Equations of viscous supersonic jets with a high degree of overexpansion

A complete system of equations determining a viscous laminar, strongly overexpanded jet is obtained; the system is formed by shortened Navier—Stokes equations, equations for the metric of a coordinate system related with the form of the jet, and equations of transition from curvilinear coordinates to Cartesian. The problem of calculating the jet is formulated as a Cauchy problem for this system. Two- and three-dimensional flows are examined. Possible swirling of the jet is taken into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137–147, March–April, 1977.     

On Asymptotics of Solutions of Semiexplicit Systems of Differential Equations

We study the problem of the existence of analytic solutions of a certain semiexplicit system of differential equations and obtain sufficient conditions for the existence of analytic solutions of the Cauchy problem in the neighborhood of a singular point.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 132–144, January–March, 2005.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–49, July–August, 1973.     

Drag of porous cylinders in a viscous fluid at low reynolds numbers

The problem of flow past a permeable cylinder at low Reynolds numbers is of interest for the solution of a number of problems in chemical technology in, for example, the design of porous electrodes and porous catalysts and in the calculation of nonstationary filtration of aerosols by fibrous filters. In the present paper, we solve the problem of transverse flow of a viscous fluid past a continuous cylinder in a porous shell and, in particular, in the case of a porous cylinder under conditions of constrained flow (system of cylinders) and an isolated cylinder at arbitrary permeability. The analogous problem of Stokes flow past permeable spheres has been solved in a number of papers [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–124, November–December, 1979.     

On the parametrization of three-point nonlinear boundary-value problems

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 395–413, July–September, 2004.     

On approximation of systems with delay and their stability

We study the problem of the approximation of differential equations with delay by a system of ordinary differential equations. We analyze the qualitative behavior of solutions of the original system and the approximating system and construct an algorithm for the investigation of the stability of solutions of systems with delay.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 208–216, April–June, 2004.     

Stability of the motion of a rigid body in an unsteady flow

The problem of rigid-body motion in an unsteady gas flow is considered using a flow model [1] in which the motion of the body is described by a system of integrodifferential equations. The case in which among the characteristic exponents of the fundamental system of solutions of the linearized equations there are not only negative but also one zero exponent is analyzed. The instability conditions established with respect to the second-order terms on the right sides of the equations are noted. The problem may be regarded as a generalization of the problem of the lateral instability of an airplane in the critical case solved by Chetaev [2], pp. 407–408.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–22, May–June, 1989.     

Cavitation flow around a plate in a channel by a stream of heavy liquid

The nonlinear problem of cavitation flow around a plate by a stream of heavy liquid is investigated in precise formulation; the plate is located on the horizontal floor of a channel when the gravity vector is directed perpendicular to the wall of the channel. Two flow systems are considered-Ryabushinskii's and Kuznetsov's system [1]. This problem was investigated in linear formulation in [2], Similar problems were considered earlier in [3–7] for unrestricted flow. Below, on the basis of a method proposed by Birkhoff [8, 9], all the principal hydrodynamic and geometric characteristics are calculated for the problem being considered.Translated from Ivestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1973.     

Nonlinear equations of elastic deformation of plates

A method for constructing nonlinear equations of elastic deformation of plates with boundary conditions for stresses and displacements at the face surfaces in an arbitrary coordinate system is proposed. The initial three–dimensional problem of the nonlinear theory of elasticity is reduced to a one–parameter sequence of two–dimensional problems by approximating the unknown functions by truncated series in Legendre polynomials. The same unknowns are approximated by different truncated series. In each approximation, a linearized system of equations whose differential order does not depend on the boundary conditions at the face surfaces which can be formulated in terms of stresses or displacements is obtained.     

Analysis of a simple model of a bubble-liquid reactor

A mathematical model which describes the operation of a two-phase bubble-liquid reactor is studied. The model reduces to investigation of the Goursat problem for a hyperbolic system of two partial differential equations. A number of exact solutions to this problem are constructed that describe the operation of the reactor from the time of ignition until the onset of the stationary regime of operation. The problem is also investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 132–140, September–October, 1979.     

Regularization of the retrospective diffusion problem and of the nonhyperbolic system of equations of barotropic two-phase flow

The procedure for regularizing the nonhyperbolic system of equations to obtain a system for which it is possible to formulate the time-inversion problem is illustrated by two examples. The procedure is based on the approximation of some of the coefficients by a random Gaussian process and subsequent averaging over its realizations.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 43–48, November–December, 1993.     

Nonisothermal viscoplastic fluid flow through porous media

The Alishaev model [1] is extended to the case of nonisothermal flow. Neglecting conductive heat transfer, it is shown that for the model in question in the plane of the complex potential not only are the problems linear but the decoupling of the thermal and hydrodynamic problems is also allowed. The latter is reduced to a mixed problem for an analytic function. This makes it possible to use the wellknown methods and results of the theory of limiting equilibrium pillars for isothermal flow [2–5]. It is also established that the solutions of the unsteady problems tend asymptotically to the solutions of the corresponding steady-state problems and can be obtained from the latter by simpler conversion. The effectiveness of the approach proposed is illustrated with reference to the problem of a source-sink system [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–122, July–August, 1990.     

Determination of relative permeabilities in a three-phase flow in a porous medium

A study is made of the problem of determining the parameters of flow described by the Buckley-Leverett system of equations by using functions that admit direct measurement. The well-known solution to the analogous problem for two-phase flow [1–3] is generalized. In contrast to [4], the general case is considered, when the fractions of the phases in the flow and the phase permeabilities depend on two variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 187–189, September–October, 1984.The author wishes to thank B. V. Shalimov for his helpful advice.     

Nonlinear Antiplane Deformation of an Elastic Body

The antiplane elastic deformation of a homogeneous isotropic prestretched cylindrical body is studied in a nonlinear formulation in actual–state variables under incompressibility conditions, the absence of volume forces, and under constant lateral loading along the generatrix. The boundary–value problem of axial displacement is obtained in Cartesian and complex variables and sufficient ellipticity conditions for this problem are indicated in terms of the elastic potential. The similarity to a plane vortex–free gas flow is established. The problem is solved for Mooney and Rivlin—Sonders materials simulating strong elastic deformations of rubber–like materials. Axisymmetric solutions are considered.     

Inverse problem of the vibrations of rods and strings of variable mass

We formulate the inverse problem for longitudinal, torsional, and transverse vibrations of rods and transverse vibrations of strings of variable mass. It is shown that if certain assumptions are made concerning the densities of the combining and separating particles, the problem reduces to the integration of independent partial differential equations and a system of ordinary differential equations of first order.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–124, September–October, 1974.     

Dynamic analysis of interacting coplanar cracks in a half space with a clamped boundary condition using boundary integral equations

The three-dimensional dynamic problem of coplanar circular cracks in an elastic half-space with a clamped boundary condition is considered. The crack faces are subjected to harmonic loads. The problem is reduced to a system of two-dimensional boundary integral equations of the type of the Helmholtz potential for unknown discontinuities in the displacements of the opposite faces of the cracks. The stress intensity factors at the crack contours are obtained and discussed.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 153–159, January–February, 2005     

Aerodynamic interaction of two plane cascades of thin lightly loaded blades in relative motion in a subsonic gas flow

The problem of subsonic ideal-gas flow over two plane cascades of thin lightly loaded blades in relative motion is solved within the framework of the linear theory of small perturbations. By means of the method of integral equations [1] the problem is reduced to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on the blades. The regularized system of integral equations for a finite number of harmonics is solved numerically by a collocation method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 168–175, May–June, 1987.     

Structure of weak shock waves with phase transitions of the first kind

An analytic solution is found to the problem of the structure of weak shock waves with phase transitions of the first kind; it is based on the use of the Mandel ' shtam—Leontovich—D ' yakov concept of second viscosity. The kinetic and thermodynamic characteristics of the problem are investigated, and also the condition of applicability of the second viscosity concept. It is shown that the obtained solution simplifies considerably if the vapor phase is regarded as a perfect gas. The results of calculations for the water—water vapor system are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–62, July–August, 1980.     

Self-similar solution of the Navier-Stokes equations for a completely ionized hydrogen plasma (the plane piston problem)

The self-similar motion of a completely ionized hydrogen plasma is considered in the two-temperature hydrodynamic approximation, i.e., we consider the plane piston problem and the problem on energy release at a fixed wall. Results obtained by numerical integration of the relevant system of ordinary differential equations are quoted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 34–39, May–June, 1969.