Oscillatory flow around the edge of a flat plate

Summary The paper deals with oscillatory flow of an incompressible viscous fluid around the edge of a flat plate. The primary interest is, in connexion with the flow in open pipes near the edge due to acoustic standing waves, in the dissipation associated with the flow around the edge. Mathematically, the problem to find the flow around the edge can be formulated as an integral equation for a dipole distribution along the plate. This can be simplified by making use of the fact that the Stokes boundary layer is thin with respect to the characteristic length scale of the flow. The simplified equation is solved by a method used recently by Boersma. With the help of this solution the dissipation is calculated. The result is compared with exact, numerical, calculation by Disselhorst. Good agreement is found.     

The Stability of a Condensate Film Moving under the Effect of Gravity and Turbulent Flow of Vapor

The linear stability of a condensate film moving along a vertical isothermal plate under the effect of gravity and turbulent vapor flow is investigated. The cases of both cocurrent and countercurrent motion of phases are treated with regard for phase transformation. An analytical solution for the distribution of film thickness along the plate taking into account the film inertia is obtained using the integral method. A two-wave equation is deduced for the film thickness, and dispersion relations are derived. The effect of moving vapor on the film stability in a wide range of flow parameters is shown.     

Flat plate drag in a micropolar fluid

Summary The drag experienced in a micropolar fluid is investigated by considering uniform streaming past a flat plate. Some recent results on the fundamental solution of the Oseenlinearization of the micropolar flow equations are used to reduce the problem to that of solving a scalar integral equation. The integral equation is analyzed by the application of both asymptotic and variational methods. Results indicate that the drag experienced in a micropolar fluid always exceeds that found in the absence of any micropolarity; however one of the parameters which characterizes a micropolar fluid can be used to minimize the drag.With 2 Figures     

Three-dimensional waves beneath an ice sheet due to a steadily moving pressure

Solutions of the nonlinear water wave equations under an ice sheet are computed using a boundary integral equation method. The ice sheet is modelled as a thin elastic plate and the fluid equations are nonlinear. Depending on the velocity of the moving disturbance generating the flow, different types of responses of the floating ice sheet are discussed.     

Pressure and flow in fluid films constrained between translating and vibrating flexible surfaces

In this paper, the dynamic pressure and flow developed in a two-dimensional, viscous fluid film constrained between flexible surfaces are analyzed. The problem formulation assumes that the response of the flexible surface is governed by linear equations of motion, and the fluid motion is governed by linearized momentum equations including the unsteady inertia. Three states of the model are developed to describe the coupled fluid-structural response problem. The fluid dynamic pressure is derived in the frequency domain as a function of the fluid impedances and the surface transverse vibrations. The perturbed, coupled problem is described by an integral equation (in state vector form) that governs the coupled responses of the flexible surfaces. The integral equation is solved by a discretization method. The analysis is applied to a rigid slider bearing with a flexible, translating plate surface under the excitation of a harmonic point load. The accuracy of the discretization method is evaluated, and numerical results for the dynamic pressure and the plate response are presented.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

Contact between an elastically supported circular plate and a rigid indenter

The axially symmetric problem of a finite circular plate loaded at its center by a smooth, rigid punch is solved by superposing an infinite layer elasticity solution with a pure bending plate theory solution. The problem is reduced to dual integral equations, which are further reduced to a single Fredholm integral equation of the second kind. The Fredholm equation is numerically solved and the results are used to compute contact stresses under the indenter as well as the overall load-deflection behavior. The problem is formulated to model a partially fixed edge around the plate's perimeter, and calculations are carried out for the limiting cases of simple supports and complete fixity. Various ratios of plate diameter to plate thickness are studied, and the results are compared to both Hertzian contact theory and standard plate theory.     

具有初曲率板弯曲的边界元分析

由于具有初曲率板弯曲问题的控制微分方程较复杂，直接求解原问题基本解推导边界积分方程较为困难。本文通过引入等效荷载，将此问题的控制微分方程化成与普通板弯曲基本方程形式相同的微分方程，利用一般求解板弯曲问题的边界元法迭代求解，建立了分析具有初曲率板弯曲问题的边界元法。算例表明本方法理论准确、精度良好。     

Ultrasonic 2D SH Crack Detection in a Layered Anisotropic Plate

The 2-D scattering problem of an internal crack in a layered anisotropic plate is considered in this paper. In the model, two ultrasonic SH probes are attached on the upper surface of the plate and the incoming displacement field is generated by one of the probes and the other probe is acting as a receiver. The transmitting and the receiving probe may be the same. The problem is solved by deriving the Green function for the layered plate and then using the integral representation for the total field to obtain an integral equation for the crack opening displacement. The integral equation is solved by expanding the crack opening displacement (COD) in Chebyshev functions. A crucial part of the method is the expansion of the Green function in a free space part, expressed in the crack coordinate system, and a reflection part, expressed in the plate coordinate system. The electrical signal response is calculated by an electromechanical reciprocity relation. Numerical examples are given for a transversally isotropic graphite-epoxy plate, where the symmetry axes are mutually perpendicular in the layers. The results are presented as A-scans, i.e. the electrical response as a function of time.     

Shear flow over a particulate or fibrous plate

Simple shear flow over a porous plate consisting of a planar array of particles is studied as a model of flow over a membrane. The main objective is to compute the slip velocity defined with reference to the velocity profile far above the plate, and the drift velocity induced by the shear flow underneath the plate. The difference between these two velocities is shown to be proportional to the thickness of the plate. When the geometry of the particle array is anisotropic, the directions of the slip and drift velocity are generally different from the direction of the overpassing shear flow. An integral formulation is developed to describe flow over a plate consisting of a periodic lattice of particles with arbitrary shape, and integral representations for the velocity and pressure are developed in terms of the doubly-periodic Green's function of three-dimensional Stokes flow. Based on the integral representation, asymptotic expressions for the slip and drift velocity are derived to describe the limit where the particle size is small compared to the inter-particle separation, and numerical results are presented for spherical and spheroidal particles of arbitrary size. The asymptotic results are found to be accurate over an extended range of particle sizes. To study the limit of small plate porosity, the available solution for shear flow over a plane wall with a circular orifice is used to describe flow over a plate with a homogeneous distribution of circular perforations, and expressions for the slip and drift velocity are derived. Corresponding results are presented for axial and transverse shear now over a periodic array of cylinders arranged distributed in a plane. Streamline pattern illustrations confirm that a negative drift velocity is due to the onset of eddies between closely-spaced particles.     

点弹性支承粘弹性矩形薄板的基频分析计算

从三维粘弹性本构关系出发,导出了具有多个点弹性支承的Kelvin型粘弹性矩形薄板的运动微分方程。针对方程中出现的二维广义d函数,采用积分方程法导出了具有多个点弹性支承的四边简支Kelvin型粘弹性矩形薄板自由振动的复特征方程,分析了材料的无量纲延滞时间、点弹性支承的弹性系数和支承位置对矩形薄板的固有频率的影响。     

Incompressible flow past oscillatory wings of low aspect ratio by the integral equations method

In the framework of the linearized perturbation theory, the pressure jump over the oscillating wing is the solution of a two‐dimensional integral equation. Performing an asymptotic expansion with respect to the aspect ratio and keeping the leading terms, we reduce the integral equation to a one‐dimensional one, obtaining a simplified method of solving the lifting surface integral equation for a class of thin wings of low aspect ratio with a straight trailing edge. The one‐dimensional integral equation is solved for the delta flat plate and the pressure coefficient field and the lift and moment coefficients are calculated. The range of validity of the new method is discussed in the final part of the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.     

A complex variable boundary element method for solving plane and plate problems of elasticity

We consider the complex variable boundary element approximation of biharmonic problem on a smooth domain with various boundary conditions. Based on the Vekua's complex integral representation of the analytic function, a new boundary integral equation is formulated. The density function appearing in the integral equation is determined directly by using the boundary element method. Some plane and plate examples are presented, and the results of the numerical solutions are accurate everywhere in the solid, including the regions near the boundary.

The approach presented is only suitable for bounded simply connected regions.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.     

Free-surface wave interaction with a thick flexible dock or very large floating platform

In this paper the recently developed semi-analytic method to solve the free-surface wave interaction with a thin elastic plate is extended to the case of a plate of finite thickness. The method used is based on the reformulation of the differential–integral equation for this problem. The thickness of the plate is chosen such that the elastic behavior of the plate can be described by means of thin-plate theory, while the water pressure at the plate is applied at finite depth. The water depth is finite.     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

厚度不连续悬臂梁板的自由振动分析

将厚度不连续梁板视为层合板, 分别应用Hamilton 正则方程半解析法建立每一层的线性方程。考虑到每两层连接界面上应力和位移的连续性, 联立各层的方程得到整个结构的特征方程, 其主要的优越性表现为: 控制方程不限制不连续梁板的厚度, 并能适合处理厚度不对称且不连续的层合板。本文中的方法可修改或扩展用来分析加筋压电材料层合板或带有压电材料传感器和驱动器块的板壳等问题。