Plane elastic solution for the second mixed boundary value problem and its application

Summary A solution is obtained for the second mixed boundary value problem with two boundaries loaded with external forces, and the other two boundaries simply supported. A rational mapping function formed as a sum of fractional expressions and complex stress functions is used for the analysis. A closed form solution can be obtained under uniform tensile and shearing loadings. These solutions are applicable to crack problems in linear elastic fracture mechanics. In order to illustrate the general solution, a double connected region is considered, in which interaction between a square hole with a crack extension and another crack is investigated. Stress distributions and stress intensity factors are obtained.

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Ebene elastinche Lösung des zweiten gemischten Randwertproblems und ihre Anwendung

Übersicht Die Lösung des zweiten gemischten Randwertproblems der ebenen Elastizität mit zwei kraftbeanspruchten und zwei gestützten Rändern wird dargestellt. Für die Analyse wird eine rationale Abbildung verwendet, die als Summe partieller Lösungen aus komplexen Spannungsfunktionen aufgebaut ist. Für gleichmäßige Zug- und Schubbeanspruchungen kann die Lösung in geschlossener Form dargestellt werden und ist verwendbar in Rißaufgaben der linear elastischen Bruchmechanik. Als Illustration der allgemeinen Lösung dient ein verzweigter Bereich der gegenseitigen Einwirkung einer quadratischen Öffnung mit Seitenkerbe und eines Risses. Die Spannungsverteilungen und Spannungsintensitäten werden dabei berechnet.

     

The exact solution of Stokes' second problem including start-up process with fractional element

The start-up process of Stokes' second problem ofa viscoelastic material with fractional element is studied. Thefluid above an infinite flat plane is set in motion by a suddenacceleration of the plate to steady oscillation. Exact solutionsare obtained by using Laplace transform and Fourier transform.It is found that the relationship between the first peakvalue and the one of equal-amplitude oscillations dependson the distance from the plate. The amplitude decreases forincreasing frequency and increasing...     

Experimental investigation on the interaction between polymer solution jet and free surface

Turbulence modification with polymer additives, i.e., Tom's effect, is a well known phenomenon. In this study, Tom's effect at a free surface was experimentally investigated. The turbulence at a free surface was generated by a horizontal liquid jet. A new specklegram technique was proposed in order to quantitatively measure the free-surface shapes caused by a turbulent jet. The specklegram method was very simple and was able to evaluate the free-surface waves accurately. The measurement confirmed that the surface of a polymer solution was less wavy than that of water. The jet beneath the free surface was measured by the LDV. The turbulence diffusion of the polymer jet was much smaller than that of the water jet. The surface turbulence was found to be modified by the polymer solution indirectly. The modification was a consequence of the Tom's effect at the shear layer around the jet.     

On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem

The present work concerns the investigation of the two-dimensional direct scattering problem of time-harmonic elastic waves from bounded anisotropic components of isotropic media. We obtain a Fourier series expansion for the elastic field in the interior of the anisotropic inclusion based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, provided that the inclusion exhibits orthotropic symmetry. This expansion is then exploited to acquire a semi-analytical solution to the associated elastic transmission scattering problem. Numerical results for several geometric configurations and varying degree of anisotropy are presented revealing the pronounced effect of the specific anisotropic character on the scattering mechanism.     

Experimental investigation of the asymptotically approached Mach reflection over the second surface in a double wedge reflection

The assumption that the Mach reflection which is formed over the second surface of a double wedge with angles _w ¹ and kw/2 approaches asymptotically the Mach reflection which would have been obtained by an identical incident shock wave over a single wedge with an angle _w = _w ² was verified experimentally. The verification of this assumption supports the shock polar analysis suggested by Ben-Dor et al. (1987) for the study of the reflection process of a planar shock wave over a double wedge. Measurements of the rate of approach to the asymptotic value are also provided.     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

On the solution of the Darboux problem

We consider the problem of determining the angular position of a rigid body in space from its known angular velocity and initial position (the Darboux problem) in quaternion setting. For an arbitrary angular velocity vector of the body, we present a solution based on Lappo-Danilevskii’s recursion relations [1]. New special cases of solvability of the Darboux problem in closed form are obtained.     

On the solution of the sector problem

A rigorous study of the sector problem is presented by using the Mellin transform technique. The stress function is obtained as an asymptotic expansion of the complex inversion integral. The number of terms of this expansion, as well as the differentiability of the stress function, depend on the differential properties of boundary conditions on the radial edges. If these boundary conditions belong toC , this asymptotic expansion is transformed to a uniformly convergent infinite series. The coefficients of the series, which depend only on the boundary conditions along the circumferential edges, are calculated by applying a bi-orthogonality condition, or, by a technique based on the Betti formula.

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Résumé En utilisant la technique de la transformation de Mellin on présente une étude rigoureuse du problème d'un secteur. La fonction des contraintes est obtenue par un développement asymptotique de l'intégral complexe d'inversion. Le nombre de termes de ce développement, ainsi que la differentiabilité de la fonction des contraintes, dépendent de proprietés différentielles des conditions aux limites sur les bords radiaux. Si ces conditions aux limites sont dansC , ce développement asymptotique est transformé en série infinie uniformement convergente. Les coefficients de cette série, qui dépendent uniquement de conditions aux limites sur le bord circulaire, sont calculés à l'aide d'une condition de bi-orthogonalité que nous démontrons, ou, à l'aide d'une technique basée sur la formule de Betti.

     

基于低雷诺数串置翼型气动特性研究

基于低雷诺数,应用二维CFD方法,对串置翼型的气动特性进行了数值模拟。比较了串置翼型布局与常规单独翼型的升阻特性。分析了翼差角度对串置翼升阻特性的影响,并且在保持鸭翼±5°偏角不变的同时,研究了两翼之间的距离以及安装的相对高度对整个串置翼型升阻特性的影响。发现上鸭翼与主翼相结合具有相对较高的最大升力系数和临界迎角, 可以显著改善串置翼型的气动特性。     

Self-similar solution of the thawing soil problem

A mathematical model of phase transitions in frozen soils containing unfrozen water is proposed. It is shown that phase transitions in frozen soils always occupy an extended zone. The problem of the interaction of frozen rock with a salt solution is solved on the assumption that the interface between the solution and the frozen rock is permeable both for the liquid and for the dissolved impurity. This problem arises, for example, in drilling wells in frozen ground, when the circulating drilling solution is an aqueous salt solution [7]. A series of natural processes is based on the interaction between groundwaters having different, possibly negative, temperatures and different degrees of mineralization and the surrounding frozen rock [8] and on the thawing of the frozen bed of northern seas in contact with saline seawater [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–142, November–December, 1988.     

One approximate solution of the Nekrasov problem

An approximate solution ω = A[ω, μ] of the nonlinear integral Nekrasov equation is obtained by successive replacement of the kernel of the integral operator by a close one. The solution is sought not directly at the bifurcation point μ₁ = 3 of the linearized equation ω = μL[ω] but at the point μ = 1 at which operator A[ω, μ], remaining nonlinear in ω, is linear in μ. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 50–56, November–December, 2007.     

Viscous solution of the triple-shock reflection problem

The reflection of a triple-shock configuration was studied numerically in two dimensions using the Navier–Stokes equations. The flow field was initialized using three shock theory, and the reflection of the triple point on a plane of symmetry was studied. The conditions simulated a stoichiometric methane-oxygen detonation cell at low pressure on time scales preceding ignition when the gas was assumed to be inert. Viscosity was found to play an important role on some shock reflection mechanisms believed to accelerate reaction rates in detonations when time scales are small. A small wall jet was present in the double Mach reflection and increased in size with Reynolds number, eventually forming a small vortex. Kelvin–Helmholtz instabilities were absent, and there was no Mach stem bifurcation at Reynolds numbers corresponding to when the Mach stem had travelled distances on the scale of the induction length. Kelvin–Helmholtz instabilities are found to not likely be a source of rapid reactions in detonations at time scales commensurate with the ignition delay behind the Mach stem.     

Experimental investigation of flow and heat transfer for the ethanol-water solution and FC-72 in rectangular microchannels

Rapid development of super scale integration circuit (IC) provides unprecedented challenge to thermal control for aviation electronic equipments. To solve the problem of cooling electronic chips and devices for aircraft avionics, this paper experimentally investigated the characteristics of single-phase forced convection heat transfer and flow resistance in rectangular microchannels with two liquid coolants. One was 30% of ethanol–water solution, the most commonly used coolant in aviation. The other was FC-72, the latest coolant for electronic equipments. Based on the experimental data collected and those available in the open literature, comparisons and analyses were carried out to evaluate the influences of liquid velocity, supercooling temperature, microchannel structures and wall temperature etc. on the heat transfer behaviors. And the correlations of flow resistance and heat transfer characteristics were provided for the ethanol–water solution and FC-72 respectively. The results indicate transition from laminar to turbulent flow occurs at the Reynolds number of 750–1,250 for FC-72, and the behaviors of flow and heat transfer in rectangular microchannels strongly depend on the kind of coolant and geometric configuration of microchannels.