No abstract

The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small twodimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders (see Ilin & Morgulis (2013)).

The flow of an ideal fluid in a domain with a permeable boundary may be asymptotically stable. Here the permeability means that the fluid can flow into and out of the domain through some parts of the boundary. This permeability is a principal reason for the asymptotic stability. Indeed, the well-known conservation laws make the asymptotic stability of an inviscid flow impossible, if the usual no flux condition on a rigid wall (or on a free boundary) is employed. We study the stability problem using the direct Lyapunov method in the Arnold's form. We prove the linear and nonlinear Lyapunov stability of a two-dimensional flow through a domain with a permeable boundary under Arnold's conditions. Under certain additional conditions, we amplify the linear result and prove the exponential decay of small disturbances. Here we employ the plan of the proof of the Barbashin-Krasovskiy theorem, established originally only for systems with a finite number of degrees of freedom. (c) 2002 American Institute of Physics.

Helical (Beltrami) flow with nonuniform coefficient is considered for the case of compressible fluid and a class of exact solutions is proposed. A paradox of helical flow is discussed and the compressibility is considered as a possible resolution of the paradox. Examples with different symmetries are given for the compressible helical flow and, in particular, the generalization of the ABC (Arnold-Beltrami-Childress) flow for the compressible case is proposed. It was shown in [31 that for incompressible fluid and nondegenerate level surfaces of the Bernoulli function qo (nondegenerate Bernoulli surfaces) the field v is integrable, which means that the Bernoulli surfaces are invariant tori and cylinders, and that motion along the surfaces is quasiperiodic.