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It is shown that the quasi-hydrodynamic (QHD) algorithm allows one to simulate viscous uncompressible flows in thermal convection problems at large Grashof numbers, including the correct description of the onset of an oscillatory process. Tests for square and rectangular cavities are described. The computations are performed by applying the QHD algorithm incorporated into the OpenFOAM package. [ABSTRACT FROM AUTHOR]

The paper presents asymptotic methods for solving the problem of small harmonic oscillations of a flat contour immersed in an incompressible viscous liquid. In the case of large values of the dimensionless viscosity parameter, asymptotic approximations up to the third order have been obtained. In the case of small values of this viscosity parameter, the main term of the asymptotic of the hydrodynamic force on an arbitrary smooth contour was constructed and its form was proved not to depend on the shape of the contour. The results obtained were confirmed by an example of the problem of oscillations of an elliptical cylinder.

The formulation of the problem, the numerical solution method, and description of the test calculation case for modeling a melt flow in the problem of crystal growth by the Czochralski (Chochralsky) method are presented. The mathematical model is based on regularized or quasi-hydrodynamic (QHD) equations to describe a viscous, incompressible heat-conducting fluid. The numerical algorithm is implemented on the OpenFOAM package within the mulesQHDFoam solver. The proposed approach allows calculations of an unsteady three-dimensional flow in a crucible with the given geometry for high Reynolds and Grashof numbers. As an example, the results for the flow in a simplified configuration are given. An asymmetric transient flow is obtained. [ABSTRACT FROM AUTHOR]

In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems. [ABSTRACT FROM AUTHOR]

The paper considers the problem on small harmonic oscillations of an elliptical contour immersed in the incompressible viscous fluid. Analytical and asymptotic methods for solving this problem have been proposed. The results obtained in the numerical implementation of the analytical method and the results of asymptotic solutions were compared. The possibilities of describing the solution in almost the entire range of values of the dimensionless viscosity parameter by joint application of the proposed methods were shown.

The explicit formulas for calculating the components of the effective tensor of relaxation kernels of a two-phase layered medium with a periodic microstructure are derived. The first phase of such a medium consists of an isotropic viscoelastic material and the second phase consists of a viscous incompressible fluid. In particular, it is established that the components of this tensor depend on the volume fraction of the fluid inside a periodicity cell and do not depend on the number of layers and the distances between them. [ABSTRACT FROM AUTHOR]

A family of two-dimensional flows of viscous incompressible fluid in a plane rectangular region with periodic boundary conditions (two-dimensional torus) is considered. The flows are induced by a force, periodic in the two spatial variables and independent of time. In the particular case of the harmonic dependence of the force on one coordinate and in the absence of average flow the well-known Kolmogorov flow is realized. In the general two-dimensional case restructurings of the stationary solutions of the Navier–Stokes equations are investigated numerically and the stability domains are determined in the space of governing physical and geometric parameters, namely, Reynolds numbers, force amplitudes, and spatial dimensions of periodicity cells. It is found that in a square region, whose side is equal to the spatial period of the external force, the main stationary flow preserves its stability against variation in the force amplitude and the Reynolds number. Contrariwise, in the cells, whose sides include several force periods, the variation in the parameters destabilizes the stationary flow. Stationary and self-oscillatory nonlinear secondary flows are considered. The effect of nonstationarity on the Lagrangian dynamics is discussed: the mechanisms of transition to the chaotic advection of passive particles depend on the commensurability of the Reynolds numbers characterizing the average flow in mutually perpendicular directions. [ABSTRACT FROM AUTHOR]

In the framework of the Navier–Stokes equations, unsteady axisymmetric flows of a homogeneous viscous incompressible fluid, in which the axial and circumferential velocities depend only on radius and time are considered, and the radial velocity is zero. It is shown that the velocity of such flows is the sum of the velocities of two flows of a viscous incompressible fluid: axial flow (radial and circumferential velocities are zero) and circumferential flow (radial and axial velocities are zero). Axial and circumferential movements occur independently, without exerting any mutual influence. This allows us to split the boundary value problems for the type of flows under consideration containing three unknown functions (pressure, circumferential and axial velocities) into two problems, each of which contains two unknown functions (pressure and one of the velocity components). In this case, the sum of pressures of the axial and circumferential flow will be the pressure of the initial flow. The discovered possibility of splitting allows using known solutions to replenish the “reserves” of axial and circumferential exact solutions. These solutions, in its turn, can be summed in various combinations and, as a result, give the velocities and pressures of new exact solutions of the Navier–Stokes equations.

The hydrodynamic action on long thin beams executing bending resonance vibrations in a fluid near a plane rigid surface (surface) is investigated. The model of quasi-two-dimensional interaction between the fluid and the beam is used, according to which the hydrodynamic action on any beam cross-section can regarded as a result of plane flow around it. The fluid flow in the planes orthogonal to the beam plane is modeled using the time-dependent system of Navier—Stokes equations, whose solution is determined numerically on the basis of the finite volume method. The solutions are determined in a wide range of the governing parameters of the oscillation process, namely, the dimensionless frequency, the vibration amplitude, and the distance to the surface. The dependences for the hydrodynamic forces thus obtained are used to assess the surface effect on the vibrations of cantilevers with actual physical parameters. [ABSTRACT FROM AUTHOR]

Longitudinal strain waves in physically nonlinear coaxial cylindrical elastic shells containing a viscous incompressible fluid both between and within them are investigated. The influence of the inertia of fluid motion on the wave amplitude and wave velocity is taken into account. If the fluid in the inner shell exerts no effect, the wave velocity and amplitude in the shells do not change. The wave profile in the accompanying coordinate system moves in the negative direction. This means that the motion velocity is subsonic. Accounting for the influence of the inertia of fluid motion in the inner shell leads to a decrease in strain wave velocity, while the viscous stress of the fluid on the shell causes the wave amplitudes to decrease. [ABSTRACT FROM AUTHOR]

The problem of deformation of a drop freely suspended in another fluid and subjected to an electric field has long been of theoretical and practical interest. In this problem, the velocity and electric fields are coupled through the boundary conditions, one of which corresponds to the nonlinear effect of surface charge convection. The balance between the electric stress and surface tension is characterized by electric capillary number CaE, whereas the effect of surface charge convection is characterized by electric Reynolds number ReE (the ratio of timescales for charge relaxation and convective flow). The drop experiences no deformation when CaE = 0, and there is no surface charge convection when ReE = 0. Finding drop steady shape even when ReE = 0 is a computationally expensive iterative procedure with generally slow convergence. However, several experimental and theoretical studies show that when drop deformation is not large, the drop steady shape is close to spheroidal. This observation and the fact that CaE-1 enters the boundary conditions linearly suggest to solve the inverse problem: Find CaE for which a spheroidal shape with given axes ratio d is as close to being steady as possible. The velocity and electric fields and CaE are expanded into series with respect to a parameter proportional to ReE, and the inverse problem is reduced to a system of successive linear problems which can be efficiently solved. Values of (CaE(d), d), obtained based on that system for various values of ReE and for various ratios of phases' electric conductivities, dielectric constants, and viscosities, are in good agreement with existing results for nonsmall deformations and nonsmall ReE. [ABSTRACT FROM AUTHOR]

The heat transfer characteristics of axisymmetric flow past a rotating torus in viscous incompressible fluid have been investigated numerically. The torus surface rotates about its centerline with constant velocity. The governing equations in a toroidal coordinate system are solved by using a finite difference method for ranges of parameters: Reynolds number 20 ≤ Re 40 ≤ Re ; Prandtl number 0.7 ≤ Pr ≤ 100; rotational speed 0 ≤ α ≤ 1.9; and fixed aspect ratio Ar = 2. The heat transfer characteristics are presented in terms of the isotherm patterns, local Nusselt numbers, and average Nusselt numbers. Variation of the local Nusselt number on the torus surface shows the effect of the Prandtl number and rotational speed on heat transfer from a torus in the flow regime. Torus rotation is an efficient Nusselt number reduction or torus surface temperature enhancement technique.

The paper is devoted to the study of the problem of high-frequency low-amplitude harmonic oscillations of a circular cylinder in a viscous incompressible fluid. The study develops the classical method of successive asymptotic expansions of a solution into internal and outer, steady and oscillating components in terms of a small parameter. The case of finite streaming Reynolds numbers is considered. Using the original method for calculation of the hydrodynamic force components from external flow characteristics, analytical expressions for four terms of the force expansion were found. The obtained results allow to evaluate not only a linear response to perturbations of flow caused by oscillations of the cylinder, but also the effect of the non-linear interaction of harmonics in the boundary layer. [ABSTRACT FROM AUTHOR]

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0-2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found. [ABSTRACT FROM AUTHOR]

The existence of a generalized weak solution is proved for the nonstationary problem of motion of a rigid body in the flow of a viscous incompressible fluid filling a cylindrical pipe of arbitrary cross-section. The fluid flow conforms to the Navier-Stokes equations and tends to the Poiseuille flow at infinity. The body moves in accordance with the laws of classical mechanics under the influence of the surrounding fluid and the gravity force directed along the cylinder. Collisions of the body with the boundary of the flow domain are not admitted and, by this reason, the problem is considered until the body approaches the boundary. [ABSTRACT FROM AUTHOR]

The problem of modeling a stationary flow of a viscous compressed fluid in the limited area is considered. The special case of the incompressible environment turns out limiting transition in the defining equations. The algorithm of calculation of a field of speeds of a viscous compressed fluid is developed. The numerical decision of a two-dimensional problem is received. Comparison with the analytical decision of the problem of the flow of a viscous incompressible fluid in the flat channel (Poiseuille flow) is provided.