MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

Analogy solutions of axisymmetrical twist problems by axisymmetrical finite element program

It used to be considered that an axisymmetrical problem and a twist problem of an axisymmetrical body cannot be simulated by each other, because the number of unknown variables in an axisymmetrical problem is greater than that in a twist problem, and the governing equations are not the same. This paper proposes a degenerated analogy method, by which the twist problems of axisymmetrical bodies can be simulated by axisymmetrical problems with finite element programs.An ordinary structural analysis method can be used to analyze an axisymmetrical problem, but a twist problem of axisymmetrical bodies is treated as a 3-dimensional problem usually. According to the method proposed in this paper, the analysis of a twist problem can be simulated by the analysis of an axisymmetrical body with a structural analysis problem. The example of analysis computation is also given. Thecomputed result is in agreement with the theoretical result.In this paper, the constitutive relation of the degenerated analogy problem is given.The authors suggest that a twist problem of a body made of any materials is simulated by an axisymmetrical problem of a body made of orthotropic material. If you have to use some program for the axisymmetrical problem to be limited to isotropic materials the penalty coefficient method can be used to solve the problem.     

弹塑性岩土介质中渗透固结问题的参数二次规划法

岩土类介质在承受荷载时,不仅产生弹塑性变形,还伴随着渗流和固结,是一个关于时间的动态过程。本文建立起处理该问题的参变量变分原理以及相应的有限元方法,这样将原问题化为求解带约束条件(本构状态方程)的二次规划问题.文中讨论了单元的选取形式及具体的实施过程,还给出了一个实例.     

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.     

Symmetry groups and similarity solutions for a free convective boundary-layer problem

The free convective boundary-layer problem due to the motion of an elastic surface into an electrically conducting fluid is studied with group-theoretical methods. The symmetry groups admitted by the corresponding boundary value problem are obtained. Particular attention is paid on the group of scaling which provides the similarity solution of the problem. Also, the admissible form of the data, in order to be conformed to the obtained symmetries, is provided. Finally, with the use of the entailed similarity solution the problem is transformed into a boundary value problem of ODEs and is solved numerically.     

Multi-Scale Asymptotic Expansion for a Small Inclusion in Elastic Media

The aim of this paper is to present an asymptotic expansion of the influence of a small inclusion of different stiffness in an elastic media. The applicative interest of this study is to provide tools which take into account this influence and correct the deformation without inclusion by additive terms that can be precalculated and which depend only on the shape of the inclusion. We treat two problems: an anti-plane linearized elasticity problem and a plane strain problem. On every expansion order we provide corrective terms modeling the influence of the inclusion using techniques of scaling and multi-scale asymptotic expansions. The resulting expansion is validated by comparing it to a test case obtained by solving the Poisson transmission problem in the case of an inclusion of circular shape using the separation of variables method. Proofs of existence and uniqueness of our fields on unbounded domains are also adapted to the bidimensional Poisson problem and the linear elasticity problem.     

On some two dimensional: potential and elastic problems involving a small parameter

An iterative method is given and used to determine stress intensity factors for a starlike arrangement of cracks where each crack is at a distance s from the centre of the star. The problem is distinguished by the fact that in the limit ε = 0 it is not possible to satisfy all the conditions of the problem. This suggests a singular perturbation procedure, however a formal iterative method is developed here from which the error at each step can be clearly seen. The method is not restricted to this problem, a rather different formulation of a similar method has been applied in [1] to determine stress intensity factors for cracks approaching or just passing through an interface. In fact the idea of the method should apply to any problem for which the limit solution ε = 0 gives a different stress-singularity to that found in the ε = 0 case.     

A Simple Hysteretic Constitutive Model for Unsaturated Flow

The present study covers the problem of rotation of a porous disk under a viscous incompressible fluid that fills the half-space above the disk, which is the generalization of the von Karman’s problem. It is found that, instead of solving the exact problem, which is rather complicated by coupling the motions of the free fluid and that contained inside the permeable disk, it is sufficient to solve a much simpler problem of the motion of the free fluid placed onto a permeable plane. Assuming the flow in the permeable disk is described by the Brinkman equations, we obtain a self-similar formulation of the problem. Employing this formulation, we also show that the boundary condition associated with continuity of the tangential strains and tangential velocity components is satisfied at the fluid–porous body interface. The coefficient for the vertical velocity component is furthermore obtained. Various extreme cases are identified.     

Hydrodynamic stability,the Chebyshev tau method and spurious eigenvalues

The Chebyshev tau method is examined; a numerical technique which in recent years has been successfully applied to many hydrodynamic stability problems. The orthogonality of Chebyshev functions is used to rewrite the differential equations as a generalized eigenvalue problem. Although a very efficient technique, the occurrence of spurious eigenvalues, which are not always easy to identify, may lead one to believe that a system is unstable when it is not. Thus, the elimination of spurious eigenvalues is of great importance. Boundary conditions are included as rows in the matrices of the generalized eigenvalue problem and these have been observed to be one cause of spurious eigenvalues. Removing boundary condition rows can be difficult. This problem is addressed here, in application to the Bénard convection problem, and to the Orr-Sommerfeld equation which describes parallel flow. The procedure given here can be applied to a wide range of hydrodynamic stability problems.Received: 4 July 2002, Accepted: 13 September 2002, Published online: 27 June 2003     

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.     

Irregular movement of ground waters with evaporation

The problem of variation in the level of ground waters between two vertical channels is examined, taking into account the evaporation, which is nonlinearly dependent on the level of the ground waters. The problem is reduced to integration of a diffusion equation with a right-hand side which is nonlinearly dependent on the unknown function and is solved by a method of successive approximations. When a certain inequality is achieved, which depends on the magnitudes entering the conditions of the problem, the method of successive approximations converges.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 102–105, July–August, 1970.     

The problem of supersonic flow over the lower surface of a triangular wing

In formulating the problem we make no assumption of smallness of the angle of attack; the attached three-dimensional compression shock which arises under the lower surface of the wing may be of arbitrary intensity, and in form is assumed to differ little from a plane shock; a finite yaw angle is allowed. We consider linear supersonic conical flow which is realized, with the exception of a characteristic linear dimension, in the portion of space bounded by the shock, the plane of the wing, and the surface of a disturbance cone with vertex at the discontinuity of the supersonic leading edge and which is a disturbance of the uniform flow behind the plane shock wave.The problem studied reduces to the homogeneous Hilbert boundary-value problem for an analytic function of a complex variable, whose real and imaginary parts are the partial derivatives of the unknown pressure disturbance with respect to the similarity coordinates.In the solution of the boundary-value problem, the effective method of Lighthill, developed with application to diffraction problems [1, 2], is generalized to the problem of an asymmetric region.The particular case of hypersonic flow about an unyawed triangular wing has been studied by Malmuth [3]; the author obtains the problem considered by Lighthill in [2] and writes out the solution contained in that work.     

Identification of the thermal and thermostressed states of a two-layer cylinder from surface displacements

The problem of identifying the law of time variation in the temperature of one boundary surface of a two-layer cylinder and its thermal and thermostressed state from the temperature and radial displacement of the other surface is formulated and solved. The inverse problem of thermoelasticity to which the problem posed is reduced is analyzed for well-posedness. The solution of the direct problem of thermoelasticity is used to numerically test the technique of solving the inverse problem __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 40–47, January 2008.     

疲劳多裂纹问题研究进展

介绍了近十年来颇受关注的疲劳多裂纹(MFC)问题的研究．其研究的对象主要是长寿命承力结构，尤其是老龄飞机；目的是建立疲劳多裂纹问题裂纹扩展的计算模型和含疲劳多裂纹结构的失效准则．该研究为长寿命承力结构的疲劳可靠性评定奠定了基础．     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Plate cascade in subsonic unsteady gas flow

Accounting for fluid compressibility creates serious difficulties in solving the problem of oscillations of a grid of thin, slightly curved profiles in a subsonic stream. The problem has been solved in [1–3] for a widely-spaced cascade without stagger whose profiles oscillate in phase opposition. The phenomenon of aerodynamic (acoustic) resonance, which may arise in a grid in the direction transverse to the stream for definite values of the stream velocity and profile oscillation frequency, was discovered in [2]. An approximate solution of the problem in which account is not taken of the effect of the vortex trails on the gas flow has been obtained in [4]. In [5, 6] Meister studied in the exact linear formulation the problem of unsteady gas motion through an unstaggered cascade of semi-infinite plates. In [7] Meister considered a grid of profiles with finite chords, but the problem solution was not completed. The problem of subsonic gas flow through a staggered lattice whose profiles oscillate following a single law with constant phase shift was solved most completely in the studies of Kurzin [8, 9] using the method of integral equations. A method of solving the problem for the case of arbitrary harmonic oscillation laws for the lattice profiles was indicated in [10]. The results of the calculation of the unsteady aerodynamic forces for the particular case of a plate cascade without stagger are presented in [9,11], and the possibility of the occurrence of aerodynamic resonance in the cascade in the directions transverse to and along the stream is indicated.Another method of solving the problem is given in [12], in which the more general problem of unsteady subsonic gas flow through a three-dimensional cascade of plates is solved. In the present study this method is applied to the solution of the problem of oscillations of staggered plate cascades in a two-dimensional subsonic gas flow. The results are presented of an electronic computer calculation of the unsteady aerodynamic characteristics of the cascade profiles, which show the essential influence of fluid compressibility on these characteristics. In particular, a sharp decrease of the aerodynamic damping in the acoustic resonance regimes is obtained.     

Geometric shapes of the interface surface of bicomponent flows between two concentric rotating cylinders

In this paper, the shape problem of interface of bicomponent flows between two concentric rotating cylinders is investigated. With tensor analysis, the problem is reduced to an energy functional isoperimetric problem when neglecting the effects of the dissipative energy caused by viscosity. We derive the associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of the second order. Moreover, by considering the effects of the dissipative energy, we propose another total energy functional to characterize the geometric shape of the interface, and obtain the corresponding Euler-Lagrangian equation, which is also a nonlinear elliptic boundary value problem of the second order. Thus, the problem of the geometric shape is converted into a nonlinear boundary value problem of the second order in both cases.     

Problem of a randomly oriented crack in a box-shaped shell

This paper deals with the stress state of a box-shaped shell formed by two semi-infinite plates joined at a right angle. The plates are homogeneous but have different thicknesses. The shell is weakened by a finite rectilinear crack of unit length which reaches one edge of the shell. The orientation of the crack and the load on its edges are arbitrarily chosen. The problem is solved with the assumption that the thickness of the plates is small compared to the length of the crack, which allows an asymptotic formulation of the problem. The problem is reduced to a special type of Riemannian vector problem in which the stress-intensity factor allows matrix factorization in accordance with Khrapkov’s scheme. The asymptotes of the resulting solution and the stress-intensity factor are examined in relation to the thickness of the shell and the angle formed by the crack and the edge of the shell. Translated from Prikladnaya Mekalinika, Vol. 34, No. 12, pp. 48–54, December, 1998.     

Theory of diffraction of weak shock waves

A numerical solution is considered to the universal nonlinear boundary-value diffraction problem which occurs in various problems of weak interaction [1, 2] in the asymptotic analysis of the flow in a region with large gradients of the parameters near the point of intersection of the incident, diffracted, and reflected waves. The analytical solutions to this type of problem usually approximately satisfy the conditions on the diffracted front, the position of which is not known beforehand, but is found along with the solution. In the present paper, the problem is solved by the numerical method of [3], which reduces the initial boundary-value problem for the system of short-wave equations with an unknown boundary to the solution of a series of boundary-value problems with a fixed boundary. The problem of the diffraction of a weak shock wave on a wedge with a finite apex angle is considered as an application of the solution. The data calculated by the asymptotic theory agree significantly better with the experimental data [5] than the theoretical data of [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 176–178, November–December, 1984.