The effect of boundary conditions on the stability of two-dimensional flows in an annulus with permeable boundary

TL;DR

This paper investigates how different boundary conditions affect the stability of two-dimensional viscous flows in an annulus with permeable boundaries, revealing increased instability under certain realistic boundary scenarios.

Contribution

It introduces a study of stability under mixed boundary conditions, showing that permeability and specific boundary prescriptions can destabilize flows previously considered stable.

Findings

Both boundary condition types increase flow instability.

Porous cylinders with normal stress boundary conditions destabilize classical flows.

Flow stability is sensitive to boundary condition specifications.

Abstract

We consider the stability of two-dimensional viscous flows in an annulus with permeable boundary. In the basic flow, the velocity has nonzero azimuthal and radial components, and the direction of the radial flow can be from the inner cylinder to the outer one or vice versa. In most earlier studies, all components of the velocity were assumed to be given on the entire boundary of the flow domain. Our aim is to study the effect of different boundary conditions on the stability of such flows. We focus on the following boundary conditions: at the inflow part if the boundary (which may be either inner or outer cylinder) all components of the velocity are known; at the outflow part of the boundary (the other cylinder), the normal stress and either the tangential velocity or the tangential stress are prescribed. Both types of boundary conditions are relevant to certain real flows: the first one - to porous cylinders, the second - to flows, where the fluid leaves the flow domain to an ambient fluid which is at rest. It turns out that both sets of boundary conditions make the corresponding steady flows more unstable (compared with earlier works where all components of the velocity are prescribed on the entire boundary). In particular, it is demonstrated that even the classical (purely azimuthal) Couette-Taylor flow becomes unstable to two-dimensional perturbations if one of the cylinders is porous and the normal stress (rather than normal velocity) is prescribed on that cylinder.