Vol 51, No 5 (2016)

An exact solution describing the convective flow of a vortical viscous incompressible fluid is derived. The solution of the Oberbeck–Boussinesq equation possesses a characteristic feature in describing a fluid in motion, namely, it holds true when not only viscous but also inertia forces are taken into account. Taking the inertia forces into account leads to the appearance of stagnation points in a fluid layer and counterflows, as well as the existence of layer thicknesses at which the tangent stresses vanish on the lower boundary. It is shown that the vortices in the fluid are generated due to the nonlinear effects leading to the occurrence of counterflows and flow velocity amplification, compared with those given by the boundary conditions. The solution of the spectral problem for the polynomials describing the tangent stress distribution makes it possible to explain the absence of the skin friction on the solid surface and in an arbitrary section of an infinite layer.

Free inertia-gravity internal waves in a plane-parallel flow are considered in the Boussinesq approximation with account of the horizontal turbulent viscosity and diffusion. The dispersion relation and the wave decay rate are derived in the linear approximation. The effect of critical layers, in which the wave frequency with the Doppler shift is equal to the inertial frequency, on the dispersion curves is considered. It is shown that the dispersion curves are cut off in the low-frequency domain due to the critical layers mentioned above. The verticalwave momentum fluxes are nonzero and can be greater than the corresponding turbulent fluxes. It is shown that the Stokes drift velocity component transverse to the direction of propagation of the wave is nonzero with account of the turbulent viscosity and diffusion.

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.

The problem of constructing uniform asymptotics for the far fields of internal gravity waves generated by a moving source of perturbations in flow of a finite-depth stratified rotating medium is considered. The solutions obtained describe the wave perturbations both inside and outside the wave fronts and can be expressed in terms of the Airy function and its derivatives. Numerically calculated wave patterns of the excited wave fields are presented.

A combined fully Lagrangian approach for meshless modeling of unsteady axisymmetric vortex flows of a gas-particle mixture with an incompressible carrier phase is developed. The approach proposed is based on the combination of a meshless vortex method for calculating axisymmetric flows of the carrier phase described by the Navier–Stokes (or Euler) equations and the full Lagrangian method for calculating the parameters of the dispersed phase. The combination of these methods reduces the problem of modeling the two phase flows to the solution of a high-order system of ordinary differential equations for the coordinates of toroidal vortex elements in the carrier phase and the particle trajectories, the velocity components, and the components of the Jacobian of transformation from the Eulerian to the Lagrangian variables in the dispersed phase. The application of the method is illustrated by modeling the behavior of an admixture of inertial Stokes particles with a small mass concentration in unsteady flows like solitary vortex rings in a viscous carrier phase and groups of vortex rings in an effectively inviscid carrier phase.

Injection of liquid carbon dioxide into a depleted natural gas field is investigated. A mathematical model of the process which takes into account forming CO₂ hydrate and methane displacement is suggested. An asymptotic solution of the problem is found in the one-dimensional approximation. It is shown that three injection regimes can exist depending on the parameters. In the case of weak injection, liquid carbon dioxide boils up with formation of carbon-dioxide gas. The intense regime is characterized by formation of CO₂ hydrate or a mixture of CO₂ and CH₄ hydrates. Critical diagrams of the process which determine the parameter ranges of the corresponding regimes are plotted.

A general schematic flow representation that explains the mechanism of inviscid gas separation in time-dependent and three-dimensional gas flows is presented. The scenario of gas flow separation from a body surface or a mixing layer is described as a vortex which induces in the flowfield a velocity opposing to that of the main flow, thus decelerating it. Within the framework of this scenario the analytical conditions of separation are obtained for conical and self-similar gas flows which coincide with the results of experimental and numerical simulations.

Viscous fluid flow induced by rotational-oscillatorymotion of a porous sphere submerged in the fluid is determined. The Darcy formula for the viscous medium drag is supplementedwith a term that allows for the medium motion. The medium motion is also included in the boundary conditions. Exact analytical solutions are obtained for the time-dependent Brinkman equation in the region inside the sphere and for the Navier–Stokes equations outside the body. The existence of internal transverse waves in the fluid is shown; in these waves the velocity is perpendicular to the wave propagation direction. The waves are standing inside the sphere and traveling outside of it. The particular cases of low and high oscillation frequencies are considered.