Multiscale Expansion Method in the Hilbert Problem

The problem of constructing an asymptotic approximation to the solution of the kinetic Boltzmann equation is considered for the hydrodynamic region of low Knudsen numbers. The problem is linearized for one-dimensional perturbations in a gas at rest. The distribution function is sought in the form of a multiscale expansion of the Hilbert asymptotic series type. The construction of a solution uniformly suitable as t is demonstrated with reference to a particular example of sonic wave propagation. It is shown that the multiscale technique makes it possible to extend the domain of applicability of the Hilbert expansion to the entire interval of dissipative relaxation.     

The effect of a singular perturbation on nonconvex variational problems

We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter 0. The technique utilizes the notion of -convergence of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.     

On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

In this paper we consider the singular perturbation boundary-value problem of thefollowing coupling type system of convection-diffusion equationsWe advance two methods:the first one is the initial value solving method,by which theoriginal boundary-value problem is changed into a series of unperturbed initial-valueproblems of the first order ordinary differential equation or system so that an asymptoticexpansion is obtained;the second one is the boundary-value solving method,by which theoriginal problem is changed into a few boundary-value problems having no phenomenon ofboundary-layer so that the exact solution can be obtained and any classical numericalmethods can be used to obtain the numerical solution of consismethods can be used to obtainthe numerical solution of consistant high accuracy with respect to the perturbationparameterε     

Three-dimensional problem of unsteady wave motions of a liquid in a region of variable depth

The three-dimensional problem of unsteady wave motions of a liquid above a plane inclined floor in the framework of a linear dispersion model was considered for the first time in [1] for the particular case =/4, where is the angle of inclination of the floor plane to the free surface of the liquid. The class of exact self-similar solutions of the problem for =/2(2m + 1), m=0, 1, 2,..., for the case of an initial perturbation of a free surface of a special type which is constant in the direction of the normal to the shoreline was found in paper [2]. The present paper is devoted to the investigation of wave motions of a liquid due to an initial perturbation of arbitrary form for the angles of inclination of the floor assumed in [2]. A complete system of eigenfunctions corresponding to the continuous and discrete spectra is found. The theorem of the expansion of an arbitrary absolutely integrable function with respect to the boundary values of the eigenfunctions is proved. An exact solution of the problem is obtained and its asymptotic analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–112, March–April, 1986.     

Energy functional derivative with respect to the length of a curvilinear oblique cut in the equilibrium problem for a Timoshenko plate

This paper describes the dependence of the solution of the equilibrium problem for a Timoshenko plate and the total energy functional of the plate on the perturbation of an oblique crack. The nonlinearity of the problem is caused by the boundary conditions in the form of inequalities (conditions such as the Signorini conditions), which describe mutual nonpenetration of the opposite crack faces. The continuous dependence of the solution of the problem on the perturbation of the crack length is established. A formula for the energy functional derivative of the perturbation of the crack length is obtained.     

Finite strains at the tip of a crack in a sheet of hyperelastic material: II. Special bimaterial cases

An asymptotic analysis of the strain and stress near-tip fields for a crack in a sheet of Generalized Neo-Hookean materials is presented in this second in a series of three papers. The analysis is based on the nonlinear plane stress theory of elasticity and concerns two special cases of the interface crack problem: in the first situation both components have the same hardening behavior; next, we investigate the particular case of a sheet of Generalized Neo-Hookean material bonded to a rigid substrate. The transition between the two special cases is studied in detail. The analytical results are also compared with a full-field finite element solution.     

On the jumping problems of a circular thin plate with initial deflection

In this paper, the jumping problems of a circular thin plate with initial deflection are studied by using the method of two variables^[3],[4] proposed by Jiang Fu-ru and the method of the normal perturbation (in this paper (1.1), (1.2)). We obtain Nth-order uniformly valid asymptotic expansion of the solution of this problem ((1.66), (1.67)). When the initial deflection vanishes the solution of a circular thinplate with initial deflection is reduced to the solution of the problems of the nonlinear bending of a circular thin plate^[6]. If the initial deflection is largish and the signs of the initial deflection with the intensity of the transverse load are opposite, when the intensity of the transverse load reaches a certain value, the circular thin plate with initial deflection should produce the jumping phenomenon^[8].     

The swirling radial free jet

Summary A general similarity solution suggested by Watson for the problem of the laminar, radial, free-jet with swirl has been previously discussed by Riley who also calculated the order to which the solution was valid. That problem is considered in more detail here and higher order terms are given. It is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ⁿ (ln ) ^m must be adopted in order to satisfy the boundary conditions. It is also shown that the general similarity solution may be obtained from the asymptotic series representing the general case with swirl for certain special values of the free constants and also for the no-swirl or free-jet problem. The asymptotic series is given to order ^–13 for the case of swirl and to order ^–29 when there is no swirl.     

Singular perturbation of initial value problem for a nonlinear second order systems

In this paper,the singular perturbation of initial value problem for nonlinearsecond order vector differential equationsε~rx″=f(t,x,x′,ε)x(0,ε)=a,x′(0,ε)=βis discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,aandβ∈R~n.Under suitable assumptions,by using the method of many-parameterexpansion and the technique of diagonalization,the existence of the solution of pertur-bation problem is proved and its uniformly valid asymptotic expansion of higher order isderived.     

Dirichlet problem of higher order elliptic equation with perturbation both in differential operator and in boundary

This paper is the continuation of article [7]. It gives further results about the asymptotic expression for the solution of higher order elliptic equation in the case of boundary perturbation combined with operator perturbation. When unperturbed problemA ₀ is not on the spectrum, the asymptotic expression for the solution of perturbation problemA may be expanded with respect to the small parameter . WhileA ₀ is on the spectrum, the asymptotic expression of the solution contains negative powers of the small parameter . The approximation of arbitrary order to the solution is considered and the recursive formula for the general term and the estimation of remainder term are given.     

Asymptotic study of viscous compressible gas flow past a blunt body

Supersonic viscous gas flow past a blunt body is examined. A method is proposed which permits constructing the asymptotic expansion of any order in the small parameter , which characterizes the viscosity and thermal conductivity coefficients. The asymptotic solution is constructed, including terras of zero, first, and second orders of . Acomparison is made with results of other authors who have studied various particular aspects of the subject problem using the method of inner and outer expansions [1–3].     

Asymptotic solution of singular perturbation problems for the fourth-order elliptic differential equations

In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik ’s method.Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.     

Rayleigh-Taylor instability at large times

The potential flow of an inviscid incompressible heavy fluid lying above a light one is investigated. The asymptotic stage is described by an unsteady discontinuity, which approximates the flow in the neighborhood of the tongue, and by a steady flow outside this narrow region. Consequences of the conservation laws which make it possible to check the accuracy of the solution of the steady-state problem are obtained. A steady-state solution is constructed for Froude numbers 0相似文献   

Wave Propagation Over the Free Surface of a Two‐Phase Medium with a Nonuniform Concentration of the Disperse Phase

The boundaryvalue problem of waves on the surface of a twophase medium with a nonuniform (exponential) distribution of the disperse phase is formulated. An asymptotic solution of the linear problem in the form of damped progressive waves is obtained. The phase velocity, frequency, and damping decrement for the waves are found. The perturbation of the admixture concentration is determined, which, unlike in the case of a uniform distribution, is manifested even in a linear approximation. Numerical calculations were performed for concrete media.     

On the initial stage of a point explosion in a radiating gas

A study is made of the initial stage of a point explosion in a radiating gray gas whose absorption coefficient is approximated by the dependenceK=x()e ^–n ,where is the density and e is the internal energy of the gas. It is shown that for n > —1/3 the initial stage of the process differs significantly from the solution of the problem in not only the classical adiabatic case [1, 2] but also in the case of a medium with nonlinear thermal conductivity [2–4]. The supply of energy to the medium at a point leads to instantaneous heating of the complete medium. The form of this heating is found analytically. The method of matched asymptotic expansions is used to investigate the behavior of the solution in the neighborhood of the center. It is found that for definite conditions at the center of the perturbed region there are formed a shock wave and a region of reverse flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–82, May–June, 1980.I should like to thank V. P. Korobeinikov for interest in the work and a helpful discussion of it.     

Asymptotic and numerical transient analysis of the free convection cooling of a vertical plate embedded in a porous medium

This paper analyzes the cooling process of a vertical thin plate originated by a fluid-saturated porous medium, taking into account the effects of both longitudinal and transversal heat conduction in the plate. Due to the finite thermal conductivity of the plate, a longitudinal temperature gradient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem from parabolic to elliptic, for large values of the suitable Rayleigh number. The energy balance equations are reduced to a system of two differential equations with two parameters: the nondimensional plate thermal conductivity and the aspect ratio of the plate . In order to obtain the evolution of the temperature of the plate as a function of time and position, the coupled balance equations are integrated numerically for several values of the parameters, including the cases of very good and poor conducting plates. The results obtained, are compared with an asymptotic analysis based on the multiple scales technique carried out for the case of a very good conducting plate. There is at the beginning a fast transient in nondimensional time scale of order followed by a slow nondimensional time scale of order unity, which gives the evolution of the cooling process. Good agreement is achieved even for values of the conduction parameter of order unity. The asymptotic solution allows us to give closed form analytical solution for the plate temperature evolution in time and space. The overall thermal energy of the plate decreases faster for smaller values of .     

On the Darcy–Brinkman Flow Through a Channel with Slightly Perturbed Boundary

The goal of this paper is to study the effects of a slightly perturbed boundary on the Darcy–Brinkman flow through a porous channel. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter \(\epsilon \) and arbitrary smooth function h. Using asymptotic analysis with respect to \(\epsilon \), the effective model has been formally derived. Being in the form of the explicit formulae for the velocity and pressure, the asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution.     

Initiation and growth characterization of corner cracks near circular hole

The finite element analysis of crack problems often incorporates the asymptotic character of the local solution into the formulation. Embedment of stress or strain singularities can impose serious restrictions on the outcome and inconsistencies in predicting crack and/or growth. These restrictions are discussed in connection with the problem of two diametrically opposite corner cracks near a circular hole subjected to remote uniform tension. Enforced in the numerical treatment is the 1/r character of the strain energy density function local to the corner crack border where r is the radial distance measured from the crack front. The tendency for the corner crack to become a through crack is predicted by assuming that each point of the crack border extends by an amount proportional to the strain energy density factor. The path would correspond to the loci of minimum strain energy density function. Numerical results are displayed graphically and discussed in connection with crack initiation and non-self-similar crack growth.     

Transonic flows around a carrying airfoil profile

Unsymmetrical gas flows around a carrying airfoil, with velocities close to sonic at infinity, are discussed. In a region situated a certain distance from the profile, an asymptotic solution of the flow problem is constructed. A detailed analysis is made of the dependence of the terms of the asymptotic sequences of the parameters characterizing the transverse and longitudinal dimensions of the body. The law of change in the lifting force as a function of the difference 1 – M, which is assumed to be small, is established. The connection between the theoretical results and experimental data is discussed in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 104–112, January–February, 1978.     

Calculating fluid pressure on piston with constant expansion rate

We consider the problem of the expansion at a constant rate of a planar, cylindrical, or spherical piston in a compressible fluid and calculate the fluid pressure on the piston as a function of its velocity. We consider the solution of the self-similar problem of piston expansion in a compressible fluid at a constant velocity. A similar problem has been solved by Kochina and Mel'nikova in On the expansion of a piston in water, PMM, vol. 23, no. 1, 1959. Results are presented of the numerical solution of this problem for certain values of the parameters characterizing the problem, and the variation of the pressure on the piston as a function of the piston velocity is approximated by several empirical formulas. For the cases of the cylindrical and spherical pistons the approximate analytic relations for the pressure on the piston as a function of the velocity are compared with the numerical solution of the self-similar problem of piston expansion.