Damping of Vertically Excited Surface Wave in Weakly Viscous Fluid

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.

Vertically forced surface wave in weakly viscous fluids bounded in a circular cylindrical vessel

Two-time scale perturbation expansions were developed in weakly viscous fluids to investigate surface wave motions by linearizing the Navier-Stokes equation in a circular cylindrical vessel which is subject to a vertical oscillation. The fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates a damping term and external excitation, was derived for the weakly viscid fluids. The condition for the appearance of stable surface waves was obtained and the critical curve was determined. In addition, an analytical expression for the damping coefficient was determined and the relationship between damping and other related parameters (such as viscosity, forced amplitude, forced frequency and the depth of fluid, etc.) was presented. Finally, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent.

Complex potentials and singular integral equation for curve crack problem in antiplane elasticity

Four types of the fundamental complex potential in antiplane elasticity are introduced: (a) a point dislocation, (b) a concentrated force, (c) a dislocation doublet and (d) a concentrated force doublet. It is proven that if the axis of the concentrated force doublet is perpendicular to the direction of the dislocation doublet, the relevant complex potentials are equivalent. Using the obtained complex potentials, a singular integral equation for the curve crack problem is introduced. Some particular features of the obtained singular integral equation are discussed, and numerical solutions and examples are given.

Instability Analysis of Nonlinear Surface Waves in a Circular Cylindrical Container Subjected to a Vertical Excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier-Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated. (c) 2005 Elsevier SAS. All rights reserved.

Using wavelet transform to study the Lipschitz local singular exponent in wall turbulence

In this paper, wavelet,transform is introduced to study the Lipschitz local singular exponent for characterising the local singularity behavior of fluctuating velocity in wall turbulence. I, is found that the local singular exponent is negative when the ejections and sweeps of coherent structures occur in a turbulent boundary layer.

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated.

Cauchy singular integral equation method for transient antiplane dynamic problems

In this paper, by use of the boundary integral equation method and the techniques of Green basic solution and singularity analysis, the dynamic problem of antiplane is investigated. The problem is reduced to solving a Cauchy singular integral equation in Laplace transform space. This equation is strictly proved to be equivalent to the dual integral equations obtained by Sih [Mechanics of Fracture, Vol. 4. Noordhoff, Leyden (1977)]. On this basis, the dynamic influence between two parallel cracks is also investigated. By use of the high precision numerical method for the singular integral equation and Laplace numerical inversion, the dynamic stress intensity factors of several typical problems are calculated in this paper. The related numerical results are compared to be consistent with those of Sih. It shows that the method of this paper is successful and can be used to solve more complicated problems. Copyright (C) 1996 Elsevier Science Ltd

Variable power singular interface elements for a crack normal to the bimaterial interface

Zero thickness crack tip interface elements for a crack normal to the interface between two materials are presented. The elements are shown to have the desired r(lambda-1) (0 < lambda < 1) singularity in the stress field at the crack tip and are compatible with other singular elements. The stiffness matrices of the quadratic and cubic interface element are derived. Numerical examples are given to demonstrate the applicability of the proposed interface elements for a crack perpendicular to the bimaterial interface.

Collapsed isoparametric element as a singular element for a crack normal to the bi-material interface

It is shown that the variable power singularity of the strain field at the crack tip can be obtained by the simple technique of collapsing quadrilateral isoparametric elements into triangular elements around the crack tip and adequately shifting the side-nodes adjacent to this crack tip. The collapsed isoparametric elements have the desired singularity at crack tip along any ray. The strain expressions for a single element have been derived and in addition to the desired power singularity, additional singularities are revealed. Numerical examples have shown that triangular elements formed by collapsing one side lead to excellent results.

Singular behaviour near the tip of a sharp V-notch in a power law hardening material

This paper presents an asymptotic analysis of the near-tip stress and strain fields of a sharp V-notch in a power law hardening material. First, the asymptotic solutions of the HRR type are obtained for the plane stress problem under symmetric loading. It is found that the angular distribution function of the radial stress sigma(r) presents rapid variation with the polar angle if the notch angle beta is smaller than a critical notch angle; otherwise, there is no such phenomena. Secondly, the asymptotic solutions are developed for antisymmetric loading in the cases of plane strain and plane stress. The accurate calculation results and the detailed comparisons are given as well. All results show that the singular exponent s is changeable for various combinations of loading condition and plane problem.

Exact quantum master equation via the calculus on path integrals

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems. (C) 2005 American Institute of Physics.

Self-sustained oscillations caused by magnetic field in a weakly-coupled GaAs/AlAs superlattice

We have investigated the influence of transverse magnetic field B up to 14 T at 1.6 K on the tunneling processes of electric field domains in doped weakly coupled GaAs/AlAs superlattices. Three regimes, i.e, stable field domains, current self-sustained oscillations and averaged field distribution are successively observed with increasing B. The mechanisms of switching-over among these regimes are due to B-induced modification of the dependence of the effective electron drift velocity on electric field. The simulated calculation gives a good agreement with the observed experimental results. (C) 2000 Published by Elsevier Science B.V. All rights reserved.

Large positive and negative lateral displacements from total internal reflection configuration with a weakly absorbing dielectric film

It is theoretically shown that the simultaneously large positive and negative lateral displacements will appear when the resonant condition is satisfied for a TE-polarized light beam reflected from the total internal reflection configuration with a weakly absorbing dielectric film. Appearance of the enhanced negative lateral displacement is relative to the incidence angle, absorption of the thin Elm and its thickness. If we select an appropriate weakly absorbing dielectric film and its thickness, the simultaneously enhanced positive and negative lateral displacements will appear at different resonant angles. These phenomena may lead to convenient measurements and interesting applications in optical devices.

Self-sustained oscillations caused by magnetic field in a weakly-coupled GaAs/AlAs superlattice

We have investigated the influence of transverse magnetic field B up to 14 T at 1.6 K on the tunneling processes of electric field domains in doped weakly coupled GaAs/AlAs superlattices. Three regimes, i.e, stable field domains, current self-sustained oscillations and averaged field distribution are successively observed with increasing B. The mechanisms of switching-over among these regimes are due to B-induced modification of the dependence of the effective electron drift velocity on electric field. The simulated calculation gives a good agreement with the observed experimental results. (C) 2000 Published by Elsevier Science B.V. All rights reserved.