Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem

In this paper we deal with three types of problems concerning the Hardy-Rellich's embedding for a bi-Laplacian operator. First we obtain the Hardy-Rellich inequalities in the critical dimension n=4. Then we derive a maximum principle for fourth order operators with singular terms. Then we study the existence, non-existence, simplicity and asymptotic behavior of the first eigenvalue of the Hardy-Rellich operator under various assumptions on the perturbation q.     

First-principle proof of the modified collision boundary conditions for the hard-sphere system

A fundamental issue lying at the foundation of classical statistical mechanics is the determination of the collision boundary conditions that characterize the dynamical evolution of multi-particle probability density functions (PDF) and are applicable to systems of hard-spheres undergoing multiple elastic collisions. In this paper it is proved that, when the deterministic N-body PDF is included in the class of admissible solutions of the Liouville equation, the customary form of collision boundary conditions adopted in previous literature becomes physically inconsistent and must actually be replaced by suitably modified collision boundary conditions.     

铸铁材料在边界润滑条件下形成薄片状磨屑的塑性流动机制

本文在铸铁材料油润滑线接触滑动磨损状态研究的基础上,对边界润滑区典型的薄片状磨屑之形成过程进行了考察。通过对磨损表面的几何形貌、磨屑的形态和内部显微组织变化的分析,提出了薄片状磨屑的塑性流动形成机制。作者认为,薄片状磨屑的形成是磨损表面材料在局部应力和摩擦热的作用下以3种方式发生塑性流动的结果:当摩擦方向垂直于磨削加工条纹时为单侧塑性流动;当摩擦方向平行于磨削加工条纹时为双侧塑性流动;而在峰顶平台(磨削加工粗糙条纹磨合后形成的微平台)较大且载荷较高时,则为平台上的材料沿着摩擦方向向前挤压流动。     

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM soluti...     

Physalis method for heterogeneous mixtures of dielectrics and conductors: Accurately simulating one million particles using a PC

Prosperetti’s seminal Physalis method, an Immersed Boundary/spectral method, had been used extensively to investigate fluid flows with suspended solid particles. Its underlying idea of creating a cage and using a spectral general analytical solution around a discontinuity in a surrounding field as a computational mechanism to enable the accommodation of physical and geometric discontinuities is a general concept, and can be applied to other problems of importance to physics, mechanics, and chemistry. In this paper we provide a foundation for the application of this approach to the determination of the distribution of electric charge in heterogeneous mixtures of dielectrics and conductors. The proposed Physalis method is remarkably accurate and efficient. In the method, a spectral analytical solution is used to tackle the discontinuity and thus the discontinuous boundary conditions at the interface of two media are satisfied exactly. Owing to the hybrid finite difference and spectral schemes, the method is spectrally accurate if the modes are not sufficiently resolved, while higher than second-order accurate if the modes are sufficiently resolved, for the solved potential field. Because of the features of the analytical solutions, the derivative quantities of importance, such as electric field, charge distribution, and force, have the same order of accuracy as the solved potential field during postprocessing. This is an important advantage of the Physalis method over other numerical methods involving interpolation, differentiation, and integration during postprocessing, which may significantly degrade the accuracy of the derivative quantities of importance. The analytical solutions enable the user to use relatively few mesh points to accurately represent the regions of discontinuity. In addition, the spectral convergence and a linear relationship between the cost of computer memory/computation and particle numbers results in a very efficient method. In the present paper, the accuracy of the method is numerically investigated by example computations using one dielectric particle, one isolated conductor particle, one conductor particle connected to an external source with imposed voltage, and two conductor/dielectric particles with strong interactions. The efficiency of the method is demonstrated with one million particles, which suggests that the method can be used for many important engineering applications of broad interest.     

Special fractal growth of dendrite copper using a hydrothermal method

Special fractal dendrite Cu nanostructures have been synthesized through a simple hydrothermal method, and the effects of the volume ratio between glycerol and water and the concentration of H₃PO₃ on the morphologies of dendrite Cu have been studied in detail. The Field emission scanning electron microscopy (FESEM), Transmission electron microscopy (TEM) and X-ray diffraction (XRD) have been used to characterize these Cu products. The results indicate that rhombic diamond and different morphologies of fractal dendrite were prepared because of the accumulation of Cu nuclei based on the diffusion-limited aggregation (DLA) and the nucleation-limited aggregation (NLA) model. Fortunately, symmetrical leaf-like dendrite Cu nanostructures different from Cu dendrites reported before have been obtained. Additionally, an explanation for the growth of fractal dendrite Cu has been discussed carefully.     

An Iterative Method of Point Mapping under Cell Reference for the Global Analysis: Theory and a Multiscale Reference Technique

In iterative method of Point Mapping under Cell Reference, a cell co-ordinate system, called cell reference, is built to identify the subregions (cells) in the state space. When the cell reference is equipped with the so-called characteristic functions, it can work as an inspector or a recorder to derive the local dynamics of the subregions from the information provided by the trajectories passing through them. This method can retain the accuracy of the Point Mapping Method but greatly reduce the computational work. In this paper, the theoretic basis for this method is first discussed and a multiscale reference technique is then devised which can select an optimal cell reference and make the method more practicable. Finally, an example for application is presented. It is shown that the present method cannot only accurately and efficiently depict the basins of attraction of a dynamical system but also potentially detect other characteristics of the system.     

DRIVING FORCE AND DEFORMATION ANALYSIS FOR DYNAMIC CRACK PROPAGATION IN GAS PIPELINES UNDER DIFFERENT BOUNDARY CONDITIONS

In the light of a growing need for fracture control of rapid crack propagation(RCP)ingas pipelines,a program PFRAC(Pipeline FRacture Analysis Code)has been developed to analysethc various events.In this paper,by using PFRAC for the simulation of axial crack propagation in gaspipelines,a number of dynamic analysis issues rclating to boundary effects for uncracked and crackedpipes are investigated.This indicates that the boundary conditions along the length of the pipe play animportant role for fracture analysis in the pipe.In contrast.the boundary condutions at thc ends of along pipeline have little effect on the dynamic fracture events.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

Numerical Study of Natural Convection Heat Transfer in an Inclined Porous Cavity with Time-Periodic Boundary Conditions

Kalabin et al. (Numer. Heat Transfer A 47, 621-631, 2005) studied the unsteady natural convection for the sinusoidal oscillating wall temperature on one side wall and constant average temperature on the opposing side wall. The present article is on the unsteady natural convective heat transfer in an inclined porous cavity with similar temperature boundary conditions as those of Kalabin et al. The inclined angle of the cavity is varied from 0° to 80°. The flow field is modeled with the Brinkman-extended Darcy model. The combined effects of inclination angle of the enclosure and oscillation frequency of wall temperature are studied for Ra* = 10³, Da = 10⁻³, , and Pr=1. Some results are also obtained with the Darcy–Brinkman–Forchheimer model and Darcy’s law and are compared with the present Brinkman-extended Darcy model. The maximal heat transfer rate is attained at the oscillating frequency f = 46.7π and the inclined angle .