Mixed Plane Problems in Linearized Solid Mechanics: Exact Solutions

The paper analyzes the exact solutions to mixed plane problems of linearized solid mechanics in cases of statics, dynamics, stability, and fracture. The exact solutions have a universal form for compressible and incompressible, elastic and plastic bodies and account for stresses and displacements expressed in terms of analytical functions of complex variables. To obtain these solutions, the use is made of complex variable theory, in particular, the Riemann–Hilbert methods and Keldysh–Sedov formula. When the initial (residual) stresses tend to zero, the exact solutions go over into the corresponding exact solutions of classical linear solid mechanics, which are based on the complex representations due to Muskhelishvili, Lekhnitskii, and Galin     

Some families of exact solutions of the equations of two-dimensional shallow water theory

Results are presented of a study of the exact solutions of the equations of two-dimensional unsteady and steady shallow water theory, based on the group properties of these equations. The first part presents the group properties of the equations in question; the second part presents the invariant solutions of these equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 62–71, November–December, 1969.The author wishes to thank L. V. Ovsyannikov and N. Kh. Ibragimov for valuable guidance in carrying out this study.     

Two-dimensional flow in porous strata with conductivity varying discontinuously along second-order curves

Solutions of boundary-value problems of two-dimensional flow in a porous medium are obtained on the basis of the theory of axisymmetric generalized analytic functions [1,2] and conversion formulas [3] for a broad class of strata whose conductivity changes abruptly along second-order curves. The singular points of these functions model arbitrary two-dimensional flows. In space the solutions describe the axisymmetric flow in porous media whose homogeneity interfaces are second-order surfaces of revolution. The solutions obtained are applied to new problems associated with environmental protection and the nonpolluting operation of water intakes under complex geological conditions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 120–128, January–February, 1993.     

Screw flow of a liquid in a cylindrical tube

A study is made of the flow of a viscous incompressible liquid with helical streamlines in an infinite cylindrical tube within which a screw rotates (auger). Generalized linearized Oseen equations are derived, and one class of the exact solutions of these is identical with the corresponding class of exact solutions of the complete Navier-Stokes equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 3–7, November–December, 1979.     

On a class of exact particular solutions of the equations of transonic gas flows

The paper presents exact particular solutions of the equations of transonic gas flows, analogous to the solutions derived in [1–3] for the case of short waves. These solutions are used to construct the flow around a body in a supersonic stream with an attached shock.     

Two-dimensional flow in porous strata with conductivity modeling by a harmonic function of the coordinates

Boundary-value problems of two-dimensional flows in porous media are investigated in finite form for a broad class of strata with harmonic conductivity. The conformal covariance of the conjugation problem formulated is demonstrated. This makes it possible to reduce it to a canonical problem whose solutions are represented by quadratures. The solutions obtained are applied to new problems associated with the operation of a well in soil strata under complex geological conditions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–112, May–June, 1995.     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Group properties and invariant solutions of equations of an electric field in the case of nonlinear ohm's law

The group properties of one-dimensional nonstationary equations of an electric field in homogeneous isotropic media with nonlinear conductivity are considered. The nonlinear Ohm's laws for which these equations have the broadest symmetry properties are determined. Ordinary differential equations determining invariants solutions are obtained; the order of the equations is lowered or they are integrated to the end.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 28–36, May–June, 1972.     

Thin-slug dynamics with allowance for diffusion and sorbtion irreversibility

For a very simple model of equilibrium partly irreversible sorbtion, a study is made of the dynamics of a thin slug of a neutral additive that is carried along with the flow with allowance for diffusion (dispersion). Several approaches to the solution of the problem are considered, and cases that admit exact self-similar solutions, which are solutions of the second kind, are identified. The asymptotic law of decay of slugs is studied.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–112, May–June, 1993.     

Exact solutions of boundary-value problems for nonlinear flow in porous media

Exact solutions for flow problems in porous media with a limiting gradient in the case when the flow region in the hodograph plane is a half-strip with a longitudinal cut [1] are known only for two models of the resistance law [2–6]. The present study gives a one-parameter family of flow laws, and argues the possibility of effective determination of exact and approximate analytical solutions on the basis of successive reduction to boundary-value problems for the Laplace equation or for the equation studied in detail in [1]. It should be noted that the characteristics of the flow are determined without additional quadratures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 107–112, May–June, 1985.     

3D Dynamic analysis of soil–tool interaction using the finite element method

Previous experimental and finite element studies have shown the influence of both soil initial conditions and blade operating conditions on cutting forces. However, most of these finite element analyses (FEA) are limited to small blade displacements to reduce element distortion which can cause solution convergence problems. In this study a dynamic three-dimensional FEA of soil–tool interaction was carried out based on predefined failure surfaces to investigate the effect of cutting speed and angle on cutting forces over large blade displacements. Sandy soil was considered in this study and modeled using the hypoplastic constitutive model implemented in the commercial FEA package, ABAQUS. Results reveal the validity of the concept of predefined failure surfaces in simulating soil–tool interaction and the significant effect of cutting acceleration on cutting forces.     

Small-amplitude nonlinear waves in a dissipative gas with excess vibrational energy

By means of the method of two-scale expansions the results of studies [3–5] are extended to the case of cylindrical and spherical waves, and an equation for the gas velocity which takes dissipative processes into account is derived for one-dimensional plane small-amplitude nonlinear waves. A number of exact particular solutions of this equation is found. Since the accuracy of solutions obtained by means of approximate methods in problems of the type in question has not previously been estimated, the limits of applicability of the analytical method employed are established by comparing the results with numerical calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 151–158, March–April, 1990.The authors are grateful to M. S. Ruderman for useful discussions.     

Mathematical Model for the Solid Electrolyte Fuel Cell Cathode

An analytically solvable mathematical model for the cathode of a solid polymer electrolyte fuel cell is proposed. The problem of diffusion in a multicomponent air-vapor mixture in a porous cathode and water transport due to hydrodynamic and electroosmotic forces is solved. The volt-ampere characteristic of the fuel cell is determined taking into account the polarization characteristics and finite conductivity of the electrolyte. An expression is obtained for the thickness of the electrochemical-reaction zone, which gives an estimate of the catalyst efficiency. It is shown that the finiteness of the rate of oxygen diffusion into the reaction zone limits the current density and the fuel cell efficiency. A comparison of the results with available theoretical and experimental data shows that the solutions obtained for the model coincide with the solutions for the more complex Bernardi and Verbrugge model.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 27–37, September–October, 2005.     

Symmetry Reductions of Equations for Solute Transport in Soil

Solute transport in saturated soil is represented by anonlinear system consisting of a Fokker–Planck equation coupled toLaplace's equation. Symmetries, reductions and exact solutions are foundfor two dimensional transient solute transport through some nontrivialwedge and spiral steady water flow fields. In particular, the mostgeneral complex velocity potential is determined, such that the soluteequation admits a stretching group of transformations that wouldnormally be possessed by a point source solution.     

Solitons in plasma with a hall dispersion

A system of equations of perfect magnetohydrodynamics is considered with allowance for Hall currents. The study of one-dimensional steady solutions which are damped at infinity can be reduced to the investigation of a Hamiltonian dynamic system with right-hand sides that are not single valued. A qualitative investigation of the system is carried out, with the determination of the region of existence of the given solutions. The solutions have the form of solitary waves — solitons. An exact solution in quadratures is obtained, which describes the structure of the solitons. The existence of two solitons of the Alfvén type is indicated. The existence domain of the corresponding solutions is analyzed. In the limiting cases of magnetosonic and Alfvén solitons, the solutions are expressed in explicit form in elementary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–168, September–October, 1985.     

One-dimensional unsteady motions of a viscous heat-conducting gas with a linear velocity distribution with respect to the coordinate

The one-dimensional motions of a perfect gas are considered in cases of spherical, cylindrical and plane symmetry, when the velocity is proportional to the distance to the center of symmetry. The solutions obtained are an extension of the known solutions of Sedov [1, 2] to the case of a viscous heat-conducting gas with a power-law temperature dependence of the coefficient of viscosity and thermal conductivity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 130–133, March–April, 1987.The author is grateful to L. I. Sedov for his interest in the work and to A. G. Kulikovskii for useful discussions.     

Similarity of flows in strongly underexpanded jets of viscous gas

The fulfillment of the conditions formulated in [1] for the similarity of flows in strongly under-expanded jets of a viscous, thermodynamically ideal gas is studied. The limits of applicability of these conditions are established on the basis of exact solutions of the one-dimensional Navier —Stokes equations and experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika zhidkosti i Gaza, No. 6, pp. 117–125, November–December, 1978.     

The uniqueness of the solution of boundary-value problems of a thermal model of discharge

The paper is devoted to a nonlinear analysis of superheating [1, 2] instability of an electric discharge stabilized by electrodes [3] in the framework of a thermal model [4] where the stability of the discharge relative to the long-wave and short-wave perturbations is proved in a linear approximation. Similar boundary-value problems arise in the theories of chemically and biologically reacting mixtures [5–7], thermal breakdown of dielectrics [8], thermal explosion [9], in the investigation of nonlinear waves in semiconductors and superconductors [10, 11], and in the investigation of Couette flow with variable viscosity [12]. The uniqueness of the one-dimensional steady solutions of the thermal model of discharge and the stability relative to the small spatial perturbations, respectively, for the exponential and step dependence of the electrical conductivity on the temperature are proved in [3, 13]. The uniqueness of the solutions in the one-dimensional case for the same electrode temperature and arbitrary dependences of the electrical and thermal conductivity on the temperature is established in paper [14]. In the present paper, the existence and uniqueness of steady solutions of the thermal model of discharge in a three-dimensional formulation for arbitrary fairly smooth electrical and thermal conductivity functions of the temperature in the case of isothermal isopotential electrodes are proved analytically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 140–145, January–February, 1986.The author expresses his gratitude to A. G. Kulikovskii and A. A. Barmin for the formulation of the problem and their discussions.     

Solution of the inverse problem of thermal conductivity for a melting slab with entrainment

This article proposes an approximate solution to the inverse problem of the Stefan type for a finite region with arbitrary boundary and initial conditions. A comparison with exact solutions is made.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 132–137, May–June, 1973.     

Nonclassical Potential Symmetry Generators of Differential Equations

We determine the nonclassical potential symmetries for a number ofequations that arise in the literature. A large number of these areobtained for some equations which only admit a single potential(classical) symmetry (e.g., the wave equation and the motion of wavesthrough some medium). However, we show that some of the exact solutionsinvariant under the nonclassical potential symmetries are equivalent toknown solutions but these solutions are not obtainable through theclassical point or potential symmetries. The Korteweg–deVries equation,it is shown, does not admit nonclassical potential symmetries – as inthe classical case.