Arnold's method for asymptotic stability of steady inviscid incompressible flow through a fixed domain with permeable boundary

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.     

Chaotic advection in compressible helical flow

Compressible helical flow with div v not equal to 0 drastically increases the area of chaotic dynamics and mixing properties when the helicity parameter is spatially dependent. We show that the density dependence on the z coordinate can be incorporated in new variables in a way that leads to a Hamiltonian formulation of the system. This permits the application of various important results like the Kolmogorov-Arnold-Moser theory and, particularly, an understanding of why and in which sense the compressible helical flow is "more chaotic" than the incompressible one. Simulation demonstrates this property for an analog of the ABC flow. An interesting type of the dynamical system with "dense" island chains is described.     

Compressible helical flows

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. ©1995 John Wiley & Sons, Inc.     

On existence of two-dimensional nonstationary flows of an ideal incompressible liquid admitting a curl nonsummable to any power greater than 1

Rostov-on-Don. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 209–212, September–October, 1992.     

Relative equilibria in the Bjerknes problem

We consider a motion of a rigid body of an arbitrary shape in a vibrating irrotational flow. A sufficient condition is established for the existence of relative equilibria of the body, i.e., of equilibria of the averaged system.