A mixed boundary value problem for Chaplygin's hodograph equation

In this paper we will prove existence, uniqueness and regularity of a classical solution to a mixed boundary value problem for Chaplygin's hodograph equation, which is degenerate elliptic on a part of the boundary. This problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane. The proof depends on Perron's method and some techniques from linear elliptic equations.     

Slip lines at the corner of the interfacial boundary of different media

A symmetric problem on the initial development of a plastic wave, simulated by two straight slip lines starting from the apex, near the corner of the interfacial boundary of different media is examined under plane strain conditions. The exact analytical solution is constructed for the Wiener-Hopf functional equation of the problem. A formula is deduced to determine the slip line length, and their slope to the interfacial boundary of the media is established.     

Approximate solution methods in mechanics analysis of a class of nonlinear seepage problems by a finite-element method

We analyze the nonlinear boundary-value problem of seepage under a subsurface hydrotechnical construction over an inclined rectilinear aquifer. The method of inverse boundary-value problems is applied, using the velocity hodograph plane in which the original problem is reduced to a linear problem. The linear problem is solved in the general case using the finite-element method. A computer program realizing the proposed algorithms has been developed. We have used this program to run a series of numerical experiments, reaching certain conclusions about the behavior of the main seepage characteristics.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 75–80, 1985.     

二元拉伐尔喷管不可压缩流动精确解

在物理平面上,仔细分析沿拉伐尔喷管中心线和喷管型线的流动,可以发现拉伐尔喷管流动的上下两半部分在速度平面中是两个相同的具有尾缘点前后错开的双尾的裂缝厚翼型。该两个翼型处在不同的黎曼面内。翼型的内部在复位势平面中可转绘成无限长的条带。利用这些结果得到了二元拉伐尔喷管内不可压缩位势流动的精确解。精确解对任意给定的收缩比n₁、扩张比n₂和喉部壁面曲率半径R*都适用。作为应用的举例,给出了一些典型的喷管型线,喷管内的流速分布以及不同瞬间流体质点的所在位置。     

Effect of magnetic field on the flow of a fourth order fluid

A study is made of the flow engendered in a semi-infinite expanse of an incompressible non-Newtonian fluid by an infinite rigid plate moving with an arbitrary velocity in its own plane. The fluid is considered to be fourth order and electrically conducting. A magnetic field is applied in the transverse direction to the flow. The nonlinear problem is solved for constant magnetic field analytically using reduction methods as well as numerically and expressions for the velocity field are obtained. Limiting cases of interest can be deduced by choosing suitable parametric values.     

A problem of steady-state electrochemical shaping with a non-Schlicht velocity hodograph

We solve the problem of the steady-state electrochemical shaping by two semi-infinite cathode plates oriented and located arbitrarily with respect to the direction of the feed motion. A characteristic feature of this problem is a non-schlicht velocity hodograph.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

理想塑性固体中逐步扩展裂纹的弹塑性场

本文从三维的塑性流动理论出发,导出了关于理想塑性固体平面应变问题的基本方程。利用这些方程,分析了不可压缩理想塑性固体的逐步扩展裂纹顶端的弹塑性场。得到了关于应力和速度的一阶渐近场。分析了弹性卸载区的演变过程和中心扇形区的发展过程。预示了出现二次塑性区的可能性。最后给出了关于应力场二阶渐近分析。     

Hydromagnetic flow near an oscillating porous flat plate

A solution in the closed form has been obtained for the flow of a viscous incompressible fluid of small electrical conductivity near an infinite insulated porous flat plate oscillating harmonically in its own plane in the presence of a transverse magnetic field of uniform strength fixed relative to the fluid. Small uniform suction has been imposed along the surface of the plate. This is a generalisation of the result of Ong and Nicholls for the hydromagnetic flow near an oscillating solid flat plate.     

The Boundary Value Problem for the Nonlinear Shallow Water Equations

We propose an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations. Our work, based on the Carrier and Greenspan [ 1 ] hodograph transformation, focuses on the propagation of nonlinear nonbreaking waves over a uniformly plane beach. Available results are briefly discussed with specific emphasis on the comparison between the Initial Value Problem and the BVP; the latter more completely representing the physical phenomenon of wave propagation on a beach. The solution of the BVP is achieved through a perturbation approach solely using the assumption of small waves incoming at the seaward boundary of the domain. The most significant results, i.e., the shoreline position estimation, the actual wave height and velocity at the seaward boundary, the reflected wave height and velocity at the seaward boundary are given for three specific input waves and compared with available solutions.     

Expansion of a wedge of non-ideal gas into vacuum

We study the problem of expansion of a wedge of non-ideal gas into vacuum in a two-dimensional bounded domain. The non-ideal gas is characterized by a van der Waals type equation of state. The problem is modeled by standard Euler equations of compressible flow, which are simplified by a transformation to similarity variables and then to hodograph transformation to arrive at a second order quasilinear partial differential equation in phase space; this, using Riemann variants, can be expressed as a non-homogeneous linearly degenerate system provided that the flow is supersonic. For the solution of the governing system, we study the interaction of two-dimensional planar rarefaction waves, which is a two-dimensional Riemann problem with piecewise constant data in the self-similar plane. The real gas effects, which significantly influence the flow regions and boundaries and which do not show-up in the ideal gas model, are elucidated; this aspect of the problem has not been considered until now.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

二维流面上的流动问题的速度图方法

对于一些特殊的流动,尤其是平面上的位势流动,速度图方法有其显著的优点．对于理想流体来说,流面总是存在的,在流面上,流动的速度向量总是在其切空间里．通过引入流函数和势函数,采用张量分析作为工具,给出了二维曲流面上位势流动的速度图方法,得到了流函数满足的速度图方程,为一些特殊的流动问题提供了一类分析方法．并且,对于得到的二维速度图方程,得到了相应的特征方程和特征根,从而可以对方程的类型进行分类．最后,给出了一些特殊流动的实例．     

Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space

This article looks into the unsteady rotating magnetohydrodynamic (MHD) flow of an incompressible second grade fluid in a porous half space. The flow is induced by a suddenly moved plate in its own plane. Both the fluid and plate rotate in unison with the same angular velocity. Analytic solution of the governing flow problem is obtained by using Fourier sine transform. Based on the modified Darcy's law, expression for velocity is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. Several existing solutions of Newtonian fluid have been also deduced as limiting cases.     

On the solution of problems of filtration with a limit gradient

The plane stable filtration of an incompressible liquid with a limit gradient is considered /1/. A special non-linear filtration law is introduced, for which the basic system of equations obtained by transformation of the hodograph /2/ has a general solution which enables the theory of functions of a complex variable to be effectively employed. As a special case the proposed law contains the law considered in /3/. Solutions of the problems of the motion produced by a source in a narrow zone, and the motion from a source-sink pair are presented.     

Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.     

The Wave Field near the Point of Incidence of the Limiting Ray

The plane scalar problem on the refraction of a high-frequency wave, given by its ray expansion, from a curvilinear interface of two media is considered. It is assumed that the velocity in the medium where the refracted wave propagates is larger than the velocity in the medium where the incident wave propagates. It is also assumed that, on the interface, there is a point on one side of which the ordinary refraction of the wave holds and on the other side of which the complete internal reflection of the wave occurs. An analytic expression of the wave field near this limiting point is found. Bibliography: 8 titles.     

Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

A new kind of analytic technique, namely the homotopy analysis method (HAM), is employed to give an explicit analytical solution of the steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. A uniform transverse magnetic field is applied normal to the surface. An explicit analytical solution is given by recursive formulae for the first-order power-law (Newtonian) fluid when the ratio of free stream velocity and stretching velocity is not equal to unity. For second and real order power-law fluids, an analytical approach is proposed for magnetic field parameter in a quite large range. All of our analytical results agree well with numerical results. The results obtained by HAM suggest that the solution of the problem under consideration converges.     

Scattering of fluid loaded elastic plate waves at the vertex of a wedge of arbitrary angle, I: analytic solution

This article is the first part of an investigation into thescattering of fluid coupled structural waves by an angular discontinuityat the junction of two plates of different material properties.These two thin elastic plates are semi-infinite in extent thereforeforming the faces of an infinite wedge, the interior of whichcontains a compressible fluid. A plane unattenuated structuralwave is incident along the lower face of the wedge and is scatteredat the apex. The edges of the elastic plates may be joined ina variety of different ways, for example, they may be pin-jointedto an external structure or welded to each other. In the formercase, the plates will experience only the usual flexural vibrationswhereas in the latter case longitudinal (in-plane) disturbanceswill be generated and will propagate away from the wedge apex. An exact explicit solution is sought in terms of a Sommerfeldintegral representation for the fluid velocity potential. Thispermits the boundary-value problem to be recast as a functionaldifference equation which is easily solved in terms of the Maliuzhinetsspecial function (Maliuzhinets, Soviet Phys. Dokl. 3 1958).The chosen ansatz for the solution is of a different form fromthat used previously by the authors for the less complicatedmembrane wedge problem. The new ansatz has several analyticand numerical advantages which enable the reflection and transmissioncoefficients to be expressed explicitly in a compact form thatis ideal for computation. In the second part of this study a full numerical investigationof the reflection and transmission coefficients will be presentedfor a variety of interesting parameter ranges and edge conditions.     

Non-local conservation laws for the equations of the irrotational isentropic plane-parallel gas motion

By introducing non-local variables, namely, the velocity potential and the stream function, and changing to the hodograph plane, the problem of finding the conservation laws for a non-linear system, describing the plane-parallel steady irrotational isentropic gas motion is reduced to the problem of finding the conservation laws for a linear Chaplygin system. The conservation laws of the zeroth and first orders for the Chaplygin system are obtained. It is established that the set of conservation laws of zeroth order that a Chaplygin system possesses consists of conservation laws that are linear in the velocity potential and the stream function, and a new non-linear conservation law. The linear conservation laws have functional arbitrariness. They produce linearity of this system and are defined by Green's operator formula. It turns out that all the conservation laws in the physical plane, obtained by Rylov, are generated by a linear combination of these linear conservation laws and trivial conservation laws. All the linear conservation laws of the first order for the Chaplygin system, generated by Green's operator formula, that are independent of the stream function, are obtained. It is shown that the Chaplygin system has no more than three first-order conservation laws, independent of the stream function, which are not a linear combination of these linear conservation laws and trivial conservation laws, and their components are quadratic in the velocity potential and its derivatives. All the Chaplygin functions for which the Chaplygin system has three such conservation laws are listed. These conservation laws are obtained.