The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Translational motion of a cylinder below the free surface of a fluid

A method of solving the problem of the translational motion of a cylinder of given shape below the free surface of an infinitely deep heavy fluid is developed. As distinct from existing techniques, the method permits the obtaining of a solution which becomes exact as the Froude number increases without bound. The solution of the problem of the motion of a circular cylinder is considered in detail. Suggestions are made concerning the characteristic properties of an exact solution of the general problem.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 9–22, November–December, 1996.     

Hydrodynamic methods in the inverse problem of gravitational prospecting

In [1, 2], a dynamical method is proposed for solving stationary inverse problems of potential theory, including the inverse problem of gravitational prospecting. It is based on analogy with the problem of establishing the interface of two immiscible fluids flowing in a porous medium. In the present paper, a system of two functional equations is derived from which one can obtain, as special cases, an equation corresponding to the method of [1, 2], and also a system of equations that enables one to propose a new and different method for solving the inverse problem of gravitational prospecting. Equations are derived in polar coordinates for plane Cauchy problems corresponding to both methods, and the results are also given of the solution of some model problems by these methods. Finally, ways of generating new methods of solution of the inverse problem of gravitational prospecting are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 63–71, July–August, 1980.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Solution of linear problems of the uniform motion of a vortex source in a multilayer fluid

A method of solving linear problems of the uniform motion of a vortex source in a multilayer fluid having an arbitrary finite number of layers is proposed. As an example, the problem of the motion of a vortex source of given intensity in a three-layer fluid is solved. Formulas for the complex velocities and hydrodynamic reactions are obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–132, May–June, 1995.     

Estimation of the hate of flow in porous media

In a number of problems of the theory of flow in porous media it is particularly important to find the integral characteristics of the flow for regions that constitute extended stream tubes. (Such a region can be imagined as the result of the continuous deformation of a cylinder whose lateral surfaces are impermeable while the bases are surfaces of constant pressure, the inlet and outlet of the flow. In the plane case the cylinder becomes a rectangle.) Usually, the flow rate Q is to be found from the difference of head H. In some cases it possible to obtain upper and lower bounds for the flow rate by varying the flow region, the flow resistance field or the form of the flow law [1–4]. The aim of the present study is to find the shape of the region of fixed volume (in the plane case area) which for given constraints realizes an extremum of the steady-state flow rate. It is shown that the extremality requirement leads to an additional local condition on the unknown part of the boundary. A class of plane problems for which the resulting boundary-value problem with an unknown boundary is effectively solved is identified. Examples are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 80–87, March–April, 1986.     

Hydrodynamic Load Associated with the Oscillations of a Cylinder at the Interface in a Two-Layer Fluid of Finite Depth

The results of numerically solving the linear problem of the small steady-state oscillations of a horizontal cylinder located at the interface between two fluids of different densities are presented. The hybrid element method is used. In this method the velocity potentials are represented by means of the finite element method in a narrow zone in the neighborhood of the body and by means of the boundary integral equations in the outer domain. The Green’s functions for an oscillating source in a two-layer fluid bounded from above by a free boundary and from below by an even horizontal bottom are derived. Numerical calculations of the apparent mass and damping coefficients are carried out for an elliptic cylinder beneath a free surface and for a cylinder with the cross-section in the form of a Lewis rib contour which floats on the free surface.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 122–131Original Russian Text Copyright © 2005 by Sturova and Syui.     

Nonlinear problem of flow past a source with the formation of a stagnant zone given different Bernoulli constants

The problem of free flow past a point source is considered for two streams with different Bernoulli constants whose encounter creates a bounded region of constant pressure. The theory and method of solving problems of plane ideal jet flows with different Bernoulli constants in the jets were developed in [1]. Here, in conformity with [1], a nonlinear system of equations is derived, the question of the construction of a high-accuracy numerical solution is considered, and certain calculation results are presented for various values of the Bernoulli and cavitation numbers, which are dimensionless parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 55–60, May–June, 1986.     

Application of the angular superposition method to the contact problem on the compression of an elastic cylinder

The angular superposition method is used to construct an approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J ₂ of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson’s photoelastic experiments and Dinnik’s computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J ₂ on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical pointM*, which separates the cylinder contraction and extension parts.The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1–5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3–5].In what follows, we propose to solve contact problems by the angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method.     

Plane problem of uniform flow of a two-layer fluid past a circular cylinder

An explicit solution is found for the problem of uniform horizontal flow of a two-layer fluid of infinite depth past a circular cylinder. The cylinder axis is perpendicular to the flow. The problem is solved within a linear formulation. The solution of the problem is expressed in the form of rapidly converging series with coefficients determined from a recurrence relation. The first seven terms of the series yield the values of the hydrodynamic loads with a relative accuracy of 10^–6. The results are in good agreement with the known values for similar problems in a homogeneous fluid. Tables of the lift and wave drag are given for homogeneous and two-layer fluids.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–97, January–February, 1996.     

Numerical solution to the problem of the complete stabilization of a supersonic boundary layer

A method is suggested for solving numerically the problem of the complete stabilization of a supersonic boundary layer. It is shown that when the surface is significantly cooled, the neutral stability curve splits into two branches. A calculation is given for the temperatures of complete stabilization for both neutral curves. A comparison of the results obtained with those derived from asymptotic calculations shows that above M = 2 (M is the Mach number) the asymptotic method gives incorrect results.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 39–43, March–April, 1972.     

Diffusion toward a cylinder in the case of an arbitrary viscous flow. Diffusive boundary layer approximation

Steady convective diffusion of a dissolved substance toward the surface of a cylinder (optionally circular) in a viscous flow is examined. An analytical solution is obtained in [1, 2] for the case of laminar flow around a curved cylinder when the freestream flow is straight and uniform. More complex hydrodynamical problems are examined in [3, 4]. In the present work an approximate analytical expression is obtained for diffusive flow of a substance toward the surface of a solid cylinder in the case of an arbitrary two-dimensional flow. Formulas are given for calculating the mass transfer at a circular cylinder in some shear flows of a viscous, incompressible fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163–166, September–October, 1976.The authors thank Yu. P. Gupalo and Yu. S. Ryazantseva for formulating the problem and their attention to the work.     

One-dimensional nonstationary motions of a liquid

An effective method is developed for solving the problem of the nonstationary motion of a liquid with plane, cylindrical, and spherical symmetry [1]. It is based on the idea of dividing the region of disturbed motion into two parts and using matched asymptotic expansions. Solutions are presented to typical problems associated with the motion of a piston, and these make it possible to obtain the solution to problems of an explosion in a liquid, oscillations of a bubble, and so forth. It is also shown that the well-known solutions for such problems given, for example, in the book of Naugol'nykh and Roi on the basis of the acoustic approximation with allowance for nonlinear terms are incorrect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1980.     

Use of the time-establishment process when solving problems relating to the steady flow of gas around a cascade of profiles

The use of the time-establishment process in solving problems relating to the steady flow of an ideal (nonviscous and heat-insulating) gas around a cascade of profiles is considered. Chief attention is paid to the role of the Chaplygin-Zhukovskii condition, which is needed [1, 2] when solving these as steady-state problems. Results obtained during the time-establishment process with and without the application of the condition in question are presented. In both cases certain time-independent distributions of the parameters satisfying the equations and conditions of the steady-state problem (or more precisely their finite-difference analog) are produced during the establishment process. Comparison of the resultant distributions shows that the difference between the two cases lies within the limits of computing error, the results of the calculations agreeing quite well with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 118–124, July–August, 1974.In conclusion, the authors wish to thank S. I. Ginzburg, G. Yu. Stepanov, and G. G. Chernov for taking part in discussions of this work.     

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.     

Stress Analysis of Biconvex Laminated Orthotropic Shells that are Shallow to a Variable Degree

The stress–strain state of biconvex laminated orthotropic shells is analyzed against the degree of shallowness and the parameters of orthotropy. Numerical values of deflections and stresses are obtained by solving two-dimensional boundary-value problems using spline-functions and the discrete-orthogonalization method. The effect of the rise of shells on the displacement and stress fields is analyzed for different parameters of orthotropy     

Parametric instability of the m = 1 mode in the dynamic stabilization of a plasma filament

We consider the parametric excitation of long-wave magnetohydrodynamic oscillations of the m = 1 type in a cylindrical plasma conductor with an alternating, longitudinal, high-frequency current. The plasma cylinder is placed in a constant longitudinal magnetic field and is enclosed in a conducting case. The problem is solved on the basis of the flexible filament model under the assumption of ideal conductivity of the plasma and the case. Hill's method is used to study the stability of the equation with periodic coefficients that describe the oscillations of the filament. Results of computer calculations of the stability increments of oscillations in the first four resonance zones for various values of the parameters of the system are given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 42–46, July–August, 1972.In conclusion the authors thank M. L. Levin for helpful discussions of the work.     

On the theory of loaded inhomogeneous cylinders

This article is concerned with the relaxed Saint-Venant’s problem in the case when the body forces and surface tractions on the lateral surface are polynomials in the axial coordinate. A new method of solving this problem is presented. The method is applied to study the problem of a uniformly loaded cylinder.     

Exploiting the limiting amplitude principle to numerically solve scattering problems

A numerical method for solving reduced wave equations is presented. The technique is basically a relaxation scheme which exploits the limiting amplitude principle. A modified radiation condition at ‘infinity’ is also given. The method is tested on two model problems: the scattering of plane shallow water waves off shoals and the scattering of plane acoustic waves off a sound-soft cylinder imbedded between two homogeneous but different half spaces. The numerical solutions exhibit correct refractive and diffractive effects at moderate frequencies.     

A realization of the moment method

The difficulties involved in solving a system of moment equations using two-sided distributions are analyzed. The properties of these distributions do not permit the realization of the moment method (in the case of a collision integral in Boltzmann form) in specific transport boundary-value problems. A method of obtaining analytic solutions of the system of moment equations for linearized transport problems is proposed. The accuracy of the method is analyzed with reference to a classical transport problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 185–188, May–June, 1991.