Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

NONLINEAR ANALYSIS OF DYNAMIC STABILITY FOR LAMINATED CIRCULAR CYLINDRICAL THICK SHELLS WITH ENDS ELASTICALLY SUPPORTED AGAINST ROTATION

Based on Timoshenko-Mindlin kinematic hypotheses and Hamilton'sprinciple,a dynamic non-linear theory for general laminated circular cylindrical shellswith transverse shear deformation is developed.A multi-mode solution for periodic in-plane loads is formulated for the non-linear dynamic stability of an anti-symmetricangle-ply cylinder with its ends elastically restrained against rotation.The resultedequations in terms of time function are solved by the incremental harmonic balancemethod.     

Analysis of the nonstationary spatial propagation of elastic waves from cavities subject to mechanical and thermal loads

A numerical analytic method is proposed to solve nonstationary coupled problems of thermoelasticity with regard to the finite velocity of thermal waves. The method is used to analyze the nonstationary spatial propagation of elastic waves from a cavity subjected on its surface to mechanical and thermal loads. The ray theory of propagation of wavefield discontinuities is used. To determine the time dependence of the field parameters behind the wavefront and to account for the relationship between the mechanical and thermal fields with prescribed accuracy, a numerical iterative procedure that employs the properties of characteristics is used. Plots are presented for the nonstationary stresses and temperature near a prolate spheroidal cavity subject to step mechanical loading and near an elliptical cylindrical cavity subject to thermal shock __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 79–88, August 2006.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

3D scattering by multiple cylindrical cavities buried in an elastic formation

This paper presents the three-dimensional scattering field obtained when multiple cylindrical circular cavities of infinite length buried in a homogeneous elastic medium, are subjected to dilatational point loads placed at some point in the medium. The solution is formulated using boundary elements for a wide range of frequencies and spatially harmonic line loads, which are then used to obtain time series by means of (fast) inverse Fourier transforms into space-time. The method and the expressions presented are implemented and validated by applying them to a cylindrical circular cavity buried in an infinite homogeneous elastic medium subjected to a dilatational point load, for which the solution is calculated in closed form. The boundary elements method is then used to evaluate the wave-field elicited by a dilatational point load source in the presence of a different number of cylindrical oval cavities. Simulation analyses using this idealized model are then applied to the study of wave propagation patterns in the vicinity of these inclusions. The amplitude of the wavefield in the frequency vs axial-wavenumber domain is presented, allowing the recognition, identification, and physical interpretation of the variation of the wavefield.     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Cycloidal pendulum with a rolling cylinder

Free vibrations of a heavy homogeneous cylinder rolling in a cylindrical cavity whose directing curve is a brachistochrone are considered. The equation of motion of the cylinder is derived and the circular frequency of free vibrations of the cylinder center of mass is determined. An analogy between the cycloidal pendulum with a rolling cylinder and the classical cycloidal pendulum in the form of a material point is obtained.     

Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation

Acoustic radiation and the dynamic field induced by a cylindrical source of infinite extent, undergoing angularly periodic and axially-dependent harmonic surface vibrations, while eccentrically suspended in a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation, are analyzed in an exact manner. This configuration, which is a realistic idealization of an acoustic logging tool suspended in a fluid-filled borehole within a permeable surrounding formation, is of practical importance with a multitude of possible applications in seismo-acoustics. The formulation utilizes the novel features of Biot dynamic theory of poroelasticity along with the translational addition theorem for cylindrical wave functions to obtain a closed-form series solution. The basic dynamic field quantities such as the resistive and the reactive components of the modal acoustic radiation impedance load on the source in addition to the radial and transverse stresses induced in the surrounding formation by an eccentric pulsating/oscillating cylinder in a water-filled borehole within a water-saturated Ridgefield sandstone medium are evaluated and discussed. Special attention is paid to the effects of source eccentricity, excitation frequency, and mode of surface oscillations on the modal impedance values and the dynamic stresses. Limiting cases are considered and good agreements with available solutions are obtained.     

Thermoelectroelastic state of a piezoceramic body with a spheroidal cavity in a uniform heat flow

An exact solution is obtained to the three-dimensional problem of thermoelectroelasticity for a piezoceramic body with a spheroidal cavity. The solutions of static thermoelectroelastic problems are represented in terms of harmonic functions. Far from the cavity, the body is in a uniform heat flow perpendicular to the axis of symmetry of the cavity __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 57–66, November 2005.     

Boundary conditions for axisymmetric circular cylinder in one-dimensional hexagonal quasicrystals

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of axisymmetric circular cylinder in one-dimensional (1D) hexagonal quasicrystals (QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, and using the reciprocal theorem and general solution, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the circular cylinder with axisymmetric deformations in 1D hexagonal QCs. The results of the present paper enable us to establish a set of correct boundary conditions, most of which are obtained for the first time. Furthermore, corresponding necessary conditions for the case of axisymmetric circular cylinder with transversely isotropic are derived subsequently, and their isotropic elastic counterparts are also obtained. The accuracy of mixed boundary conditions of isotropic axisymmertic circular cylinder is proved at last.     

Small oscillations of a liquid in a partitioned cylindrical vessel

The equations of motion of a rigid body whose cavity is partially filled with an ideal fluid have been obtained in works of Moiseev [1, 2, 3], Okhotsimskii [4], Narimanov [5], and Rabinovich [6]. All the equation coefficients have been calculated for a cavity in the form of a circular cylinder or two concentric cylinders.The problem of fluid motion in a partitioned cylindrical cavity was considered by Rabinovich [7]. It was also considered by Bauer [8], who analyzed the particular case of vessel motion in the plane of one of the partitions.In the following we consider the two-dimensional motion of a cylinder with radial and annular baffles, and a definition is given of the velocity potential in the case of arbitrary positioning of the radial baffles with respect to the motion plane. Formulas are obtained for determining the parameters of a mechanical analog of the wave oscillations, which consists of two mathematical pendulum subsystems.     

波浪与外圆弧开孔壁双圆筒柱的相互作用

应用比例边界有限元法(SBFEM)研究了短峰波与圆筒外接圆弧开孔结构物的相互作用. 求解时将外接圆弧延伸构建一个虚拟圆, 该圆的孔隙影响系数可由矩阵G_0统一进行表达. 整个流场可划分成一个有限域和一个无限域. SBFEM只需对虚拟圆边界进行离散, 使空间维数降低一阶, 在圆的半径方向保持解析, 并且无限域处的辐射边界条件能自动满足. 利用变分原理推导SBFEM方程, 有限域和无限域分别采用贝塞尔函数和汉克尔函数作为基函数来求得对应域的解. 将计算结果与解析解和其他数值方法进行了比较, 验证了该方法是一种用很少单元便能得到精确结果的高效算法. 进一步研究了诸如短峰波波向、结构的几何、材料参数等因素对结构所受波浪载荷及绕射波轮廓的影响, 并进行了分析.      

SH波入射时垂直半空间中双相介质界面附近圆孔的动力分析

利用复变函数和Green函数法研究了垂直半空间中双相介质界面附近圆孔对SH波的散射与动应力集中问题。该问题的解答采用镜像法,首先构造出含有圆孔的直角平面区域出平面问题的Green函数,然后利用契合技术,并根据界面处位移连续性条件将解答归结为具有弱奇异性的第一类Fredholm积分方程组的求解,结合散射波的衰减特性,直接离散该方程组,把积分方程组转化为线性代数方程组可得到该问题的数值结果。最后,通过算例分析了圆孔的动应力集中情况。结果表明,与全空间中界面附近圆孔对SH波的散射相比,由于垂直半空间自由边界的存在,孔边动应力集中系数明显增大;另外,入射波由硬介质(波速大)进入到软介质(波速小)时,与均匀介质相比,孔边动应力集中更显著,最不利的参数组合,孔边动应力集中系数几乎提高了一倍,入射波由软介质进入到硬介质时,情况相反。     

Analytical solutions for finite cylindrical dynamic cavity expansion in compressible elastic-plastic materials

Analytical solutions for the dynamic cylindrical cavity expansion in a com-pressible elastic-plastic cylinder with a finite radius are developed by taking into account of the effect of lateral free boundary, which are different from the traditional cavity expan-sion models for targets with infinite dimensions. The finite cylindrical cavity expansion process begins with an elastic-plastic stage followed by a plastic stage. The elastic-plastic stage ends and the plastic stage starts when the plastic wave front reaches the lateral free boundary. Approximate solutions of radial stress on cavity wall are derived by using the Von-Mise yield criterion and Forrestal’s similarity transformation method. The effects of the lateral free boundary and finite radius on the radial stress on the cavity wall are discussed, and comparisons are also conducted with the finite cylindrical cavity expansion in incompressible elastic-plastic materials. Numerical results show that the lateral free boundary has significant influence on the cavity expansion process and the radial stress on the cavity wall of metal cylinder with a finite radius.     

SH波对双相介质界面附近圆形孔洞的散射

建立了求解平面SH波对双相介质界面附近圆形孔洞散射与动应力集中的一种分析方法．利用复变函数与多极坐标的方法构造了一个Green函数，它是在含有圆形孔洞的弹性半空间的水平面上任一点上作用时间谐和的出平面线源荷载的位移解．利用“契合”模型，并根据界面上位移连续性条件，建立了求解SH波对双相介质界面附近圆形孔洞散射的具有弱奇异性的第一类Fredholm型积分方程．给出了圆孔周边上动应力集中系数的表达式．作为算例，分析了在界面一侧或界面两侧附近具有圆形孔洞时SH波的散射，并讨论了入射波波数、不同的材料组合以及孔心至界面的距离对动应力集中的影响．     

封闭圆柱腔内旋转点声源声压的近场频域解

在圆柱腔壁面为声学硬壁面的假定下，利用圆柱腔体空间内的格林函数导出了圆柱腔内旋转运动点声源空间声压的近场频域解；进而利用该频域解计算分析了旋转简谐单极子点源的声学特性。计算分析表明：由于考虑了壁面的影响，在圆柱腔体内，点声源旋转频率和源频率的变化将改变声压的空间指向性；源频率和旋转频率的增加伴随明显的多普勒效应，同时基频声压沿圆柱腔体轴线的分布从比较平缓变为明显波动；沿半径方向基频声压也波动变化。     

Stokes flow between two confocal rotating spheroids with slip

The steady axisymmetric flow problem of a viscous fluid confined between two confocal spheroids that are rotating about their axis of revolution with different angular velocities is considered. A linear slip, of Basset type, boundary condition on both surfaces of the spheroidal particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in prolate and oblate spheroidal coordinates. The boundary conditions on the particle’s surface and spheroidal container are satisfied by a collocation technique. The torque exerted on the spheroidal particle by the fluid is evaluated with good convergence for various values of the slip parameters, the relative angular velocity and aspect ratios of the spheroids. The limiting case of no-slip is in good agreement with the available values in the literature.     

Development of a computer program for the prediction of flow in a rotating cavity

A computer program has been developed to predict laminar source-sink flow in a rotating cylindrical cavity. Although the program is based on a standard finite difference technique for recirculating flow, it incorporates two novel features. Step changes in grid size are employed to obtain sufficient resolution in the boundary layers and special treatment is given to the solution of the pressure correction equations, in the ‘SIMPLE’ algorithm, in order to improve the convergence properties of the method. Results are presented both for the flow in an infinite rotating cylindrical annulus and a finite rotating cylindrical cavity, with the inner cylindrical surface acting as a uniform source and the outer cylinder as a sink. These show good agreement with existing analytical solutions and illustrate some of the problems associated with the computation of rapidly rotating flows.     

Nonstationary Plane Elastic Contact Problem for Matched Cylindrical Surfaces

An approach is proposed to study a collision of a long cylinder with the inside surface of a circular cylindrical cavity in an elastic medium. The problem is solved in plane formulation. A nonstationary mixed initial–boundary-value problem with unknown boundaries moving with a variable velocity is formulated and then reduced to an infinite system of Volterra integral equations of the second kind or, in a simplified formulation, to a sequence of Volterra integral equations. The penetration velocity is determined as a function of the cylinder mass and initial conditions. It is established that the reaction force peaks instantaneously and then dies out     

A light cylinder under horizontal vibration in a cavity filled with a fluid

The behavior of a light cylindrical body of circular cross-section under horizontal vibration in a rectangular cavity filled with a fluid is experimentally investigated. At critical vibration intensity the body is repelled from the upper side of the cavity and takes up a stable suspended position, in which the gravity field is balanced by the vibrational repulsive force, executing longitudinal oscillations. As the vibrations are intensified, the gap between the cylinder and the wall widens. A new form of instability, namely, the excitation of the tangential motion of the body along the vibration axis, is found to exist on the supercritical range. The cylinder is at a finite distance from the upper side of the cavity and the tangential motion is due to the loss of symmetry of the oscillating motion. The transition of the cylinder to the suspended state and its return to the wall, as well as the excitation of the average longitudinal motion and its cessation, occur thresholdwise and have a hysteresis. The body dynamics are studied as a function of the dimensionless vibration frequency.