Force interaction of a sphere and a viscous liquid in the presence of a wall

The problem of the force interaction of a vibrating sphere and a viscous liquid bounded from outside by a rigid wall at rest is studied under the assumption that the largest displacement of the sphere is small compared to its radius and the radius of the sphere is small compared to the distance between the sphere and the wall surface. The liquid flow and the force exerted by the liquid on the sphere are determined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol 41, No. 1, pp. 57–62, January–February, 2000.     

Rotation of a sphere in an unbounded gas

The problem of the slow rotation of a sphere in an unbounded gas is solved at arbitrary Knudsen numbers. The kinetic equation with collision integral in the form of the Bhatnagar-Gross-Krook model (BGK-model) describing the state of the surrounding medium is solved by the Lees method, all the moments of the distribution function which ensure the asymptotic passage of the distribution function into the Chapman-Enskog distribution function at large distances from the sphere being taken into account. In the particular case of moderately large spherical particles a value of the isothermal slip coefficient similar to the exact value is obtained. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–171, January–February, 1997.     

On the self-excited vibrations of bluff bodies suspended with a narrow clearance in a vertical pipe

The results of an experimental study of the flow past a sphere suspended with a narrow clearance in a circular cylindrical pipe and allowed only lateral movement, are presented. The study was limited to the case of intense vibrations accompanied by impacts between the sphere and the pipe wall. The experiments were conducted in steel and copper pipes at Reynolds numbers varying from 10⁴ to 10⁵ and several values of the sphere mass and the clearance between the sphere and the pipe wall. The self-excited vibration frequency was measured by a newly developed method based on registering the frequency only in the resonance modes which set in when the sphere and pipe vibration eigenfrequencies coincide. The flow was photographed and filmed at various values of the water flow rate. The major nondimensional parameters characterizing the phenomenon considered were revealed, and a universal relation between them and the nondimensional self-excited vibration frequency was established with acceptable accuracy. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 136–143, March–April, 2000. The research was carried out with financial support from the Russian Foundation for Basic Research (project No. 98-01-00152).     

Temperature field of a three-layer sphere

This paper presents a solution of the problem of unsteady heat transfer in a three-layer hollow sphere in a central-symmetric formulation with various time-dependent boundary conditions on the inner and outer surfaces. In each layer of the sphere there is heat release of known intensity which depends on the radial coordinate and time. The solution is obtained by a finite integral transform on the radial coordinate. A numerical solution is presented for one version of the boundary conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 78–84, January–February, 2009.     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

Steady-state frictional heat generation on axisymmetric sliding contact of a thermosensitive sphere and a fixed, thermally insulated base

An approach to the construction of a calculation scheme for finding the contact area dimensions, pressure, and temperature for a thermosensitive sphere sliding along a fixed, thermally isolated base is suggested. The well-known solution for the constant thermal properties of the sphere material follows as a particular case of the solution obtained. Polytechnical University, Byalystok 15333, Poland. Franko L'vov State University, L'vov 290602. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 174–183, January–February, 1999.     

Waves in a heavy two-layer liquid excited by an oscillating sphere

An approximate solution of an initial-boundary-value problem appropriate for the semiaxist>0 (t is time) is constructed for a system of integrodifferential equations which describes the waves excited in an initially stationary unbounded heavy two-layer fluid by a vertically oscillating sphere located at a distance from the interface that is significantly greater than its radius. The shape of the steady-state wave is found by passing to the limit as time increases indefinitely. The wave resistance experienced by the sphere during the transient process and in the steady-state regime is studied as a function of frequency. Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–133, March–April, 1998.     

One example of magnetohydrodynamic stokes flow around a self-propelled sphere

Magnetohydrodynamic flow around a sphere equipped with an internal source of electromagnetic fields in the form of a variable megnetic dipole is investigated in the Stokes approximation. This dipole is capable of imparting translational motion to the sphere relative to the liquid. The properties of streamline flow in the self-propelled mode of operation of the source due to the influence of distributed volumetric forces on the character of the flow are demonstrated. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 3–11, July–August, 1998.     

Deformation of an ellipsoidal ferrogel sample in a uniform magnetic field

The elongation of a ferroelastic material sample (whose initial shape is a sphere or an ellipsoid of revolution) under the action of an external magnetic field is studied in an in approximation of small strains. For a sphere, there is a classical estimate obtained under the assumption that elongating in the direction of the field, it becomes a spheroid and the stress and strain fields remain uniform. In the present calculation, it is assumed that the body is an ellipsoid (a sphere in a particular case) only in the absence of an external field; the shape of the sample in the presence of a field is not specified in advance but is found from the condition of balance of surface forces (elastic and magnetic). For the spherical case, the problem is solved exactly: it is shown, that the contour of the deformed body is described by a third-order algebraic equation. The case where the initial configuration is an ellipsoid of revolution is studied numerically. It is shown that in all versions, the refined solution leads to an appreciable increase in the elongation of the sample compared to the classical estimate. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 153–164, May–June, 2005.     

New analysis of contact melting of phase change material around a hot sphere

The process of contact melting of the solid phase change material (PCM) around a hot sphere, which is driven by the temperature difference between the PCM and the sphere, is analyzed in this paper. Considering the difference of the normal angle between the sphere surface and the solid–liquid interface of the melting PCM, the fundamental equations of the melting process are derived with the film theory. The new film thickness and pressure distribution inside the liquid film and the variation law of the normal angle of the solid–liquid interface and the melting velocity of the sphere are also obtained. It is found that (1) while normal angle at sphere surface φ is within a certain value φ₀, which is related to Ste number and the outside force F, it has no obvious effect on the pressure distribution inside the liquid film and the numerical results by the present model are in accordance with the analytical results in the published literature, (2) the film thickness at φ = ±90° is constringent to a certain value and not the infinity, (3) the analytical results can be employed approximately to analyze the contact melting process except for the film thickness at φ = ±90°.     

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.     

Vertical impact on a body floating on the surface of an ideal incompressible fluid layer of finite depth

The asymptotics of the most important impact characteristics, namely, the apparent mass and the apparent moments of inertia, are found for large depths. Specific computational formulas are given for a sphere and a degenerate torus, a solid obtained by rotation of a circle about its tangent. Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–81, January–February, 1999.     

Study on aerodynamic force acting on a sphere with and without boundary layer trips around the critical Reynolds number with a magnetic suspension and balance system

Aerodynamic force acting on a sphere for five kinds of boundary layer trips around the critical Reynolds number, together with the force on a smooth sphere, was successfully measured. This was achieved using JAXA’s 60-cm Magnetic Suspension and Balance System after performing detailed simulations and adjusting the sphere mass and its control parameters. The minimum drag coefficient of a smooth sphere was evaluated around 0.19 in the support-interference-free condition. No hysteresis was observed for the drag coefficient in the critical range for tested sphere with boundary layer trips. Using three serially connected 2nd-order Butterworth low-pass filters, an inertia force oscillating at less than 15 Hz was evaluated from the measured model position, and the unsteady aerodynamic force acting on the sphere was also evaluated with reasonable accuracy. Two kinds of oscillatory aerodynamic forces appeared in the critical range depending on the sphere surface condition: a force rotating around an axis parallel to the uniform flow for both a smooth sphere and a sphere with axially symmetric 0.17-mm-high backward step, and an oscillating force in the plane including the axis parallel to the flow for a sphere with axially symmetric step implemented with 0.35–mm-thick tape with wrinkles acting as small vortex generators. There was also observed a force irregularly rotating through less than 180° in the range about a sphere axis parallel to the flow for a smooth sphere in the supercritical range.     

Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and general properties

A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 24–36, March–April, 2009.     

Flow of a dispersed phase in the Laval nozzle and in the test section of a two-phase hypersonic shock tunnel

An unsteady gas-particle flow in a hypersonic shock tunnel is studied numerically. The study is performed in the period from the instant when the diaphragm between the high-pressure and low-pressure chambers is opened until the end of the transition to a quasi-steady flow in the test section. The dispersed phase concentration is extremely low, and the collisions between the particles and their effect on the carrier gas flow are ignored. The particle size is varied. The time evolution of the particle concentration in the test section is obtained. Patterns of the quasi-steady flow of the dispersed phase in the throat of the Laval nozzle and the flow around a model (sphere) are presented. Particle concentration and particle velocity lag profiles at the test-section entrance are obtained. The particle-phase flow structure and the time needed for it to reach a quasi-steady regime are found to depend substantially on the particle size. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 102–113, September–October, 2008.     

The acoustic field in a rigid cylindrical vessel excited by a sphere oscillating by a definite law

The problem on the interaction between a spherical body oscillating by a definite law and a rigid cylinder filled with an ideal compressible liquid is formulated. The geometrical center of the sphere is located on the cylinder axis. The solution is based on the possibility of representing the particular solutions of the Helmholtz equation in cylindrical coordinates in terms of particular solutions in spherical coordinates, and vice versa. As a result of satisfaction of the boundary conditions on the surfaces of the sphere and cylinder, an infinite system of linear algebraic equations is obtained to determine the coefficients of expansion of the potential of liquid velocities into a Fourier series in terms of Legendre polynomials. The use of the reduction technique for solving the infinite system obtained is substantiated. The hydrodynamic characteristics of the liquid filling the cylindrical cavity are determined and compared with the case of a sphere vibrating in an infinite liquid. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 88–97, June, 2000.     

Hypersonic flow in an MHD-acceleration facility and under full-scale conditions

A formulation of the problem, a method of solution, and calculated results are considered for flow past the windward side of a sphere under conditions of an aerodynamic facility (a wind tunnel) with magnetohydrodynamic (MHD) acceleration of the air flow and the corresponding full-scale flight conditions in the Earth's atmosphere. Calculations were performed on the basis of numerical solution of Navier-Stokes equations taking into account thermochemical nonequilibrium of air and catalytic properties of the body surface. Results of the mathematical simulation of flow around a sphere in an MHD-acceleration wind tunnel are compared with experimental data obtained at the Central Aerohydrodynamic Institute (TsAGI). The problem of recalculation of experimental data for full-scale conditions is analyzed. Central Aerohydrodynamic Institute, Zhukovskii 140160. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 91–102, March–April, 1998.     

The flow of viscoplastic material between two concentric spheres

The motion of a viscoplastic medium between two concentric spheres is considered upon rotation of one sphere with constant angular velocity. This problem is solved by an heuristic iterative method. The boundary of the stagnation zones is found and its specific shape is shown. The flow characteristics versus the parameter of the medium are obtained. Voronezh State Engineering Academy, Voronezh 394017. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 133–139, January–February, 1999.     

Magnetosonic analysis and the method for diagnostics of expansion of a plasma cloud in a magnetized background

The problem of sub-Alfvén expansion of a superconducting plasma sphere in a homogeneous magnetized background is considered. The specifics of a self-consistent model of a low-frequency linear MHD approximation that we used in the present paper is the simultaneous allowance for the energy necessary for maintaining the field and plasma equilibrium at a moving boundary and the additional perturbation of a decelerating field generated by the currents induced in a background plasma. This has allowed us to clarify significantly the dependence of the radiated magnetohydrodynamic energy on the Mach-Alfvén number. We found and calibrated universal dynamic characteristics on the basis of which we developed new techniques for determining the initial energy and the velocities of expansion of an explosive plasma cloud with the use of the peak values of magnetic signals in the near (quasistatic), transient, and wave zones. The possibility of effective application of these techniques in experiments on laser-plasma cloud generation in a vacuum homogeneous magnetic field is shown. Institute of Laser Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 3–13, May–June, 1998.