylinders having a ratio of inner to outer cylinder radii, β, and driven by the axial motion of the inner cylinder is studied in the low Reynolds number (<1) regime. The inner wall of the outer cylinder is lined by a deformable neo-Hookean solid layer of dimensionless thickness H. This flow configuration is encountered in medical procedures such as thread-injection and angioplasty, where the inserted needle is surrounded by the deformable wall of blood vessels. In stark contrast to the configuration with rigid cylinders, we predict the existence of finite- and short-wave linear instabilities even in the creeping-flow limit, driven by the deformable nature of the outer cylinder. Interestingly, these instabilities exist for arbitrary β, and even for non-axisymmetric perturbations, in parameter regimes where the flow is stable for the configuration with a rigid outer cylinder. For the finite-wave instability, the axisymmetric mode is the most critical mode of the instability, while the non-axisymmetric mode with azimuthal wavenumber n = 4 is the critical mode for the short-wave instability. By replacing the outer rigid boundary surrounding the deformable wall by an “unrestrained” stress-free boundary, we demonstrate that the flow becomes significantly more unstable. Thus, the present study shows that sliding Couette flow with a deformable wall can be linearly unstable at an arbitrarily low Reynolds number, in direct contrast to the stability of the same configuration with a rigid cylinder.