The Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

Displacement of oil by solutions of an active additive from a uniform stratum with allowance for gravity

Self-similar solutions describing the displacement of oil by solutions of an adsorbed active additive have been obtained and investigated [1–3] in the framework of a one-dimensional flow model with neglect of diffusion, capillary, and gravity effects. In the present paper, a self-similar solution is constructed for the problem of oil displacement by an aqueous solution of an active additive from a thin horizontal stratum with allowance for gravity under the assumption that there is instantaneous vertical separation of the phases. This makes it possible to estimate the effectiveness of flooding a stratum by solutions of surfactants and polymers in the cases when gravitational segregation of the phases cannot be ignored.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–92, January–February, 1984.     

Steady states of a continuous-flow adiabatic chemical reactor

Flow reactors are widely used in the chemical industry for purposes of catalytic reactions [1,2]. Calculation of reactors of this type, even in one-dimemional approximation, is complicated and possible only with the use of numerical methods [1, 3]. Such calculations make it possible to find the steady-state distribution of temperature and concentration in the chemical reactor if one exists; in general, however, there may be other steady-state regimes which may be preferable from the standpoint of obtaining a different degree of conversion of the starting product, operating stability, etc.In this connection special interest attaches to the question of the existence and number of steady-state solutions of the system of equations describing the reactor process.This problem was previously considered in [4–7]. Thus, in [4, 5] it was pointed out that in certain special cases more than one steady-state regime may exist. In [6, 7] the question of sufficient conditions of uniqueness was investigated. In [7] it was shown that the steady-state regime is unique in the ease of short reactors or a dilute mixture of reactants. In [8] the problem of the existence and uniqueness of the steady-state regime was examined for a chain reaction model with direct application of the general theorems of functional analysis.The present paper includes an analysis of a very simple mathematical model of an adiabatic chemical reactor in which an exothermic or endothermie reaction takes place. It is established that in the case of an endothermic process a unique steady-state regime always exists. In the exothermic case the problem of the steady-state regime also always has a solution which, however, may be nonunique; the possibility of the existence of several steady-state regimes, associated with the form of the temperature dependence of the heat release rate, is substantiated.The authors thank G. I. Barenblatt, A. I. Leonov, L. M. Pis'men, and Yu. I. Kharkats for discussing and commenting on the work.     

Method of outer and inner asymptotic expansions in the theory of Brownian motion of aerosol particles

We examine the Brownian motion of particles in a gaseous medium, complicated by the influence of inertial forces. The equation for the distribution function in phase space describing motion of this type was obtained in [1]. Also presented in [1] are the solutions of this equation for certain simple particular cases. The approximate equations of motion of aerosol particles in coordinate space were first obtained in [2] and solved for certain concrete problems in [3,4]. More exact equations of motion in coordinate space, and also the limits of applicability of the equations of [2], are presented in [5].     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]     

New model of gas flow problem in multi-layered gas reservoir and application

In this paper,the new model of the real gas filtration problem has been presented multi-layered gas reservoir,when a gas well output and wellbore storage may be variable,and have obtained the exact solutions of pressure distribution for each reservoir bed under three kinds of typical out-boundary conditions.As a special case,according to the new model have also obtained the exact solutions of presssure distribution in homogeneous reservoir and is given important application in gas reservoir development.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

Passage of a supersonic flow over intersecting surfaces

Using the linear formulation, the problem of passage of a supersonic flow over slightly curved intersecting surfaces whose tangent planes form small dihedral angles with the incident flow velocity at every point is considered. Conditions on the surfaces are referred to planes parallel to the incident flow forming angles 0≤γ≤2π at their intersection [1]. The problem reduces to finding the solution of the wave equation for the velocity potential with boundary conditions set on the surfaces flowed over and the leading characteristic surface. The Volterra method is used to find the solution [2]. This method has been applied to the problem of flow over a nonplanar wing [3] and flow around intersecting nonplanar wings forming an angle γ=π/n (n=1, 2, 3, ...) with consideration of the end effect on the wings forming the angle [4]. In [5] the end effect was considered for nonplanar wings with dihedral angle γ=m/nπ. In the general case of an arbitrary angle 0≤γ≤2π the problem of finding the velocity potential reduces to solution of Volterra type integrodifferential equations whose integrands contain singularities [1]. It was shown in [6] that the integrodifferential equations may be solved by the method of successive approximation, and approximate solutions were found differing slightly from the exact solution over the entire range of interaction with the surface and coinciding with the exact solution on the characteristic lines (the boundary of the interaction region, the edge of the dihedral angle). The solution of the problem of flow over intersecting plane wings (the conic case) for an arbitrary angle γ was obtained in terms of elementary functions in [7], which also considered the effect of boundary conditions set on a portion of the leading wave diffraction. In [8, 9] the nonstationary problem of wave diffraction at a plane angle π≤γ≤2π was considered. On the basis of the wave equation solution found in [8], this present study will derive a solution which permits solving the problem of supersonic flow over nonplanar wings forming an arbitrary angle π≤γ≤2π in quadratures. The solutions for flow over nonplanar intersecting surfaces for the cases 0≤γ≤π [6] and π≤γ≤2π, found in the present study, permit calculation of gasdynamic parameters near a wing with a prismatic appendage (fuselage or air intake). The study presents a method for construction of solutions in various zones of wing-air intake interaction.     

Growth of a main crack under the influence of gas moving inside it

The motion of a gas or liquid in a growing main crack is examined in connection with the problem of the hydraulic fracture of an oil-bearing bed [1, 2] and evaluation of the quantity of gaseous products escaping from the cavity formed by the underground explosion into the atmosphere by way of the crack [3]. The studies [1, 2] formulated and solved a problem on the quasisteady propagation of an axisymmetric crack in rock under the influence of an incompressible fluid pumped into the crack. An exact solution was obtained in [4] to the problem of the hydraulic fracture of an oil-bearing bed with a constant pressure along the crack. The Biot consolidation theory was used as the basis in [5] for an examination of the growth of a disk-shaped crack associated with hydraulic fracture of a porous bed saturated with fluid. A numerical solution to a similarity problem on the motion of a compressible gas ina plane crack was obtained in [6]. Here we examine the problem of the propagation of a main crack (plane and axisymmetric) under the influence of a gasmoving away from an underground cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 116–122, July–August, 1986.We thank V. M. Entova for his remarks, which helped to improve the investigation.     

Solving three-dimensional problems of two-phase filtration using averaging

The present work examines a method of solving three-dimensional problems of two-phase filtration based on averaging the equations and introducing functions reflecting the layering of the flow in collectors inhomogeneous over the thickness. These functions are constructed from the calculation of the two-dimensional flow in the plane of the vertical cross section of the bed. This approach is generalized to the case of the displacement of petroleum by aqueous solutions of chemical reagents. Inhomogeneous problems of multiphase filtration may only be solved numerically. However, in the case of three-dimensional flow, even the use of an effective difference scheme is beset with considerable difficulties, dueprimarily to the increased requirements for memory and speed of operation of the computer. In [1] a principle of approximate integration of the filtration equations in a thin inclined bed was proposed. Assuming a hydrostatic law of vertical pressure distribution, the equations were averaged, and successive solution of problems of ever smaller dimensionality was carried out, not with the initial curves of phase permeability, but with curves of averaged phase permeability. Subsequently, there was further development of this principle of capillary-gravitational equilibrium, and others, to deal with the case of flooding of an inhomogeneously laminar bed [2, 3]. A significantly different approach to the determination of the auxiliary functions, not involving assumptions to as the relations between the viscous drag and the capillary and gravitational forces, is to use the solution of the two-dimensional problem of the replacement of petroleum by water in the plane of the vertical cross section of the bed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 71–75, May–June, 1977.     

Asymptotic method for analysis of a conical shell under local loading

Differential equations of the general theory describing the stress-strain state of conical shells are very complicated, and when computing the exact solution of the problem by analytic methods one encounters severe or even so far insurmountable difficulties. Therefore, in the present paper we develop an approach based on the method of asymptotic synthesis of the stressed state, which has already proved efficient when solving similar problems for cylindrical shells [1, 2]. We essentially use the fourth-order differential equations obtained by Kan [3], which describe the ground state and the boundary effect. Earlier, such equations have already been used to solve problems concerning force and thermal actions on weakly conical shells [4–6]. By applying the asymptotic synthesis method to these equations, we manage to obtain sufficiently accurate closed-form solutions.     

Application of the perturbation method in mechanics of deformable solids

Although the solutions of the classical problems of continuum mechanics have been studied sufficiently well, the smallest deviations, for example, of the body boundary or of the material characteristics from the traditional values prevent one from obtaining exact solutions of these problems. In this case, one has to use approximate methods, the most common of which is the perturbation method. The problems studied in [1–6] belong to classical problems in which the perturbation method is used to study the behavior of deformable bodies. A wide survey of studies analyzing the perturbations of the body boundary shape caused by variations in its stress-strain state is given in [5, 6]. In numerous studies, it was noted that the problem on the convergence of approximate solutions and hence the studies of the continuous dependence of the solution of the original problem on the characteristics of perturbations (“imperfections”) play an important role. In the present paper, we analyze the forms of mathematical models of deformable bodies by studying whether the solution of the original problem continuously depends on the characteristics of the perturbed shape of the body boundary on which the boundary conditions are posed in terms of stresses and on the characteristics of the material properties. We use the results of this analysis to conclude that, when using the perturbation method, one should state the boundary conditions in terms of stresses on the boundary of the real body in stressed state.     

Well interaction in strata with dual porosity

The basic equations describing the process of the filtration of a homogeneous liquid in media with dual porosity were obtained in [1]. Analogous equations are used in the study of filtration of a homogeneous liquid in strata separated by a low-permeability membrane [2]. In the present study we solve the problem of the interference of wells in such media for an areal system of well locations. Particular attention is devoted to the study of the solution for periodic boundary conditions, which are characteristic for the cyclic methods of oil extraction.The author wishes to thank V. L. Danilov and A. A. Bokserman for posing the problem and for suggestions during discussions of the results.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Polymer flooding of oil formations with bottom water

A quasi-one-dimensional model describing the process of polymer flooding of oil formations underlain by bottom water is proposed. The model is based on the assumption of instantaneous gravitational phase separation along the vertical and is a generalization of the hydrodynamic model previously considered in [1]. The self-similar solutions constructed show that in the case of polymer flooding of an oil formation the presence of bottom water leads to qualitative changes in the saturation and concentration distribution and has an important influence on both the running and the final oil output. The results obtained are consistent with the data of two-dimensional numerical modeling of the process [2] and indicate the possibility of more efficient exploitation of water-oil zones as a result of pumping thickened water into the formation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–90, May–June, 1986.     

Problems in determination of the dimensions of limitingly equilibrium blocks with the displacement of viscoplastic petroleum by water

The review article [1] formulated the problem of the determination of the limitingly equilibrium form of blocks with the displacement of a viscoplastic liquid (petroleum) by a viscous liquid (water). The problem arising for the Laplace equation in a region with an unknown boundary has been discussed previously in a somewhat different physical interpretation in [2]. In [2], as well as in [3–5], a number of solutions are given to the problem of determining the dimensions of blocks, for the simplest typical schemes of the arrangement of the sources and sinks. The solutions obtained find application in the analysis of the effect of a limiting pressure gradient on the indices of the exploitation of petroleum deposits, using flooding. There are discussed below some new flows of this same class, having a lesser degree of symmetry, and therefore reducing to more complex boundary-value problems.