Resonant oscillations in open tubes with a mean flow

Summary The resonant response of a jet of gas to periodic pressure fluctuations near the inlet-cross-section of a tube is investigated. It is assumed that the cross-section of the jet just matches the inlet cross-section. It is found that the pressure disturbances within the tube can grow very large and also the acoustic power which is indicated from the outlet of the tube can be much larger than the power input owing to the forcing pressure oscillation.

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Zusammenfassung Die resonante Anregung eines Gasstrahls durch periodische Druckstörungen nahe des Eintrittsquerschnittes eines Rohres wird untersucht. Es wird angenommen, dass der Querschnitt des Strahls mit dem Eintrittsquerschnitt übereinstimmt. Man findet, dass die Druckstörungen innerhalb des Rohres sehr gross werden können und dass die vom Ende des Rohres abgestrahlte akustische Leistung viel grösser sein kann als die Leistung, die auf Grund der Druckstörungen beim Rohreintritt eingespiesen wird.

     

On the flow of a cosserat fluid through a curved pipe

Summary A theoretical analysis is made of the flow of a Cosserat fluid in a curved pipe under a pressure gradient. It is assumed that the curvature of the pipe is small, that is the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross-section. Following Dean [2] a solution is developed by the method of successive approximation. The paths of the particles in the central plane and the projection of the streamlines on the cross-section of the pipe are compared with those of a Newtonian fluid. It is observed that in the theory of Cosserat fluids the curvature of the streamlines in the central plane increases and the motion is slower in the cross-section of the pipe. It is also shown that the rate of flow of a Cosserat fluid through a curved pipe is decreased due to the curvature of the pipe.

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Résumé On fait une analyse théorique de l'écoulement d'un fluide de Cosserat dans un tube sous un gradient de pression. On suppose que la courbature du tube est faible, c'est-à-dire que le radius du cercle qui fait la ligne du centre du tube est fort par rapport au radius de la coupe transversale.D'après Dean [2], on développe une résolution par approximations successives. On fait la comparaison des trajectoires des particules dans le plan central et la projection des lignes d'écoulement sur la coupe transversale du tube avec celles d'un fluide de Newton.On note que dans la théorie des fluides de Cosserat, la courbature des lignes d'écoulement dans le plan central augmente, et que la motion est ralentie dans la coupe transversale du tube. On démontre ensuite que le taux d'écoulement d'un fluide de Cosserat dans un tube courbe se diminue à raison de la courbature.

     

Dusty viscous flow through a cylinder of triangular cross-section

Unsteady laminar flow of a dusty viscous, incompressible fluid through a cylindrical tube of triangular cross-section is considered in two cases: (i) when the pressure gradient varies harmonically with time and (ii) when it varies exponentially. The velocity fields for the fluid and dust particles have been determined. Flux and skin-friction drag on the walls of the cylinder have been calculated and particular cases discussed.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg–de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Joined dissimilar thin elastic tubes containing steady perfect flow

Two tubes of circular cross-section and of the same radius initially, but composed of different elastic materials, are joined together to form a single straight tube with discontinuous properties. Stretching in the axial direction causes radial displacement that varies along the length of the tube. In a perfect inviscid incompressible flow through the tube of variable cross-section the internal pressure varies as described by Bernoulli's equation, and the variable pressure also causes variation of the radial displacement. The equations of coupled finite deformation and fluid pressure problem can be integrated explicitly (using membrane theory for the tube) for arbitrary material properties, but determination of the integration constants is not trivial. The results are interpreted numerically for Mooney materials. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical illustrations are obtained for various upstream velocities. The results obtained here thus solve the problem of steady internal pressure loading of this type in extended dissimilar thin isotropic tubes. The tube will become unstable if the fluid velocity is too large. Applications to engineering structures are possible.     

Geometrically induced discrete spectrum in curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same cross-section and that the discrete spectrum is not empty.     

蠕动流中圆球阻力系数的入口效应

本文研究了圆球在半无穷长圆管入口处的蠕动流。得到了速度分布,压力分布和流函数的无穷级数形式的分析解.采用配置法将无穷级数截断并确定出级数中各项系数.求出了均匀入口流绕静止圆球以及圆球以瞬时速度在管内静止流体中运动这两种情形下圆球的阻力系数以及圆球表面上的应力分布.结果表明,当圆球在入口处运动时会遭受到较无穷圆管内为大的阻力.本文还对配置法的收敛性进行了数值实验.试验证明,该法具有好的收敛性.     

Complementary Variational Principles for Knudsen Flow Rates

A collisionless gas flows through the interior of a tube ofcircular cross-section, having both an entrance and an exitand which reflects molecules diffusely. Suitable complementaryvariational principles are used to obtain upper and lower boundsfor the transmission probability of the tube. The numericalresults, believed to be the most accurate to date, are comparedwith those of other authors.     

Long Wavelength Acoustic Scattering by a Torus of Arbitrary Aspect Ratio

Both the near and far field scattering potentials are calculatedto fourth order in the small parameter ratio of dimension ofbody to incident wavelength for the scattering of axial, planeacoustic waves by a rigid torus of arbitrary aspect ratio (theratio of major to minor radii). The near field potentials aredetermined from sets of toroidal harmonics and biharmonics which,although complete, are non-orthogonal. Consequently the boundaryconditions generate recurrence relations between successiveamplitudes, which are solved using difference equation theory.The far field potentials follow from the near field potentialsthrough Rayleigh's matching procedure. The dependence of fieldquantities and the scattering cross-section on aspect ratiois illustrated. The cylindrical limit of toroidal geometry isdemonstrated.     

Beschleunigung eines Kolbens in einem Stosswellenrohr mit veränderlichem Querschnitt

Zusammenfassung Es wird ein vereinfachtes, eindimensionales Modell beschrieben für die Beschleunigung eines Kolbens in einem Stosswellenrohr, dessen Querschnitt sich nach einer gewissen Lauflänge sprunghaft erweitert. Numerische Rechnungen werden mit Messungen im Stosswellenrohr verglichen.

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Summary A simplified, one-dimensional model is developed to describe the acceleration of a piston in a shock tube having a discontinuous increase in cross-section. Numerical calculations are compared with shock tube experiments.

     

On the Number of Tubes Touching a Sphere or a Tube

A problem is formulated about how many unit-radius tubes can touch a ball of given radius from the outside and from the inside. Upper bounds for the maximum numbers of contacts are obtained for both interior and exterior contacts. It is also shown that the maximum number of unit-radius tubes touching the same orthogonal cross-section of a particular tube of radius P is [π (arcsin(P+1)⁻¹)⁻¹] and if the number of contacts takes on its maximum, then all tubes are locally aligned.     

Eigenvalues and eigenfunctions for general graetz problems

A method is derived for the determination of the eigenvalues and the corresponding eigenfunctions which arise in the problem of forced convection of heat through an infinite tube of arbitrary cross-section. The solution is obtained in terms of the eigenvalues and eigenfunctions of the related reduced wave equation, and involves the calculation of the eigenvalues of a suitable symmetric matrix.     

弯曲血管涡旋分离区的确定

弯曲血管中动脉粥状硬化和涡旋区的产生密切相关.本文在牛顿粘性不可压缩流体作定常层流运动的前提下,采用动量积分关系式方法和准常曲率的概念.对曲率小,曲率变化不太大的弯曲血管,得到了确定分离点和再附点位置的方程.据此可确定最早分离的地方及对应的临界雷诺数以及对不同方位角分离点和再附点依赖于雷诺数R_e的曲线.分析所得结果表明,分离首先发生在曲率变化最剧烈的最外侧(或最内侧),随着R_e数的增加分离区向主流的上下游.方位角和矢径方向延伸从而形成三维的分离泡,分离泡在三个方向上的尺寸随着R_e数的增加不断地扩大本文所得的结果还从理论上阐明了实验观察到的下述饶有趣味的事实:对于对称的弯曲血管,如果在上游外侧有一分离泡,则在下游内侧与其对称的地方必有另一分离泡存在.     

一般旋转壳在轴对称变形下的复变量方程

本文在Love-Kirchhoff的假定下,求得了一般旋转壳在轴对称变形下的复变量方程.当旋转壳是圆截面环壳时,这些方程简化为F.T?lke(1938)[3],R.A.Clark(1950)和B.B.Новожилов(1951)[3]的方程.当平均半径R比环截面半径a大得很多时,求得了细环壳的复变量方程,当这个细环壳的截面是圆形时,简化作为作者(1979)[6]的圆截面的细环壳复变量方程,我们列出了椭圆截面的细环壳复变量方程.当椭圆截面近似于圆截面时,该方程在形式上和圆细环壳方程基本相同.     

Asymptotics of the solution to the Neumann problem in a thin domain with sharp edge

An asymptotic expansion of the solution to the Neumann problem for a second-order equation in a thin domain with peak-like edge is constructed and justified. Owing to the sharpness of the edge, the procedure of dimension reduction leads to a degenerate limit equation on the longitudinal cross-section of the domain and a solution has irregular behavior near the boundary. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 193–219.     

Asymptotic spectral analysis in semiconductor nanowire heterostructures

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between material composition and geometric parameters like curvature and torsion. First, the macroscopic behaviour of a nanowire made of composite fibres with microscopic periodic texture is analysed, which amounts to determining the asymptotic behaviour of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross-section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm–Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d–1d dimension reduction for the two-scale problem in a variational framework by means of Γ-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross-section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.     

Software for interpolating and modelling unknown functions

Software has been developed for fitting a stochastic process model to multi-dimensional data. Applications include contouring, cross-section plotting and optimization. The behaviour of the variance and the gradient of the interpolating function in the near neighbourhood of closely adjacent data points has been investigated.     

Scattering of a spherical wave by a small ellipsoid

A point generated incident field impinges upon a small triaxialellipsoid which is arbitrarily oriented with respect to thepoint source. The point source field is so modified as to beable to recover the corresponding results for plane wave incidencewhen the source recedes to infinity. The main difficulty insolving analytically this low-frequency scattering problem concernsthe fitting of the spherical geometry, which characterizes theincident field, with the ellipsoidal geometry which is naturallyadapted to the scatterer. A series of techniques has been usedwhich lead finally to analytic solutions for the leading twolow-frequency terms of the near as well as the far field. Incontrast to the near-field approximations, which are expressedin terms of ellipsoidal eigenexpansions, the far field is furnishedby a finite number of terms. This is very interesting becausethe constants entering the expressions of the Lamé functionsof degree higher than three are not obtainable analyticallyand therefore, in the near field, not even the Rayleigh approximationcan be completely obtained. On the other hand, since only afew terms survive at the far field, the scattering amplitudeand the scattering cross-section are derived in closed form.It is shown that, in practice, if the source is located a distanceequal to five or six times the biggest semiaxis of the ellipsoidthe Rayleigh term of the approximation behaves almost as theincident field was a plane wave. The special cases of spheroids,needles, discs, spheres as well as plane wave incidence arerecovered. Finally, some theorems concerning monopole and dipolesurface potentials are included.     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.     

Segregation of Carbon Paste in a Smelting Electrode

Requests for reprints to A. C. Fowler An industrial problem concerning segregation in carbon pasteis described. Billets of paste, which are a mixture of cokeand pitch, are fed into a steel tube in a furnace, and the tubeis lowered at a rate of about 0.75 m day^–1. In the tube,the pitch is defonnable, and the paste is essentially a veryviscous two-phase participate medium. The grain size distributionis essentially bimodal, consisting of coarse ({small tilde}10mm) particles in a distribution of finer (100 µm) particles.Segregation occurs near the walls of the tube, leading to alower content of coarse particles; such segregation is unwanted,and so the length of this segregated region is required to beminimized. It is suggested that segregation is due to the stiffnessof the coarse particulate mixture, which is less able to deformthan the pitch/fines mixture. Consequently, the length of thesegregated region is controlled by the coarse mixture viscosity,and an expression is derived for the segregation length.