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.

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.

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.

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.

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.

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.

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.