Exact solutions of model BGK Boltzmann equation in temperature jump and weak evaporation problems

An exact solution to the model Boltzmann equation with Bhatnagar-Gross-Krook (BGK) collision operator is obtained in the problems of weak evaporation and temperature and density jumps of a rarefied gas in a half-space. Case's method is used to find generalized eigenvectors of the corresponding characteristic equation. An existence and uniqueness theorem for the solution of the posed problems with boundary conditions on a flat surface and far from it is proved. For this, we develop a formalism of diagonalization and factorization of the vector Riemann-Hilbert boundary-value problem with matrix coefficient whose diagonalizing matrix has branch points in the complex plane.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–171, January–February, 1992.     

THE EXACT INTEGRAL EQUATION OF HERTZ’S CONTACT PROBLEM

This paper presents the exact integral equation of Hertz‘s contact problem,which isobtained by taking into account the horizontal displacement of points in the contactedsurfaces due to pressure.     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997     

Influence of temperature on the slip velocity of semidilute xanthan gum solutions

We obtain experimental evidence of the influence of temperature in the range 12-32°C on the slip phenomena of two different 0.3% xanthan solutions in a glass capillary rheometer. Enhancement of the slip velocity was observed for both samples around the corresponding thermally induced order-disorder transition temperature. Intrinsic viscosity measurements were performed to find the conformation change of both samples. Concentrations of 0.15% and 0.2% were analyzed for one sample, showing absence of slip at 0.15%.Slip velocity measurements were determined with the traditional Mooney method for a L/D ratio of the capillaries (640) enough to neglect entry head losses. Comparisons were done with the method developed by Piau et al. (1990) and with the one developed by Hatzikiriakos and Dealy (1992). The resulting behavior of the slip velocity with the capillary diameters, calculated with the method of Hatzikiriakos and Dealy, was contrary to the behavior experimentally found by other authors. The observed differences in the slip velocity, measured with the other two methods, were proportional and nearly independent of temperature and diameter of the capillaries.     

The velocity slip of polar fluids

Summary The theory of polar fluids, briefly reviewed and shown to be identical to the theory of micropolar fluids, is applied to five different viscometric flows and to the flow around a slowly rotating sphere. Comparing theoretical predictions with experimental results for the viscometric flows it is demonstrated that the theory may serve as a model for the flow of suspensions if a velocity slip along a solid wall is accounted for.In the important limit, in which the length scale of the microstructure is much less than a characteristic geometric length theNavier-Stokes equations subject to a velocity slip boundary condition are all one has to work with for the evaluation of the apparent fluidity; a positive slip coefficient guarantees the sigma phenomenon to result. For other flows one may also adopt this procedure as long as one is interested only in average macroscopic quantities. In this way it turns out that the classical expressions for the hydrodynamic force and torque on a slowly translating and rotating sphere may be employed if the geometric radius is replaced by an apparent one.But even for the evaluation of local quantities, where the full system of equations has to be used, one needs a velocity slip boundary condition. Only then does a qualitative agreement with the experimentally determined details of the flow field result. It is demonstrated that such a requirement is a natural and physically expected supposition for any continuum approximation of a fluid with internal structures.

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Zusammenfassung Die Theorie der polaren Flüssigkeit, welche kurz referiert und für die gezeigt wird, daß sie identisch ist mit der Theorie der mikropolaren Flüssigkeit, wird auf 5 stationäre Schichtenströmungen sowie auf die Strömung um eine langsam rotierende Kugel angewendet. Durch Vergleich von theoretischen Resultaten mit experimentellen Ergebnissen für Schichtenströmungen wird gezeigt, daß die Theorie durchaus als Modell für die Strömung von Suspensionen geeignet ist, vorausgesetzt, daß eine Geschwindigkeitsgleitung entlang einer Wand mitberücksichtigt wird.In dem entscheidenden Grenzfall, in dem die Größe der Mikrostruktur klein ist im Vergleich zu charakteristischen Makrodimensionen, hat man zur Bestimmung der scheinbaren Fluidität nur dieNavier-Stokes-Gleichungen zu lösen, allerdings unter der Randbedingung einer Geschwindigkeitsgleitung; ein positiver Gleitungskoeffizient führt dann zum Sigma-Effekt. Auch auf andere Strömungen ist diese Methode anwendbar, solange man sich nur für irgendwelche Mittelwerte interessiert. Auf diese Weise zeigt sich, daß die klassischen Ausdrücke für die hydrodynamische Kraft und das Drehmoment auf eine sich langsam bewegende starre Kugel ihre Gültigkeit behalten, wenn man den geometrischen Radius durch einen scheinbaren ersetzt.Aber sogar zur Berechnung lokaler Größen, für die man das vollständige Gleichungssystem benötigt, muß eine Geschwindigkeitsgleitung an der Wand angenommen werden. Denn nur dann erhält man eine qualitative Übereinstimmung mit dem experimentell bestimmten Strömungsprofil. Es wird gezeigt, daß eine derartige Annahme für jede Approximation einer Flüssigkeit mit innerer Struktur durch ein Kontinuum physikalisch zu erwarten ist.

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With 2 figures     

On the uniqueness of boundary integral equation for the exterior Helmholtz problem

From the point of view of energy analysis,the cause that the uniqueness of theboundary integral equation induced from the exterior Helmholtz problem does not hold isinvestigated in this paper.It is proved that the Sommerfeld’s condition at the infinity ischanged so that it is suitable not only for the radiative wave but also for the absorptive wavewhen we use the boundary integral equation to describe the exterior Helmholtz problem.Therefore,the total energy of the system is conservative.The mathematical dealings toguarantee the uniqueness are discussed based upon this explanation     

SOLVING VIBRATION PROBLEM OF THIN PLATESUSING INTEGRAL EQUATION METHOD

ＳＯＬＶＩＮＧＶＩＢＲＡＴＩＯＮＰＲＯＢＬＥＭＯＦＴＨＩＮＰＬＡＴＥＳＵＳＩＮＧＩＮＴＥＧＲＡＬＥＱＵＡＴＩＯＮＭＥＴＨＯＤ￥(许明田，程德林)ＸｕＭｉｎｇｔｉａｎ；ＣｈｅｎｇＤｅｌｉｎ（Ｄｅｐａｒｔｍｅｎｔ．ｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＰｈｙｓｉ...     

三维Helmholtz积分方程外问题几乎奇异积分的半解析算法

边界元法求解声场Helmholtz外问题时,由于简单闭合曲面外的无限域边界积分方程与原边值问题(外问题)不完全等价,从而会在某些激励波数(与相应内问题的特征波数重合)下不能获得唯一解。文章引入一种计算几乎奇异积分的半解析算法,结合CHIEF点法,在较宽的波数范围内计算了声场外问题近场和远场内的声压。计算结果表明,该算法不仅有效地克服了频域内解的非唯一问题,而且与单纯的CHIEF点法相比能够显著提高计算精度。     

Intrinsic instability of the lattice BGK model

Based on the stability analysis with no linearization and expansion, it is argued that instability in the lattice BGK model is originated from the linear relaxation hypothesis of collision in the model. The hypothesis stands up only when the deviation from the local equilibrium is weak. In this case the computation is absolutely stable for real fluids. But for flows of high Reynolds number, this hypothesis is violated and then instability takes place physically. By performing a transformation a quantified stability criteria is put forward without those approximation. From the criteria a sufficient condition for stability can be obtained and serve as an estimation of the limited Reynolds number as high as possible.     

Numerical solution of multiple hole problem by using boundary integral equation

This paper studies a numerical solution of multiple hole problem by using a boundary integral equation. The studied problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. One numerical example is provided to check its accuracy.     

Computational model of boundary integral equation in solid mechanics

In the first part of the paper, the computational model of boundary integral equation in solid mechanics is presented while in the second part the model is used in the solution of two problems of solid mechanics.     

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.     

Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

A singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick–slip flow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coefficients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coefficients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made. Copyright © 2005 John Wiley & Sons, Ltd.     

Hypersingular integral equation for multiple curved cracks problem in plane elasticity

The complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations. These hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor, SIF, for the problem considered. Numerical examples for double circular arc cracks are presented.