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Several problems of motion of a viscous incompressible fluid with a point source in the flow region are considered. The corresponding initial-boundary-value problems for the Navier-Stokes equations have no solutions in the standard class of functions because the flow velocity field contains an infinite Dirichlet integral. Problem regularization allows one to prove its solvability under certain constraints on the initial data.

The development of perturbations of a tangential discontinuity surface separating two stationary flows of an ideal incompressible fluid slowly varying in space is studied taking into account surface tension. Perturbations are described using the complex Hamilton equations. The dependences of the amplitude of the perturbations on the coordinate and time are obtained.

A presentation of the general solution of the equations of dynamics of a transversely isotropic thermoelastic medium is obtained in the case where the Carrier-Gassmann condition is satisfied with due allowance for the additional expression relating the temperature stress coefficients to the elasticity moduli. The displacements are expressed via three resolving potentials satisfying three inhomogeneous quasi-wave equations. The potentials are related by the heat conduction equation. A presentation of the solution with the use of the stress and displacement functions is provided. Two displacement functions are determined by solving the system of two homogeneous equations, which do not involve the temperature. After these displacement functions are determined, the temperature can be found from the third equation. The resultant presentation of the solution also yields the solution of the static equations of thermoelasticity.

Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electroconducting fluid are considered. Classes of exact solutions of these equations are described. With the use of the natural curvilinear coordinate system, where the streamlines and magnetic force lines are coordinate curves, the model equations are partially integrated and converted to the form that is more convenient for the description of the magnetic lines and streamlines of particles. As the coordinate system used is related to the initial coordinate system by a nonlocal transformation, the group admitted by the system can change. An infinite-dimensional (containing three arbitrary functions of time) group of symmetries is calculated for the system in the natural coordinates. An optimal system of subgroups of dimensions 1 and 2 is constructed for this group. For one of the optimal system subgroups, an invariant exact solution is found, which describes the electroconducting fluid flow of the vortex source type with swirling magnetic lines and streamlines.

A long-wave approximation that describes running solitary waves is considered within the framework of the model of a weakly stratified two-layer fluid. It is demonstrated that wave regimes occur near the boundary of the parametric domain of shear instability in the stratified flow. This fact offers an explanation for the mechanism of intense mixing in deep near-bottom layers.

A possibility of using the 4D Qflow protocol, which is commonly used for medical diagnostics in magnetic resonance imaging, for determining the structure of the three-dimensional fluid flow in the human blood circulation system is considered. Specialized software is developed for processing DICOM images obtained by the magnetic resonance scanner, and the retrieved unsteady three-dimensional velocity field is analyzed. It is demonstrated that magnetic resonance measurements allow one to detect the existence of the flow in blood vessel models and also to study the degree of its swirling (helicity) both qualitatively and quantitatively.

This paper describes an inverse problem for determining the right side of a parabolic equation with a nonstandard growth condition and integral overdetermination condition. The Galerkin method is used to prove the existence of two solutions of the inverse problem and their uniqueness, one of them being local and the other one being global in time. Sufficient blow-up conditions for the local condition for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary. The blow-up of the solution is proven using the Kaplan method. The asymptotic behavior of the inverse problem solutions for large time values is investigated. Sufficient conditions for vanishing of the solution for a finite time are obtained. Boundary conditions ensuring the corresponding behavior of the solutions are considered.

A problem of changing of the orientation of a solid in a space by means of motion of the internal mass is under consideration. It is shown that it is possible for a solid to be arbitrarily reoriented due to special motions of the internal mass. Approaches to controlling the internal motions ensuring this reorientation are proposed.

This paper presents a review of theoretical, experimental, and numerical studies of geometric attractors of internal and/or inertial waves in a stratified and/or rotating fluid. The dispersion relation for such waves defines the relationship between the frequency and direction of their propagation, but does not contain a length scale. A consequence of the dispersion relation is energy focusing during wave reflection from inclined walls. In a limited volume of fluid, focusing leads to the concentration of wave energy near closed geometrical configurations called wave attractors. The evolution of the ideas of wave attractors attractors from ray-theory predictions to observations of wave turbulence in physical and numerical experiments is described.

The nonstationary motion of a spherical layer of an ideal fluid is investigated taking into account the adiabatic distribution of gas pressure in the internal cavity. The existence of nonlinear oscillations of the layer is established, and their period is determined. It is shown that there is only one equilibrium state of the layer. Amplitude equations taking into account the action of capillary forces on the surfaces of the layer in a linear approximation are obtained and used to study the stability of nonlinear oscillations of the layer. The limiting cases of a spherical bubble and soap film are considered.