The principle of exchange of stabilities for Couette flow

Abstract

Abstract.The eigenvalue problem for the linear stability of Couette flow between two rotating concentric cylinders to axisymmetric disturbances is considered. It is proved that the principle of exchange of stabilities holds when the cylinders rotate in the same direction and the circulation decreases outwards. The proof is based on the notion of a positive operator which is analogous to a positive matrix. Such operators have a spectral property which implies the principle of exchange of stabilities.

Exchange of stabilities in Couette flow between cylinders with Navier-slip conditions

“…This is the principle of exchange of stabilities (PES) which in the early years made it much easier to uncover the instability. As the theory has developed, this is also vital in describing the approach to nonlinear instability as well in the "Rayleigh-unstable" case where μ < η 2 [6]. However, because this work involves an instability primarily associated with slip, we are interested in the case where μ > η 2 , and A < 0 in (2.8), while B > 0 in (2.9); still the Taylor numberT > 0.…”

“…Dimensionless parameters. The more recent approach ( [2], [6]) has been to scale the system so that r 1 = η and r 2 = 1, while in [14] Synge scaled the inner radius to unity. The rotation rates are taken as Ω 1 and μΩ 1 , respectively.…”

Strong exchange of stabilities in rapidly rotating parabolic Poiseuille flow

“…Since that time, it has been developed quite conclusively that for Spiral-Couette flow and for the more general Spiral-Poiseuille flow, that though a complicated competition of modes takes place [10,11], steady helical vortices are observed. In this context, a special problem is treated in this work, as part of a series of studies on mathematical issues arising from the onset of fluid instabilities [5,12,13]. The basic flow under consideration occurs between two infinitely long coaxial cylindrical pipes, with inner and outer radii a and b, respectively.…”

The small magnetic Prandtl number approximation suppresses magnetorotational instability

“…That is, we prove that Chandrasekhar's reduced system of equations predict stability when ∂(r 2 Ω) 2 /∂r > 0, at least for insulating magnetic boundary conditions, which are particularly relevant to experiments. Our proof makes use of insights and techniques developed by Herron and Ali ( [11]) (see also [10]). …”