The dynamics of a compressible viscous liquid (review). II

This article is the second part of a review of the dynamics of rigid and elastic bodies in a compressible viscous liquid in a linearized formulation. The following processes are investigated: the forced harmonic vibrations of rigid bodies in moving and resting compressible viscous liquids, the nonstationary motion of rigid bodies in a compressible viscous liquid at rest, the movement of rigid bodies in a resting compressible viscous liquid under the action of radiation forces that are due to the interaction with propagating acoustic harmonic waves, the propagation of harmonic waves in thin-walled cylindrical elastic shells containing a compressible viscous liquid, and the propagation of harmonic waves in hydroelastic systems consisting of a resting compressible viscous liquid and elastic compressible or incompressible bodies with initial stresses. Publications concerning the above problems are analyzed. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 3–30, March, 2000.     

Elastic waves in prestressed bodies interacting with a fluid (survey)

Conclusion The above survey of different studies and analysis of results obtained widiin the framework of linearized three-dimensional theory show that use of the given model makes it possible to account for fluid viscosity and initial stresses in elastic bodies. Both of these factors play a significant role in actual media. The model also permits determination of the effect of fluid viscosity and initial stresses on the wave processes in hydroelastic systems. The use of an approach based on representations of general solutions of linearized problems of aerohydroelasticity for bodies with uniform initial strains and a compressible viscous fluid makes it possible to obtain dispersion relations in a general form diat is invariant relative to different types of elastic potential and valid for arbitrary compressible and incompressible materials. The approach also allows researchers to study the main classes of problems encountered in practice, conduct numerical experiments, and use the results to find new properties, laws, and mechanical effects that are characteristic of the investigated wave processes and reflect the mutual effects of the fields of initial and dynamic stresses, as well as the interaction of elastic bodies with viscous fluids. Translated from Prikladnaya Mekhanika, Vol. 33, No. 6, pp. 3–39, June, 1997.     

Interaction of differently shaped bodies in a potential flow of perfect compressible fluid: Axisymmetric internal problem

An approach is proposed to investigate an axisymmetric system consisting of an infinite thin elastic cylindrical shell that contains a potential flow of perfect compressible fluid and a periodically vibrating spherical inclusion. The approach emerged as part of a project devoted to developing methods to bring plugged oil wells back into production by the Vibration Theory Department of the S. P. Timoshenko Institute of Mechanics. This mathematical approach allows transforming the general solutions to equations of mathematical physics from one coordinate system to another to obtain an exact analytical solution (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 16–31, September 2006.     

Impact of a solid on a liquid surface: Asymmetric plane case

The vertical impact of a rigid elliptical cylinder on a compressible liquid surface is considered in a plane asymmetric formulation. Solution of the boundary problem reduces to solving an infinite system of linear Volterra integral equations of the second kind. The results are analyzed in terms of the initial asymmetry angle. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 75–84, September, 1999.     

Theory of normal contact of rigid bodies

The theory of normal contact of rigid bodies with allowance for Coulomb friction is developed. The Boussinesq principle is generalized to contact problems with friction. 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. 128–132, January–February, 2000.     

Resonant phenomena in a cylindrical shell containing a spherical inclusion and immersed in an elastic medium

A method is proposed to investigate the behavior of an axisymmetric system consisting of an infinite thin elastic cylindrical shell immersed in an infinite elastic medium, filled with a perfect compressible fluid, and containing an oscillating spherical inclusion. The system is subjected to periodic excitation. The task is to detect so-called resonant phenomena, to establish conditions that cause them, and to examine the possibilities of using the characteristic parameters of such a hydroelastic system to influence these conditions. The method allows transforming the general solutions of mathematical physics equations from one coordinate system to another to obtain exact analytic solutions (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 82–97, July 2006.     

Method of calculating steady-state flows of a viscous fluid with free boundary in vortex-stream function variables

Computations of the motion of a viscous incompressible fluid with a free boundary in vortex-stream function variables entail difficulties in the implementation of boundary conditions on the free surface. A new approach to the formulation of boundary conditions, which takes into account the specifics in giving these conditions on the free boundary is proposed. An efficient numerical method of calculating steady-state flows of a fluid is developed and implemented. Model computations for problems that have exact solutions are performed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 59–68, March–April, 1998.     

Dynamics of particles near a plane boundary in the radiation field of an acoustic wave

The problem of interaction of a rigid body and a boundary of a viscous fluid under acoustic wave propagation is investigated. An example of the motion of a cylinder in the vicinity of a flat rigid wall caused by acoustic field forces is examined. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 74–79, October, 1999.     

Dynamic problems of the mechanics of the brittle fracture of materials with initial stresses for moving cracks. 1. Problem statement and general relationships

This paper studies the brittle fracture of materials with initial stresses when the stresses act only along the cracks. Dynamic problems for moving cracks are considered when the plane crack infinite in one direction moves with constant velocity. General formulas are presented for compressible and incompressible elastic bodies with an arbitrary structure of elastic potentials. The stresses and displacements are presented as analytical functions of complex variables. Some general relationships may be used in order to obtain exact information on the singularity order of the crack tip for dynamical problems under consideration in the general formulation. Translated from Prikladnaya Mechanika, Vol. 34, No. 12, pp. 3–15, December, 1998.     

Modeling of supersonic flows near flying vehicle elements

Supersonic flows near flying vehicle elements are calculated in the approximation of the full Navier-Stokes equations for a viscous compressible heat-conducting gas with different values of free-stream. Mach and Reynolds numbers and angles of attack. The main laws of the flow near the lifting surface. and in the inlet are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 98–108, March–April, 2009.     

Convergent laminar fluid flow between two rotating disks

A stationary flow from the periphery to the center in a hollow between two coaxial, closely located rotating disks is studied by the iterative method of solving a system of equations of the dynamics of a viscous incompressible fluid. The existence and uniqueness of the approximate solution are shown. Moscow State University of the Food Industry, Moscow 125080. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 77–83, March–April, 2000.     

Generation of beams of three-dimensional periodic internal waves by sources of various types

The energy and force characteristics of periodic internal wave beams in a viscous exponentially stratified fluid are analyzed. The exact solutions of linearized problems of generation obtained by integral transformations describe not only three-dimensional internal waves but also the associated boundary layers of two types. The solutions not containing empirical parameters are brought to a form that allows a direct comparison with experimental data for generators of various types (friction, piston, and combined) of rectangular or elliptic shape. The stress tensor and force components acting on the generator are given in quadratures. In the limiting cases, the solutions are uniformly transformed to the corresponding expressions for the problems in a two-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 12–23, May–June, 2006     

Radiation of nonstationary waves by a thin-walled piezoceramic cylinder containing a viscous compressible fluid

The transient operation conditions of a thin-walled piezoradiator containing a viscous compressible fluid and immersed into an acoustic medium are considered. The problem is solved using the integral Laplace time transformation. Originals of the general solution of the linearized Navier-Stokes equations are constructed for the interior of the domain. These originals are used to find the unknowns from a system of integral Volterra equations. Numerical examples are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 63–71, December, 1999.     

On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 56–67, September–October, 2006.     

Dynamics of a fluid sheet on a curved rigid wall with allowance for phase transformation on the free surface

The flow of a three-dimensional sheet on a curved wall is considered. Gravity and surface tension forces act on the sheet while a droplet stream falls on its free surface. The systems of equations of viscous incompressible fluid dynamics on a curved rigid surface and the boundary conditions with allowance for the falling droplet stream are formulated. The problems of steady axisymmetric motion of the sheet on cylindrical and conical surfaces are considered. The effect of the curvature of the rigid wall on the solution is examined. Kharkov. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–50, July–August, 1994.     

Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load

Complex potentials in common form for compressible and incompressible elastic bodies are used to formulate and solve the problem of stationary motion of a prestressed two-layer elastic half-space under a moving surface load. The results presented are similar to those obtained earlier using the Fourier transform __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 3–15, May 2008.     

The stability of the interface between two bodies compressed along interface cracks. 1. Exact solutions for the case of unequal roots

The buckling of the interface between two bodies is considered in the case where the interface contains several plane cracks and the bodies are compressed along them (along the interface of two different materials). The investigation is carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. Complex variables and potentials of the mentioned linearized theory are applied. This problem is reduced to the problem of linear conjugation of two analytical functions of complex variables. The exact solution is derived for the case of unequal roots of the basic equation. Some mechanical effects are analyzed for the general formulation of problems (elastic, elastoplastic, compressible, incompressible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 69–79, April, 2000.     

Influence of fluid viscosity on waves in an initially deformed,compressible, elastic layer interacting with a fluid medium

The propagation of harmonic waves along the interface of an initially stressed, compressible layer and a viscous, compressible fluid half-space is investigated. A dispersion relation that does not depend on the form of the elastic potential is derived on the basis of the three-dimensional linearized elasticity equations for elastic bodies with uniform initial deformations and on the linearized Navier-Stokes equations for a viscous Newtonian fluid at rest. The phase velocities and attenuation coefficients of the elastic modes are determined numerically as functions of the thickness of the elastic layer using a Murnaghan-type three-invariant elastic potential.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 3–9, September, 1994.     

Modeling of the hydrodynamics of a facility for removing mechanical impurities from oil

The separation of a viscous fluid flow by means of a device operating on the centrifugal principle was modeled. Parameters of the device with a specified separation size were obtained in a series of numerical experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 108–112, July–August, 2008.     

Forces and torques exerted on bodies sliding in a fluid

The motion of a system of bodies along a plane in a viscous fluid in the presence of flow shear is considered. It is demonstrated that a main torque, linearly proportional to the velocities of the bodies, is exerted by the fluid on the system of the bodies and the plane. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1997.