On the long-time limit of the Garvin–Alterman–Loewenthal solution for a buried blast load

The Garvin–Alterman–Loewenthal solution refers to the problem of a line blast load suddenly applied in the interior of an elastic half-space. It is expected that the long-time asymptotic limit of this solution should be equal to the solution of a related static problem. This expectation is justified here. First, the solution of the static problem is constructed. Then, the asymptotic limit of the transient problem is found, correcting previously published results.     

Hertz接触问题的解的唯一性条件

讨论了接触面为圆面的Ｈｅｒｔｚ接触问题。若压力分布是轴对称的，则该接触问题的解必是唯一的。且在上述条件下，该接触问题的积分方程可化为两个推广的Ａｂｅｌ积分方程组，此方程组的解便给出此接触问题的解。     

On the uniqueness and stability of solutions to some continuum mechanics problems

One-dimensional shear flows of a perfect fluid, a perfect rigid-plastic body, a Newtonian viscous fluid, and a viscoplastic medium in a plane layer and the compression problem for a Euler beam are considered as examples to discuss a relation between the nonuniqueness of a solution to the basic continuum mechanics problem and the instability of the unique solution to this problem. When the basic solution is not unique, the question is raised on the mechanical interpretation of the linearized stability problem for this solution.     

An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem

In this paper we discuss,an initial-boundary value problem of hyperbolic type with firstderivative with respect to x.The asymptotic solution is constructed and its uniform validityis proved under weader compatibility conditions.Then we develop an exponentially fitteddifference scheme and establish discrete energy inequality.Finally,we prove that thesolution of difference problem uniformly converges to the solution of the original problem.     

Spreading of a paraboloid elevation of liquid over a horizontal base

The literature contains studies [1–4] of the problem of the spreading of an axisymmetric elevation of ground water with conservation of the initial mass of the liquid and under the condition that some of the liquid remains in the previously occupied volume. The investigations used the Boussinesq equation with constant and discontinuous (at the point where ?h/?t = O, where h is the height of the elevation) permeability of the medium. For the first problem, there is an exact analytic solution of the type of an instantaneous source; the solution to the second problem was sought in the form of a self-similar solution of the second kind as an asymptotic solution for the corresponding Cauchy problem. In the present paper, the solution to the problem of the spreading of an axisymmetric elevation over a horizontal base is generalized to the case of an elevation having the shape of an elliptic paraboloid.     

One-dimensional non-Darcy flow in a semi-infinite porous media: a multiphase implicit Stefan problem with phases divided by hydraulic gradients

One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from exper-imental results can always be described by a piecewise linear function,and the problem can be equivalently transformed to a multiphase implicit Stefan problem.The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients,not the excess pore water pressures.Using the similarity transformation technique,an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities.A similar Ste-fan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions,is established. Meanwhile,the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated.In the end,computational examples of the solution are presented and discussed.The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.     

含有裂纹和夹杂的复合柱体的扭转

本文根据Saint-Venant扭转理论,提出了一种能用于扭转分析的线夹杂模型,并得到了它的基本解,进而将此解与的单层势函数解及单裂纹基本解结合,对同时带有裂纹和夹杂的复合柱体的扭转作了讨论,最后将问题归为解一组混合型积分方程,并建议了数值解法。文中通过问题的退化,证明本文提出的夹杂模型在数学和力学上都是正确的,最后作了若干数值例子的计算,其结果令人满意。     

Existence of the Frontal Regime of Water-Steam Phase Transition in Hydrothermal Reservoirs

The problem of steam extraction from a water-saturated hydrothermal reservoir is considered. A parameter range in which there exists a noncontradictory solution of the problem with a sharp evaporation front is found. A critical curve distinguishing the region of existence of the frontal solution is constructed and an approximate solution of the problem corresponding to low evaporation surface velocities is found.     

Pressure distribution in the vicinity of a circular battery of wells operating in the elastic-water-drive filtration regime

We pose the problem of determining the unsteady pressure during combined operation of a circular battery of wells (sinks) in a concentric, uniform, horizontal, circular reservoir under elasticregime conditions, which admits solution by integral-transform methods. An exact solution of the posed problem is presented. A simplified approximate solution of the problem is recommended on the bases of analysis of the exact solution and an example calculation.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

On penalty function methods in the finite-element analysis of flow problems

In this paper the penalty function method is reviewed in the general context of solving constrained minimization problems. Mathematical properties, such as the existence of a solution to the penalty problem and convergence of the solution of a penalty problem to the solution of the original problem, are studied for the general case. Then the results are extended to a penalty function formulation of the Stokes and Navier-Stokes equations. Conditions for the equivalence of two penalty-finite element models of fluid flow are established, and the theoretical error estimates are verified in the case of Stokes's problem.     

The existence of the solution for linear complementary problem

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＴＨＥＳＯＬＵＴＩＯＮＦＯＲＬＩＮＥＡＲＣＯＭＰＬＥＭＥＮＴＡＲＹＰＲＯＢＬＥＭＫｏｕＳｈｕｓｈｕｎ（寇述舜）（Ｄｅｐｔ．ＯｆＭａｔｈ．．ＴｉａｎｌｉｎＵｎｉｖｅｒｓｉｔｙ．Ｔｉａｎｊｉｎ．３０００７２,．Ｐ．Ｒ．Ｃｈｉｎ...     

Diffraction of elastic waves in the plane multiply-connected region and dynamic stress concentration

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions. The complete function series which approach the solution of the problem and general expressions for boundary conditions are given.’ Then the problem is reduced to the solution to infinite series of algebraic equations and the solution can be directly obtained by using electronic computer. In particular, for the case of weak interaction, an asymptotic method is presented here, by which the problem ofp waves diffracted by a circular cavities is discussed in detail. Based on the solution of the diffracted wave field the general formulas for calculating dynamic stress concentration factor for a cavity of arbitrary shape in multiply-connected region are given.     

Elastodynamic wave scattering by an incompressible elastic inclusion

The problem of scattering of time-harmonic elastodynamic waves by an incompressible elastic inclusion is solved by means of the null field approach. The solution is obtained both directly and as a limit of the solution to the corresponding problem for a compressible inclusion. It is also demonstrated that the null field solution to the problem of scattering by a rigid movable scatterer can be obtained from a null field solution for the incompressible scatterer by taking the limit of infinite shear modulus. Some numerical results for spherical and spheroidal inclusions are given.     

Singular integral equation for antiplane-wave scattering by a semi-infinite crack

The problem of antiplane-wave scattering by a semi-infinite crack is reduced to a singular integral equation of the Cauchy type. This equation is obtained by treating the problem as the limiting case of a sequence of problems for which the crack-opening displacements decay exponentially at infinity, and by using real-variable (as opposed to complex-variable) Fourier-transform methods. An integral identity is used to obtain the solution of the singular integral equation. The solution is shown to coincide with the classical Wiener-Hopf solution of the problem.     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

Diffusional growth of cloud particles: existence and uniqueness of solutions

Diffusional growth of cloud particles is commonly described by a coupled system of parabolic equations and ordinary differential equations. The Dirichlet boundary condition for the parabolic equation is obtained from the solution of the ordinary differential equations, but this solution itself depends on the solution of the parabolic equations. We first present the governing equations describing diffusional growth of cloud particles. In a second step, we consider a simplified model problem, motivated by the diffusional growth equations. The main difference between the simplified model problem and the diffusional growth equations consists in neglecting the dependence of the domain for the parabolic equations on the solution. For the model problem, we show unique solvability using a fixed point method. Finally, we discuss application of the main result for the model problem to the diffusional growth equations and illustrate these equations with the help of a numerical solution.     

The simply supported rectangular plate: An exact,three dimensional,linear elasticity solution

An exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness is presented within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a pointwise manner and is in the form of a double Fourier sine series. The classical Navier solution for the problem is shown to be the limit of the present solution as the plate thickness aspect ratio approaches zero. It is noted that the solution presented provides a benchmark against which approximate theories of transversely loaded plates may be measured. The new elasticity solution also provides a heuristic basis for a novel theory of thick plates of arbitrary planform and edge support recently given by the author.     

Theoretical solution of the boundary problem of subsonic aerodynamics

A general theoretical solution of the boundary problem of aerodynamics of high subsonic velocities is presented. The solution of the partial differential equation for the velocity potential is carried out in the physical plane in streamline co-ordinates. The principle of the solution is the representation flow of a compressible fluid around a given profile to a hypothetical flow of an incompressible fluid around a different associated profile. In other words, the problem of compressible flow is transformed to the problem of incompressible flow, which can easily be solved. The results of this solution show very good agreement with solutions of other authors and with experiments.     

Multipoint problem for higher-order parabolic equations in a parallelepiped

We study the well-posedness of a problem for a Petrovskii-parabolic equation with coefficients depending on the space coordinates and with multipoint conditions with respect to the time variable. We establish conditions for the existence and uniqueness of the classical solution of the problem. For the proof of the existence of a solution of the problem, the method of divided differences is used. We prove a metric-type theorem on lower bounds for the small denominators that appear in the construction of the solution.