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Vol. 4, No. 4

Vol. 4, No. 4, 2008

Borisov A. V.,  Mamaev I. S.,  Kilin A. A.
Abstract
The paper contains the review and original results on the dynamics of liquid and gas self-gravitating ellipsoids. Equations of motion are given in Lagrangian and Hamiltonian form, in particular, the Hamiltonian formalism on Lie algebras is presented. Problems of nonintegrability and chaotical behavior of the system are formulated and studied. We also classify all known integrable cases and give some hypotheses about nonintegrability in the general case. Results of numerical modelling are presented, which can be considered as a computer proof of nonintegrability.
Keywords: liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior
Citation: Borisov A. V.,  Mamaev I. S.,  Kilin A. A., Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 363-406
DOI:10.20537/nd0804001
Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M.
Abstract
The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems on the three-dimensional sphere. Canonical variables for the reduced system are constructed both on two-dimensional and three-dimensional spheres. The method is illustrated by applying it to the two-body problem on a sphere (the bodies are assumed to interact with a potential that depends only on the geodesic distance between them) and the three-vortex problem on a two-dimensional sphere.
Keywords: Poisson structure, Lie algebra, subalgebra, Andoyer variables
Citation: Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M., Algebraic reduction of systems on two- and three-dimensional spheres, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 407-416
DOI:10.20537/nd0804002
Cheremnykh O. K.
Abstract
The paper deals with the motion of an axisymmetric vortex ring in an incompressible media whose velocity $\vec{v}$ and density $ρ$ satisfy the equations $div\,\vec{v}=0$, $\vec{v}\cdot grad\, ρ=0$. The second equation allows us to consider the case when the density varies across the ring. It is shown that the media’s density can vary only in the vicinity of the flow possessing vorticity and must be constant if the flow is potential. Thus, the ring’s velocity and the shape of its atmosphere depend not only on the size of the vortex core and circulation but also on the spatial distribution of the density across the ring.
Keywords: incompressible media, vortex rings, Maxwell vortex, distribution of the density across a vortex ring
Citation: Cheremnykh O. K., On the motion of vortex rings in an incompressible media, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 417-428
DOI:10.20537/nd0804003
Gudimenko A. I.
Abstract
Motion of three point vortices in a perturbed singular configuration is studied numerically and analytically. Several cases of the motion are analyzed according to location of the vorticity center to the orbit of one of the vortices. For each of these cases the trajectories of vortices are calculated.
Keywords: point vortices, asymptotic behavior
Citation: Gudimenko A. I., Dynamics of perturbed singular configuration of three point vortices, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 429-441
DOI:10.20537/nd0804004
Izmaylova K. K.,  Chupakhin A. P.
Abstract
We investigate the partial invariant solution of the system of the equations of the magneto hydrodynamics (MHD). This solution describes plane, steady motions of infinitely conducting gas in attendance of a magnetic field. The key-equation is the Bendikson equation type with degenerated singular point. We research topology of integral curves in a neighborhood of this singular point and infinity applying method of Frommer. There are two regimes of gas motions.
Keywords: magneto hydrodynamics, partial invariant solution, distributed source in a cross-section magnetic field, Bendikson equation, method of Frommer
Citation: Izmaylova K. K.,  Chupakhin A. P., Gas flow from the distributed source in a cross-section magnetic field., Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 443-465
DOI:10.20537/nd0804005
Konovalova N. I.,  Martynov S. I.
Abstract
The problem of non-stationary viscous flow around of two spheres is considered. Hydrodynamic interaction of particles is taken into account. The solution of problem was obtained in terms of small parameter. The forces and torques exerting on spheres are calculated. Results were used for analysis of possibility to obtain the expressions for average force and torque in mixture in terms of volume concentration of hight degree. The general solution of problem for viscous flow around more than two spheres is given.
Keywords: non-stationary viscous flow, two spheres, hydrodynamic interaction, general solution
Citation: Konovalova N. I.,  Martynov S. I., Non-stationary viscous flow around of two spheres, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 467-481
DOI:10.20537/nd0804006
Ganiev R. F.,  Reviznikov D. L.,  Ukrainsky L. E.
Abstract
The paper studies wave-mixing processes in plane and axisymmetric flowing channels. Basing on the original dipole method we suggest an efficient approach to mathematical modeling of such processes. Extensive simulations show that instead of traditional mechanical mixers homogeneous mixtures can be produced in the intense vortex wakes downstream of a body. The paper shows that axisymmetric tangential swirl channels hold much promise.
Keywords: wave stirring, vortex streets, admixture homogenization, numerical methods, dipole method
Citation: Ganiev R. F.,  Reviznikov D. L.,  Ukrainsky L. E., Wave mixing, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 483-496
DOI:10.20537/nd0804007
Borisov A. V.,  Gazizullina L.,  Ramodanov S. M.
Abstract
Citation: Borisov A. V.,  Gazizullina L.,  Ramodanov S. M., E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 497-513
DOI:10.20537/nd0804008
Abstract
Citation: In memory of George M. Zaslavsky, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 515-516
DOI:10.20537/nd0804009
Abstract
Citation: New books of the Scientific and Publishing Center «Regular and Chaotic Dynamics» and Institute of Computer Science (Moscow-Izhevsk), Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp. 517-520

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