Indirect boundary element method for shear flow over prolate and oblate hemispheroidal protuberances on plane walls

The indirect boundary element method was used to study the hydrodynamics of longitudinal shear flow and cross flow with a longitudinal rate of shear over prolate and oblate hemispheroidal protuberances projecting from a plane wall. Analytic techniques such as Fourier analysis, spheroidal co-ordinates, and the method of images were used to make the numerical methods more efficient. A novel method for computing the hydrodynamic torque was used—instead of directly calculating the torque from the weightings of the Green's functions (a method that is only valid when the weightings have physical significance) the hydrodynamic torque was computed indirectly using a Green's function for torque that derived here. As a test of this method, the present scheme was applied to determine the hydrodynamic torque of full spheroids, where exact solutions are known, and excellent results were obtained. Our results for hemispheroids projecting from plane walls were, except for extremely wide oblate hemispheroids, within a factor of two of those of full spheroids. Our results also agreed with those of previous study of oblate hemispheroidal protuberances. © 1998 John Wiley & Sons, Ltd.     

Generalized Timoshenko–Reissner model for a multilayer plate

A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.     

Large deflections of an inclined cantilever with an end load

An elastica is weighted on one end and hinged on the other end. The hinged end is then turned by a torque. The solutions depend on the angle turned and a non-dimensional parameter K which represents the relative importance between the end load and flexural rigidity. Approximate solutions for small and large K. and perturbations about the almost vertical cases are found. The results compare well with exact numerical integration. For given K the torque is an extremely non-linear function of the angle turned. Non-uniqueness and hysteresis loops occur when K#62;π/2. The slightly inclined column is also studied.     

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.     

Interaction between a simplified soft palate and compressible viscous flow

Fluid–structure interaction in a simplified 2D model of the upper airways is simulated to study flow-induced oscillation of the soft palate in the pharynx. The goal of our research has been a better understanding of the mechanisms of the Obstructive Sleep Apnea Syndrome and snoring by taking into account compressible viscous flow. The inspiratory airflow is described by the 2D compressible Navier–Stokes equations, and the soft palate is modeled as a flexible plate by the linearized Euler–Bernoulli thin beam theory. Fluid–structure interaction is handled by the arbitrary Lagrangian–Eulerian formulation. The fluid flow is computed by utilizing 4th order accurate summation by parts difference operators and the 4th order accurate classical Runge–Kutta method which lead to very accurate simulation results. The motion of the cantilevered plate is solved numerically by employing the Newmark time integration method. The numerical schemes for the structure are verified by comparing the computed frequencies of plate oscillation with the associated second mode eigenfrequency in vacuum. Vortex dynamics is assessed for the coupled fluid–structure system when both airways are open and when one airway is closed. The effect of mass ratio, rigidity and damping coefficient of the plate on the oscillatory behavior is investigated. An acoustic analysis is carried out to characterize the acoustic wave propagation induced by the plate oscillation. It is observed that the acoustic wave corresponding to the quarter wave mode along the length of the duct is the dominant frequency. However, the frequency of the plate oscillation is recognizable in the acoustic pressure when reducing the amplitude of the quarter wave mode.     

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term简

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.     

Flutter of a Wide Strip Plate in a Supersonic Gas Flow

The stability of an elastic plate in the form of a wide strip in a supersonic inviscid gas flow is investigated in the linear approximation. An expression for the dependence of the pressure on the plate deflection, asymptotically exact for wide plates, is used. Two qualitatively different instability types are obtained: flutter with respect to a single oscillatory mode due to negative aerodynamic damping and flutter of a related type due to the interaction of oscillatory modes. For each type the stability criterion and the frequency at which the oscillation amplitude grows most intensely are found.     

弹性力学状态变量体系下带有初应力的振动问题

通过在Hellinger-Reissner广义势能中引入应变的非线性项,推导出了弹性力学Hamilton体系下的具有初应力的振动方程,并运用精细积分给出了两端简支的梁、组合梁和四边简支板及组合板在初应力下振动频率。本文结果是严格弹性力学意义(没有引入任何几何变形假设)下的精确解,为衡量各种计入剪切变形的薄板、中厚板理论的准确性提供了一个标准。     

On canonical bending relationships for plates

In recent years, a series of papers have appeared on algebraic relationships between the solutions (e.g., deflections, buckling loads and frequencies) of a given higher-order plate theory and the classical plate theory. The bending relationships, for example, can be used to generate the transverse deflection of a plate according to the particular higher-order theory from the known deflection of the same problem according to the classical plate theory. In the present study relationships between the bending solutions of several higher-order plate theories and the classical plate theory are obtained in a canonical form (i.e., one set of relationships contain several theories and they can be specialized to a specific theory by assigning values to the constants appearing in the relationships). Numerical examples of bending solutions for rectangular plates with various boundary conditions are presented to show how the relations can be used to determine the deflections and bending moments for various theories. The relationships are validated by comparing the numerical results obtained using the relationships for the Mindlin plate theory against those computed using the ABAQUS finite element program.     

A high-accuracy temporal–spatial pseudospectral method for time-periodic unsteady fluid flow and heat transfer problems

A temporal–spatial pseudospectral (TSP) method is proposed for the high-accuracy solutions of time-periodic unsteady fluid flow and heat transfer problems. In this method, both the spatial and temporal derivative terms in the governing equations are computed by pseudospectral method. The spatial derivatives are computed through Chebyshev and Lagrange polynomials while the time derivatives are computed by Fourier series. The TSP method is capable of directly finding out the periodic state solutions without the necessity to resolve the initial transient state solutions, hence holds high computational efficiency and high numerical accuracy properties for the time-periodic problems. This method is validated by three 2D benchmark problems: the time-periodic incompressible flow with exact solutions; the natural convection in enclosure with time-periodic temperature on one sidewall, and on both sidewalls. The TSP results fit well the exact solutions or the benchmark solutions and the TSP accuracy is much higher than the time marching spatial pseudospectral accuracy. Some time-dependent fluid flow and heat transfer characteristic parameters are analysed. The proposed TSP method could be further extended to more complex time-periodic unsteady fluid flow and heat transfer problems where high-accuracy results are required.     

COUPLED VIBRATORY CHARACTERISTICS OF A RECTANGULAR CONTAINER BOTTOM PLATE

The natural frequencies of an elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface are studied. The fluid is assumed to be incompressible, inviscid and irrotational, and the effect of surface waves is neglected. An analytical-Ritz method is developed to study the vibratory characteristics of the plate in contact with the fluid. First of all, the exact expression of the motion of the fluid is obtained, in which the unknown coefficients are determined by using the method of separation of variables and the method of Fourier series expansion. Then, the Ritz approach is used to obtain the frequency equation of the system. The vibrating beam functions are adopted as the admissible functions for the wet-mode expansion of the plate, and the added virtual mass incremental (AVMI) matrices are obtained for plates with arbitrary boundary conditions. Finally, a convergence study is carried out and some numerical results are given. The accuracy of AVMI factor solutions is discussed by comparing with the more accurate analytical-Ritz solutions presented in this paper. Furthermore, It is seen that the present method is also suitable for the vibration analysis of rectangular plates in contact with infinite fluid by taking the finite, but larger size fluid domain as an approximation in the computation.     

An approximate method for magnetic boundary layers

An approximate method is derived for obtaining the flow in magnetic boundary layers with aligned fields which were discussed by Stewartson (1965). On comparing with exact, computed solutions the method is found to be accurate when applied to forward stagnation point flow for the whole, allowable range of the magnetic field strength parameter. It is then applied to the layer that develops from the forward stagnation point on a circular cylinder placed in a uniform stream and values obtained for the point at which it separates from the surface.     

Viscous dissipation with fluid inertia in oscillatory shear flow

For liquids with high viscosity and low thermal conductivity, viscous dissipation can cause appreciable errors in rheological property measurements. Here, the influences of both viscous dissipation and fluid inertia on the property measurements in oscillatory sliding plate rheometry are investigated. For Newtonian fluids, Bird (1965) solved the combined problem analytically, but only for high frequencies. Here his solution is extended to any frequencies. Also, the equations of motion and energy are solved for linear viscoelastic fluids, and new analytical solutions for the velocity and temperature profiles are given. In both Newtonian and linear viscoelastic fluids, the temperature rise in the gap increases with frequency. The location of the maximum temperature shifts from the mid-plane at low frequency towards the moving wall at high frequency. The fluid inertia increases the viscous dissipation in both fluids. By solving the combined problem, this paper simplifies rheometer design by providing one unified criterion for avoiding measurement errors. Operating limits are presented graphically for minimizing the effects of both fluid inertia and viscous dissipation in oscillatory sliding plate rheometry.     

Interparticle force and torque on rigid spheroidal particles in acoustophoresis

The movement of the particles in acoustophoresis is driven by the acoustic radiation force acting on the particles. Particles with positive contrast factor tend to agglomerate once they are pushed by the primary force to the vicinity of the pressure node. The main driving force of this agglomeration is the interparticle force. In this study, the boundary element method is used to calculate the interparticle force and torque acting on a pair of spheroidal particles. The numerical results show that the interparticle force is dominant over the primary force when the spheroids are near the pressure nodal plane, similar to the case of two spheres. On contrary, the interparticle torque is insignificant compared to the primary torque, even when the spheroids are close to each other. The results also provide a preliminary study about how biological cells, which are mostly not spherical in shape, agglomerate and orient themselves in the vicinity of the pressure node.     

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.     

Unsteady flow in a circular sector duct

Analytic solutions for the unsteady flow in a circular sector duct are found using series sums of Bessel integrals. For starting flow due to a step pressure gradient, the velocity profile is initially flat, then approaches the rounded steady state shape in a time scale proportional to the square of opening angle of the sector. For oscillatory flow, the velocity is quasi-steady for low frequencies, but shows “annular effect” at large frequencies. Increased opening angle increases the amplitude and the phase lag. In all cases, the shear stress at the apex is zero for acute sector angles but becomes infinite for obtuse sector angles.     

滑动最小二乘插值函数在板弯曲问题上的应用

 利用滑动最小二乘插值函数作为加权残值法的试函数，分析了 该试函数的拟合特性，对试函数中的基函数以及权函数的选取提出了 建议；采用最小二乘配点法求出试函数中的系数，进而可得到定解问 题的近似解；利用该试函数对薄板的挠曲、中厚板的弯曲两个例子进 行了数值计算，并与理论结果或其它数值结果进行对比，结果表明， 该试函数适用于多种边值问题，且精度高. 该法简化了选择试函数的 过程，尤其适用于工程中的各种数值计算.     

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method （SSDQM）. The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model （ALM） is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

A novel slightly compressible model for low Mach number perfect gas flow calculation

By analyzing the characteristics of low Mach number perfect gas flows, a novel Slightly Compressible Model (SCM) for low Mach number perect gas flows is derived. In view of numerical calculations, this model is proved very efficient, for it is kept within thep-v frame but does not have to satisfy the time consuming divergence-free condition in order to get the incompressible Navier-Stokes equation solution. Writing the equations in the form of conservation laws, we have derived the characteristic systems which are necessary for numerical calculations. A cell-centered finite-volume method with flux difference upwind-biased schemes is used for the equation solutions and a new Exact Newton Relaxation (ENR) implicit method is developed. Various computed results are presented to validate the present model. Laminar flow solutions over a circular cylinder with wake developing and vortex shedding are presented. Results for inviscid flow over a sphere are compared in excellent agreement with the exact analytic incompressible solution. Three-dimensional viscous flow solutions over sphere and prolate spheroid are also calculated and compared well with experiments and other incompressible solutions. Finally, good convergent performances are shown for sphere viscous flows. The project supported by the Basic Research on Frontier Problems in Fluid and Aerodynamics in China and the National Natural Science Foundation of China (19772069)