Effect of heat generation on transient flow of micropolar fluid in a porous vertical channel

The present work is performed to study the effect of heat generation on fully developed flow and heat transfer of micropolar fluid between two parallel vertical plates. The rigid plates are assumed to exchange heat with an external fluid by convection. The governing equations are solved by using Crank–Nicolson implicit finite difference method. The effects of governing parameters such as transient, heat generation, micropolar parameter, Prandtl number, Biot number, and Reynolds number on the velocity and temperature profiles are discussed. It is found that the presence of heat generation enhances the velocity and temperature of the micropolar fluid at the middle of the channel.     

Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach

This paper brings into focus some of the interesting effects arising from the non-linear motion of the liquid free surface, due to sloshing, in a partially filled laminated composite container along with the associated coupling due to fluid-structure interaction effects. The finite element method based on two-dimensional fluid and structural elements is used for the numerical simulation of the problem. A numerical scheme is developed on the basis of a mixed Eulerian-Lagrangian approach, with velocity potential as the unknown nodal variable in the fluid domain and displacements as the unknowns in the structure domain. The FE formulation based on Galerkin weighted residual method along with an iterative solution procedure are explained in detail followed by a few numerical examples. Numerical results obtained by the present investigation for the rigid containers are first compared with the existing solutions to validate the code for non-linear sloshing without fluid-structure coupling. Thereafter the computational procedures are advanced to obtain the coupled interaction effect of non-linear sloshing in laminated composite containers.     

Coupled vibrations of cantilever cylindrical shells partially submerged in fluids with continuous, simply connected and non-convex domain

The approach developed in the present paper is applied for the coupled-vibration analysis of a cantilever cylindrical shell partially submerged in a fluid with a continuous, simply connected and non-convex domain. The shell is partially and concentrically submerged in a rigid cylindrical container partially filled by a fluid which is assumed to be incompressible and inviscid. The velocity potential for fluid motion is formulated in terms of eigenfunction expansions using the collocation method. The interaction between the fluid and the structure takes into account by using the compatibility requirement along the wet surface of the shell and the Rayleigh-Ritz method is used to calculate natural frequencies and modes of the coupled system. The validity of the developed theoretical method is verified by comparing the results with those obtained from the finite element analysis. Furthermore, the effects of submergence depth, radial distance between shell and container, and circumferential wavenumbers on the natural frequencies and modes of the coupled system are investigated.     

Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation

The paper describes a method for the prediction of dynamic characteristics of rectangular plates with cutouts. The method is based on the use of variational principles in conjunction with finite difference technique. A concept of interlacing grids has been developed to express the strain energy of nodal subdomains into which the plate is divided. The use of this idea has been demonstrated in relation to internal and boundary nodes. Natural frequencies and corresponding mode shapes of rectangular plates with one and two cutouts have been predicted and experimentally verified.     

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.     

A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates

A finite element is presented to analyze the three-dimensional (3-D) vibration of piezoelectric coupled circular and annular plates. The proposed finite element is a modification of a conventional axisymmetric finite element and is capable of conducting both axisymmetric and nonaxisymmetric vibration analysis of circular and annular laminated plates, with piezoelectric layers therein. The present formulation, a two-dimensional model itself, can investigate 3-D vibration of those plates for a preselected number of nodal diameters, and is therefore more economical than the conventional 3-D finite element analysis, yet still has almost the same accuracy and versatility as the 3-D analysis. In cases such as analysis of stators of traveling wave ultrasonic motors where only vibration modes with particular numbers of nodal diameters are of interest, the proposed approach is very convenient and useful.     

约束条件下胶球间排空力的研究

用MonteCarlo方法对处于两平行硬板约束下三个浓度的大小胶球系统进行了模拟,通过对大胶球表面小胶球密度的统计,由密度积分公式获得了大胶球所受的排空力.研究结果显示,因为平行硬板的存在或当改变两平行硬板的距离时,同浓度下,排空力在硬板距离小的时候最明显;三个浓度中,浓度高的,排空力受硬板距离影响最大;有硬板约束比无该约束的时候,排空力效果更显著.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

Pitch change during reverberant decay

The natural frequencies and mode shapes of a number of box beams are calculated by using the finite element displacement method. The structures are considered as assemblages of plates, and in general it is necessary to consider both the in-plane and transverse motion of the plates. A method of representing these two types of motion in the analysis of the vibrations of box beams is presented. A number of box beams of varying sectional parameters are analysed as systems of plates and the results compared with the predictions of Euler and Timoshenko beam theories. The comparisons show that for short beams constructed of thin plates, the new method can successfully represent the localized plate deformations, which cannot be described by beam theory.     

周期多孔板的面内振动衰减域及其优化

针对Bragg散射型周期多孔板难以实现较低起始频率并维持较宽衰减域的问题,优化设计了一种含菱形孔的周期多孔板。采用有限元法结合周期边界条件,并运用COMSOL对周期多孔板的面内弹性波频散关系进行计算,通过ANSYS模拟有限尺寸周期多孔板的频率响应,将周期多孔板悬吊进行了正弦波激励的振动试验。研究结果表明,含菱形孔的周期多孔板相比于含圆形和六边形孔的周期多孔板具有更宽的衰减域;材料属性对衰减域影响较大,丁晴橡胶和硅橡胶易于获得低频衰减域;孔隙率的增大有利于获得低频宽带的衰减域;增大菱形孔水平夹角能获得较宽的衰减域。对衰减域的形成机理分析发现,含菱形孔的周期多孔板同时具有Bragg散射型和局域共振型声子晶体的特性,表明两种衰减域机理具有内在的联系。优化设计的周期多孔板存在一条5281.76 Hz至8824.30 Hz的完全衰减域,经过至少2个周期,振动即得到较明显衰减。数值和试验得到的衰减区具有较高的一致性。该研究为减振降噪板的开发提供了新的思路,且由于制作过程便捷,在改善建筑声环境中具有潜在的应用前景。      

Vibro-acoustic behaviour of laminated double glazing enclosing a viscothermal fluid cavity

This paper presents a new numerical model to investigate the vibro-acoustic behaviour of two laminated glass plates enclosing a thin viscothermal fluid cavity. The aim of this work is to develop an original five layer (two skins plies, two adhesive films and a core ply) laminated plate finite element by mixing Kirchhoff and Mindlin plate’s theory. The formulation is based on the theory that accounts for the transverse shear in the adhesive films and in the core. The acousto-elastic model is established in dimensionless appropriate form including the effects of viscosity and thermal conductivity of fluid and by taking into account the fluid-structure interaction. The discretization of the energy functional by finite element method gives after minimisation a symmetrical coupled matrix system in which the acoustic matrices are frequency dependent. Therefore, an iterative procedure is derived to determine the eigenmodes of the coupled system. The modal approach is adopted to determine the vibro-acoustic system’s response. Then, the validation of the new laminate finite element model is achieved by comparing the sandwich plate results against data obtained from literature. Subsequently, predicted responses, such as the vibration transmissibility and the transmission loss of the coupled system, for a given laminated double glazing under an imposed homogeneous pressure are presented and discussed. Numerical results show the importance of both lamination and viscothermal fluid effects on double glazing vibro-acoustic behaviour.     

Analytical Proof That There is no Effect of Confinement or Curvature on the Maxwell–Boltzmann Collision Frequency

The number of Maxwell–Boltzmann particles that hit a flat wall in infinite space per unit area per unit time is a well-known result. As new applications are arising in micro and nanotechnologies there are a number of situations in which a rarefied gas interacts with either a flat or curved surface in a small confined geometry. Thus, it is necessary to prove that the Maxwell–Boltzmann collision frequency result holds even if a container’s dimensions are on the order of nanometers and also that this result is valid for both a finite container with flat walls (a rectangular container) and a finite container with a curved wall (a cylindrical container). An analytical proof confirms that the Maxwell–Boltzmann collision frequencies for either a finite rectangular container or a finite cylindrical container are both equal to the well-known result obtained for a flat wall in infinite space. A major aspect of this paper is the introduction of a mathematical technique to solve the arising infinite sum of integrals whose integrands depend on the Maxwell–Boltzmann velocity distribution.     

Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

In studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.     

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.     

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.     

Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Deterministic Motion of the Controversial Piston in the Thermodynamic Limit

We consider the evolution of a system composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adiabatic wall (the adiabatic piston). We study the thermodynamic limit for the piston where the area A of the cross-section, the mass M of the piston, and the number N of particles go to infinity keeping A/M and N/M fixed. The length of the container is a fixed parameter which can be either finite or infinite. In this thermodynamic limit we show that the motion of the piston is deterministic and the evolution is adiabatic. Moreover if the length of the container is infinite, we show that the piston evolves toward a stationary state with velocity approximately proportional to the pressure difference. If the length of the container is finite, introducing a simplifying assumption we show that the system evolves with either weak or strong damping toward a well-defined state of mechanical equilibrium where the pressures are the same, but the temperatures different. Numerical simulations are presented to illustrate possible evolutions and to check the validity of the assumption.     

二维平板通道中流动与传热的格子-Boltzmann模拟

格子-Boltzmann数值模拟方法(LBM),在最近十几年来得到迅速发展。本文发展了LBM的流动与传热模型,并对二维平板通道中的流动与传热进行了模拟,采用密度分布函数得到速度场,用单独的内能分布函数得到温度场,并与传统FVM方法所得到的多个特征量结果进行了比较,模拟结果与FVM解均吻合很好。鉴于LBM边界条件处理简单和易于实施等特点,该方法可望成为求解流动与传热的一种有效数值模拟手段。     

Theoretical analysis of hot-image effect in a high-power laser system with cascaded nonlinear medium

We present a universally applicable and quick method to forecast the intensity and location of the hot-image effects in a high-power laser system structured by cascaded Kerr medium plates. The analytical expressions for the locations and the peak intensity of the hot-images are deduced by using propagation matrix method. The results are useful for laser designers to estimate and minimize the threat of optical damage. Theoretical analysis demonstrates that a maximum of N hot-images may appear in a laser system structured by N cascaded Kerr medium plates and the distance between two adjacent hot-images is two times the interval between two adjacent Kerr medium plates. The number and locations of the hot-images are related with the number of the Kerr medium plates, the distance from the scatterer to the front of the first Kerr medium plate, and the interval between two adjacent Kerr medium plates. The peak intensity of the hot-images depends on the number of the Kerr medium plates, the B-integral of each Kerr medium plate, the amplitude and phase modulation coefficients of the scatterer and the peak intensity of the input beam. The hot-image effects in a laser system with cascaded Kerr medium plates from two to eight are discussed in detail and numerically analyzed. Numerical simulation results are in agreement with the theoretical results.     

A direct comparison between volume and surface tracking methods with a boundary-fitted coordinate transformation and third-order upwinding

A Volume Tracking (VT) and a Front Tracking (FT) algorithm are implemented and compared for locating the interface between two immiscible, incompressible, Newtonian fluids in a tube with a periodically varying, circular cross-section. Initially, the fluids are stationary and stratified in an axisymmetric arrangement so that one is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). A constant pressure gradient sets them in motion. With both VT and FT, a boundary-fitted coordinate transformation is applied and appropriate modifications are made to adopt either method in this geometry. The surface tension force is approximated using the continuous surface force method. All terms appearing in the continuity and momentum equations are approximated using centered finite differences in space and one-sided forward finite differences in time. In each time step, the incompressibility condition is enforced by a transformed Poisson equation, which is linear in pressure. This equation is solved by either direct LU decomposition or a Multigrid iterative solver. When the two fluids have the same density, the former method is about 3.5 times faster, but when they do not, the Multigrid solver is as much as 10 times faster than the LU decomposition. When the interface does not break and the Reynolds number remains small, the accuracy and rates of convergence of VT and FT are comparable. The well-known failure of centered finite differences arises as the Reynolds number increases and leads to non-physical oscillations in the interface and failure of both methods to converge with mesh refinement. These problems are resolved and computations with Reynolds as large as 500 converged by approximating the convective terms in the momentum equations by third-order upwind differences using Lagrangian Polynomials. When the volume of the core fluid or the Weber number decrease, increasing the importance of interfacial tension and leading to breakup of the interface forming a drop of core fluid, the FT method converges faster with mesh refinement than the VT method and upwinding may be required. Finally, examining the generation of spurious currents around a stationary “bubble” in the tube for Ohnesorge numbers between 0.1 and 10 it is found that the maximum velocity remains approximately the same in spite mesh refinements when VT is applied, whereas it is of the same order of magnitude for the coarsest mesh and monotonically decreases with mesh refinement when FT is applied.