THE METHOD OF CONTINUOUS DISTRIBUTION OF SINGULARITIES TO TREAT THE STOKES FLOW OF THE ARBITRARY OBLATE AXISYMWETRICAL BODY

This paper deals with the Stokes flow of the arbitrary oblate axisymmetrical body by means of constant density and quadratic distribution function approximation for the method of continuous distribution of singularities. The Sampson spherical infinite series arc chosen as fundamental singularities. The convergence, accuracy and range of application of both two approximations are examined by the unbounded Stokes flow past the oblate spheroid. It is demonstrated that the drag factor and pressure distribution both conform with the exact solution very well. Besides, the properties, accuracy and the range of application are getting belter with the improving of the approximation of the distribution function. As an example of the arbitrary oblate axisymmetrical bodies, the Stokes flow of the oblate Cassini oval are calculated by these two methods and the results are convergent and consistent. Finally, with the quadratic distribution approximation the red blood cell, which has physiologic meaning, is considered and for the first time the (orresponding drag factor and pressure distribution on the surface of the cell are obtained.     

The stokes flow of an arbitrary prolate axisymmetric body towards an infinite plane wall

By distributing continuously the image Sampsonlets with respect to the plane and by applying the constant density, the linear and the parabolic approximation, the analytic expressions in closed form for flow field are obtained. The drag factor of the prolate spheroid and the Cassini oval are calculated for different slender ratios and different distances between the body and the plane. It is demonstrated that the proposed method is satisfactory both in convergence and accuracy. Comparison with existing results in the case of prolate spheroid shows that the coincidence is quite good.     

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.     

轴对称体扭转问题的轴对称有限元模拟解

轴对称体的轴对称问题与扭转问题一向被认为是两个互相不能模拟的问题.前者的未知量与方程多于后者,形式也不相同.本文提出一种退化模拟方法.能够把扭转问题模拟为轴对称问题的一类特殊情况来解.一般的结构分析程序都能够分析轴对称问题,但轴对称体的扭转问题通常作为三维问题处理.按本文提出的方法,可用结构分析程序的轴对称分析功能模拟扭转分析.本文还给出模拟计算的算例.计算结果表明与理论解完全一致.本文对退化模拟的材料本构关系进行了研究,建议在数值计算时以各向异性材料的轴对称问题模拟任何材料的扭转问题.当限定用各向同性材料的轴对称问题来模拟时,采用了罚系数法.     

The creeping motion of multiple spherical liquid drops

This paper deals with the drag factor of the multiple spherical liquid drops in the creeping motion by means of the Sampson singularities and collocation technique. The drag factors of the drops are calculated under distinct conditions: different number of liquid drops in the chain and different sphere spacing. From the results the influence of the viscosity ratio on the shielding effect and end effect are revealed. The convergence of the method is also studied in this paper.In this paper the collocation technique developed by Gluckman et al. in treating the rigid sphere case is applied to deal with the creeping motion of multiple spherical liquid drops which has improtant applications in bioengineering and chemical engineering. Writing the general solutions in inner and outer regions of the spheres and satisfying the kinematic and dynamic matching conditions at the collocation points on the interfaces, a set of linear algebraic equations is obtained to determine the unknown coefficients in the solutions. By means of any matrix inversion technique the approximate solutions are presented. In the first section of this paper the mathematic formulation of the problem is given and then in the second section the numerical results are introduced and analysed.     

Visualization of Unsteady Creeping Viscous Flows Using Vector Products

A finite element method for analyzing unsteady incompressible creeping flows is presented. Marker particles are introduced to analyze the flow motions. To determine the marker position in the element, vector products are used. By checking the signs of the product, the marker position during the transient analysis can be determined in a simple manner. A benchmark-type problem for which an analytical solution is available and the filling process of a simple axisymmetrical mould shape are solved to illustrate this method.     

A coupled fluid-elasticity model for the wave forcing of an ice-shelf

A mathematical model for predicting the vibrations of ice-shelves based on linear elasticity for the ice-shelf motion and potential flow for the fluid motion is developed. No simplifying assumptions such as the thinness of the ice-shelf or the shallowness of the fluid are made. The ice-shelf is modelled as a two-dimensional elastic body of an arbitrary geometry under plane-strain conditions. The model is solved using a coupled finite element method incorporating an integral equation boundary condition to represent the radiation of energy in the infinite fluid. The solution is validated by comparison with thin-beam theory and by checking energy conservation. Using the analyticity of the resulting linear system, we show that the finite element solution can be extended to the complex plane using interpolation of the linear system. This analytic extension shows that the system response is governed by a series of singularities in the complex plane. The method is illustrated through time-domain simulations as well as results in the frequency domain.     

按位移推导平面轴对称问题的一般性解答

根据平面轴对称问题的物理概念, 将平面轴对称问题分为轴对称应力问题和轴对称位移问 题, 给出了这两种轴对称问题的基本方程, 并指出平面轴对称位移问题是平面轴对称应力问 题的特例. 在此基础上, 分别按位移推导了平面轴对称应力问题和平面轴对称位移问题的一 般性解答. 按位移推导平面轴对称问题, 可以考虑体力分量, 从而可避免按应力函数推导平 面轴对称应力问题时不能考虑体力分量的局限性.     

Creeping flow past and within a permeable spheroid

Flow past and within an isolated permeable spheroid directed along its axis of symmetry is studied. The flow velocity field is solved using the Stokes creeping flow equations governing the fluid motion outside the spheroid, and the Darcy equation within the spheroid. Expressions for the hydrodynamic resistance experienced by oblate and prolate spheroids are derived and analyzed. The limiting cases of permeable circular disks and elongated rods are examined. It is shown that the spheroid’s resistance varies significantly with its aspect ratio and permeability, expressed via the Brinkman parameter.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation dynamics simulation in finite body: 1. General theory

A unified approach, originating from Cauchy integral theorem, is presented to derive boundary integral equations for two dimensional elasticity problems. Several sets of boundary integral equations are derived and their relations are revealed. Explicit expressions for materials with different symmetry planes are listed. Special attention is given to the formulation that is based on the tractions and the tangential derivatives of displacements along solid boundary, since its integral kernels have the weakest singularities. The formulation is further extended to include singular points, such as dislocations and line forces, in a finite body, so that the singular stress field can be directly obtained from solving the integral equations on the external boundary, without involving the linear superposition technique that was often used in the literature. Its application in simulating discrete dislocation motion in a finite solid body is discussed.     

Conditions for balancing a rotating body in an isolated system with automatic balancers

Conditions are established for automatic balancers with rigid bodies to balance a rotating body that undergoes spatial motion and is a component of an isolated system. It is discovered that a rotating oblate body can be statically balanced when the balance plane is rather close to the center of mass of the system. It is also established that if the rotating body is prolate, the nutation angle can be initially decreased by balancing the body; however, it will then increase because of the dissipation of energy in the system __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 113–120, November 2007.     

细长椭球体在水中自由下落的运动特性实验研究

固体颗粒在液体中的运动现象在日常生活和工业应用领域广泛存在, 其中因蕴含着丰富的流体力学现象而受到学者们的广泛关注. 本文通过实验研究了细长椭球体在水中受浮力影响的下落特性. 实验中采用带有两台相互垂直的高速摄像机和光源组成的运动跟踪平台并结合荧光染色技术对细长椭球体下落过程中的运动轨迹和尾涡结构进行研究. 文中选用的细长椭球体与环境流体的密度比为1.2, 其长短轴比范围为2$\sim $10, 相应的阿基米德数范围为400$\sim$1400, 对应实现的终态雷诺数范围为120$\sim$1350. 实验过程中我们观察到细长椭球体在水中下落过程中产生的5种典型路径, 分别为: 小振幅不规则运动、小振幅高频振荡运动、大振幅低频振荡运动、高度非线性运动以及直线运动, 并得到了对应的速度振荡以及倾斜角的演化规律. 进一步地, 分析了细长椭球体运动过程中受到的阻力系数与雷诺数之间的关系. 随后采用荧光可视化技术清晰获得了颗粒下落过程中的尾涡结构特性, 并结合颗粒的运动状态详细分析了涡脱落过程对颗粒运动状态转捩的影响. 最后, 通过对比前人关于圆柱体下落的运动特性的相关结果, 获得了细长椭球体和细长圆柱体运动特性之间的异同点以及其潜在的物理机理.      

Finite element analysis of axisymmetric elastic body problems

Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used. As the locations of element close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integration have been suggested by various authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper a new form function is suggested. In this new form function, the radial displacementu vanishes as radial coordinatesr approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of element is useful for the calculation of axisymmetric elastic solid body problems.     

The singularity expansion method as applied to the elastodynamic scattering problem

A basic problem in non-destructive testing of materials is the classification of defects with respect to size and geometry. Utilizing elastrodynamic scattering, ultrasonic test methods reduce this problem either to the application of reconstruction algorithms or to the parametrization of experimental data in terms of quantities being related to the physical scattering mechanisms. The so-called Singularity Expansion Method (SEM), originally developed for broadband electromagnetic scattering by arbitrarily shaped targets proves essentially useful to predict and understand impulsive scattering of ultrasound; in addition, even though not yet fully solved in practice, it seems possible to parametrize experimentally obtained time records in a sense which is physically intuitive.

SEM starts either with the eigenfunction expansion in the complex Laplace domain for canonical objects such as a sphere or spheroids, or with a corresponding integral equation formulation for more arbitrarily shaped defects. The essential point is then an expansion in terms of the singularities of the scattered field in the Laplace domain similar to the expansion of transfer functions in linear system theory. It turns out that the location of these singularities is characteristic for the geometry of the scattering body; therefore, it might be a useful tool to parametrize size and shape of the defects.

Several theoretically derived singularity patterns are presented for various body shapes and material compositions, which yield a thorough and physically intuitive interpretation in terms of distinct creeping wave modes. They are compared with first experimental results for a spherical void in a steel test specimen.     

Determining the Parameters of Hydrodynamic Singularities in a Two-Dimensional Flow from the Data on its Free Surface

A technique for determining the coordinates and intensities of hydrodynamic point singularities from the singularity-induced disturbances on the free surface of a heavy inviscid infinite-depth fluid is developed. Steady-state flow past singularities and the motion of singularities of variable intensity in accordance with an arbitrary law are considered. A calculation rule based on the classical least-squares method is formulated for the cases of one and two singularities; the rule can be used for the direct processing of experimental data.     

The solution for axisymmetrical shells with abrupt curvature change (corrugated shells) by the finite element method

It has been noted in the present paper that the finite element method using linear elements for solving axisymmetrical shells cannot be applied to the analysis of axisymmetrical shells with abrupt curvature change, owing to the fact that the influence of the curvature upon the angular displacements has been neglected. The present paper provides a finite element method using linear elements in which the influence of curvature is considered and the angular displacements are treated as continuous parameters. This method has been applied to the calculation of corrugated shells of the type C, and compared with the experimental results obtained by Turner-Ford as well as with the analytical solution given by Prof. Chien Wei-zang. The comparisons have been proved that this theory is correct.     

Stability of a planar resonance satellite motion under spatial perturbations

The satellite motion relative to the center of mass in a central Newtonian gravitational field on an elliptic orbit is considered. The satellite is a rigid body whose linear dimensions are small compared with the orbit dimensions. We study a special case of planar motion in which the satellite rotates in the orbit plane and performs three revolutions in absolute space per two revolutions of the center of mass in the orbit. Perturbations are assumed to be arbitrary (they can be planar as well as spatial). In the parameter space of the problem, we obtain Lyapunov instability domains and domains of stability in the first approximation. In the latter, we construct third- and fourth-order resonance curves and perform nonlinear stability analysis of the motion on these curves. Stability was studied analytically for small eccentricity values and numerically for arbitrary eccentricity values.     

Viscous-fluid outflow from a vessel with account for jet formation

The results of a numerical simulation of slow free-boundary viscous-fluid outflow from a vessel are presented with account for jet formation. The problem is formulated in the creeping motion approximation. For solving the problem, a numerical algorithm for plane geometry, based on an indirect variant of the boundary-element method, is used. As a result of parametric studies, the evolution of the free surface inside the vessel and the jet shape is determined for different values of the governing parameters. Flow regimes with rapid funneling and film formation on the vessel walls are detected. The existence of an asymptotic flow regime is demonstrated using dimensional analysis and confirmed by calculation.     

Creeping flow analyses of free surface cavity flows

Two industrially important free surface flows arising in polymer processing and thin film coating applications are modelled as lid-driven cavity problems to which a creeping flow analysis is applied. Each is formulated as a biharmonic boundary-value problem and solved both analytically and numerically. The analytical solutions take the form of a truncated biharmonic series of eigenfunctions for the streamfunction, while numerical results are obtained using a linear, finite-element formulation of the governing equations written in terms of both the streamfunction and vorticity. A key feature of the latter is that problems associated with singularities are alleviated by expanding the solution there in a series of separated eigenfunctions. Both sets of results are found to be in extremely good agreement and reveal distinctive flow transformations that occur as the operating parameters are varied. They also compare well with other published work and experimental observation.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998