Propagation of SH‐wave in an initially stressed orthotropic medium sandwiched by a homogeneous and an inhomogeneous semi‐infinite media

The paper presents a study of propagation of shear wave (SH‐wave) in an orthotropic elastic medium under initial stress sandwiched by a homogeneous semi‐infinite medium and an inhomogeneous half‐space. The technique of separation of variables has been adopted to get the analytical solutions for the dispersion relation in a closed form. The propagation of SH‐waves is influenced by inhomogeneity parameters and initial stress parameter. Velocities of SH‐waves are calculated numerically for different cases. As a special case when the intermediate layer and half‐space are homogeneous, computed frequency equation coincides with general equation of Love wave. To study the effect of inhomogeneity parameters and initial stress parameter, we have plotted the velocity of SH‐wave in several figures and observed that the velocity of wave decreases with the increases of non‐dimensional wave number. It can be found that the phase velocity decreases with the increase of inhomogeneity parameters. We observed that the velocity of SH‐wave decreases with the increases of initial stress parameter in both homogeneous and inhomogeneous media. GUI has been developed by using MATLAB to generalize the effect of the parameters discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

The end zone of a shear crack propagating at an intersonic velocity

An asymptotic solution of the problem for a shear crack propagating at an intersonic velocity is given that determines the size of the end zone, the distribution of the displacement jump in it and the dependence of the propagation velocity on an effective stress intensity factor, introduced in this paper. Numerical data are presented for the case of linear softening in the end zone and a comparison is made with the results for the Leonov–Panasyuk–Dugdale model. It is established using an effective stress intensity factor that the propagation is unstable at velocities close to the velocity of transverse waves; it becomes stable when approaching the velocity of longitudinal waves.     

Unsteady flow past a flat plate with suction

A solution is given for the transient response for laminar boundary layer flow past a flat plate to a step-function change in suction velocity. An arbitrary but constant suction velocity normal to the plate is allowed prior to step-change. Using the Laplace transform technique the solutions for the unsteady velocity profile and shear stress are obtained and are graphically sketched when the suction velocity doubles in the stepchange. The results show clear evidence of boundary-layer contraction when suction velocity is increased.     

拟牛顿流的一种三变量域模型的有限元方法的数值分析

0．引言目前，涉及高温条件下材料蠕变性质的粘弹性流动问题已引起人们广泛的研究兴趣，不少文章讨论了如何对其进行数值求解（见［1］－－［41），首先，人们研究了较简单的仅以速度，压力两个变量来表述此现象的模型问题（如［1，2］）等．鉴于应力变量在材料性质方面的特殊重要性，最近J．Baxanzer等人在［3］中首次对应力满足幂函数规律的蠕变流研究了包含应力、速度和压力三种变量的模型问题的有限元逼近，当粘性的牛顿部分为零时（详见下述）在假定速度与应力、速度与压力有限元空间之间同时满足两种＊＊B条件以后，证明了有限元解…     

Kernel-based fuzzy clustering and fuzzy clustering: A comparative experimental study

In this study, we present a comprehensive comparative analysis of kernel-based fuzzy clustering and fuzzy clustering. Kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering, however, the effectiveness of this extension vis-à-vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis. Our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering. Generic Fuzzy C-Means (FCM) and Gustafson–Kessel (GK) FCM are compared with two typical generalizations of kernel-based fuzzy clustering: one with prototypes located in the feature space (KFCM-F) and the other where the prototypes are distributed in the kernel space (KFCM-K). Both generalizations are studied when dealing with the Gaussian kernel while KFCM-K is also studied with the polynomial kernel. Two criteria are used in evaluating the performance of the clustering method and the resulting clusters, namely classification rate and reconstruction error. Through carefully selected experiments involving synthetic and Machine Learning repository (http://archive.ics.uci.edu/beta/) data sets, we demonstrate that the kernel-based FCM algorithms produce a marginal improvement over standard FCM and GK for most of the analyzed data sets. It has been observed that the kernel-based FCM algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters.     

Turbulent Poiseuille flow with near-critical wall transpiration

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress with the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. The case of near-critical transpiration, when the shear stress at the injection wall vanishes, is considered. It is shown that the maximum point on the mean velocity profile lies in a thin sublayer near the suction wall in this case. A formula for the position of the maximum point as a function of the transpiration factor is obtained. The mean velocity profiles near the suction wall are calculated. A friction law for Poiseuille flow with near-critical transpiration is found, which makes it possible to describe the relation between the shear stress at the wall, the Reynolds number, and the transpiration velocity by a single function of one variable. Direct numerical simulation of the flow for some transpiration factors is performed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On exact solutions of Stokes second problem for a Burgers�� fluid, II. The cases �� =?��2/4 and �� > ��2/4

In this study, the exact solutions of the Stokes second problem for a Burgers?? fluid are presented when the relaxation time satisfies the conditions ?? =???²/4 and ?? >???²/4. The velocity field and the associated tangential stress, when only one initial condition is necessary for velocity, are determined by means of the Laplace transform. The physical interpretation for the emerging parameters is discussed with the help of graphical illustrations. The similar solutions for the Stokes?? first problem are obtained as the limiting cases of our solutions.     

Nonlinear mathematical analysis for blood flow in a constricted artery under periodic body acceleration

This study analyses the pulsatile flow of blood through mild stenosed narrow arteries, treating the blood in the core region as a Casson fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is employed to solve the resulting coupled implicit system of non-linear partial differential equations. The expressions for shear stress, velocity, wall shear stress, plug core radius, flow rate and longitudinal impedance to flow are obtained. The effects of pulsatility, stenosis depth, peripheral layer thickness, body acceleration and non-Newtonian behavior of blood on these flow quantities are discussed. It is noted that the plug core radius, wall shear stress and longitudinal impedance to flow increase as the yield stress and stenosis depth increase and they decrease with the increase of the body acceleration, pressure gradient, width of the peripheral layer thickness. It is observed that the plug flow velocity and flow rate increase with the increase of the pulsatile Reynolds number, body acceleration, pressure gradient and the width of the peripheral layer thickness and the reverse behavior is found when the yield stress, stenosis depth and lead angle increase. It is also recorded that the wall shear stress and longitudinal impedance to flow are considerably lower for the two-fluid Casson model than that of the single-fluid Casson model. It is found that the presence of body acceleration and peripheral layer influences the mean flow rate and mean velocity by increasing their magnitude significantly in the arteries.     

Numerical evaluation of the suppression effect of a free-to-rotate triangular fairing on the vortex-induced vibration of a circular cylinder

The suppression of vortex-induced vibration (VIV) of a circular cylinder with a free-to-rotate triangular fairing in the Reynolds number range of Re = 1100–6100 is numerically investigated using computational fluid dynamics. The unsteady Reynolds-averaged Navier–Stokes equations and the shear stress transport k–ω turbulence model coupled with an improved fourth-order Runge–Kutta method are used to solve the wake flow, the structure's vibration, and the fairing's rotation. The computational model is validated with the available experimental results for a cylinder with an attached short-tail fairing. The numerical results indicate that the triangular fairing has a positive role in suppressing vibration when it achieves a stable position deflected from the flow direction. The suppression effect is sensitive to the incoming flow velocity. The fairing shifts from a stable state to an unstable one when the flow velocity varies. Therefore, maintaining the hydrodynamic stability of the fairing is the key to achieving success in vibration suppression, and the stability is dependent on the characteristic length and the rotational friction. Although the strong flapping of the 70° triangular fairing excites a more vigorous vibration, it may be used as an amplifier of VIV for energy harvesting.     

The stress–strain state and disintegration of a meteoroid moving through the atmosphere

The solution of the problem of the stress–strain state of an elastic body of spherical shape when it enters the Earth's atmosphere at a superorbital velocity is obtained in the form of a series in Legendre polynomials, in the case of a viscous gas at hypersonic velocity, in the quasi-stationary formulation. Using the Hubert–von Mises–Hencky criterion the limit stresses corresponding to the start of disintegration are obtained. This enables the nature of the disintegration of the body to be judged and also enables the heights at which destruction of known meteoroids begins to be estimated.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

Exact solutions of accelerated flows for a Burgers’ fluid II. The cases γ = λ2/4 and γ > λ2/4

The purpose of this study is to provide the exact analytic solutions of accelerated flows for a Burgers’ fluid when the relaxation times satisfy the conditions γ = λ²/4 and γ > λ²/4. The velocity field and the adequate tangential stress that is induced by the flow due to constantly accelerating plate and flow due to variable accelerating plate are determined by means of Laplace transform. All the solutions that have been obtained are presented in the form of simple or multiple integrals in terms of Bessel functions. A comparison between Burgers’ and Newtonian fluids for the velocity and the shear stress is also made through several graphs.     

The motion of a solid with large deformations

We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates… Therefore it is wise to choose a third-gradient theory for the body. Stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time because it may be interrupted by crushing, resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.     

梯形复式断面均匀流水力计算

对由主槽、斜边坡以及边滩三部分组成的梯形复式断面渠道均匀流的水力计算进行了研究,采用表观切应力表示各子断面间的横向动量交换,结合各部分力的平衡关系式,推导出了主槽、斜边坡和边滩平均流速的计算公式．采用UK-FCF的实验数据对公式进行了验证,滩槽相对水深为0.057~0.4的7组工况计算值与实测值的比较表明,不考虑表观切应力的计算值与实测值的一致性都较差．比较而言,斜边坡部分平均流速在水深较小时计算值大于实测值,在水深较大时计算值小于或接近实测值,表观切应力对边滩部分平均流速的影响不明显,而主槽部分若不考虑表观切应力的影响,则计算值与实测值差别较大．运用三段式模型与刘沛清等提出的两段式模型分别进行了有边坡梯形断面复式渠道的水力计算,结果表明对于有边坡段的梯形断面渠道（河道）,三段式模型更加有效．最后,将计算的表观切应力与实测表观切应力进行了比较,两者的一致性表明采用的动量输运系数是合适的．     

The plane problem of the extrusion of a viscoplastic medium by parallel plates

A solution of the problem of the plane parallel flow of viscoplastic medium between two parallel plates when they approach (separate) at a specified velocity is given within the framework of the Bingham model in the inertialess thin-layer approximation for arbitrary values of the coefficient of viscosity and the yield stress. Analytic expressions are obtained for the velocity and pressure fields. The boundary of the flow kernel, where the shear stress on the areas of the parallel planes of the plates is less than the yield stress and the component of the velocity, parallel to the plates, does not change in a transverse direction, is determined. A single similarity parameter which defines the kinematic and dynamic flow characteristics is found. For a specified law of motion of the plates, a general expression is obtained for the force acting on plates of finite size in terms of a dimensionless function of a single dimensionless parameter. The law of approach (separation) of the plates under a constant force is found.     

Non-Newtonian characteristics of peristaltic flow of blood in micro-vessels

Of concern in the paper is a generalized theoretical study of the non-Newtonian characteristics of peristaltic flow of blood through micro-vessels, e.g. arterioles. The vessel is considered to be of variable cross-section and blood to be a Herschel–Bulkley type of fluid. The progressive wave front of the peristaltic flow is supposed sinusoidal/straight section dominated (SSD) (expansion/contraction type); Reynolds number is considered to be small with reference to blood flow in the micro-circulatory system. The equations that govern the non-Newtonian peristaltic flow of blood are considered to be non-linear. The objective of the study has been to examine the effect of amplitude ratio, mean pressure gradient, yield stress and the power law index on the velocity distribution, wall shear stress, streamline pattern and trapping. It is observed that the numerical estimates for the aforesaid quantities in the case of peristaltic transport of blood in a channel are much different from those for flow in an axisymmetric vessel of circular cross-section. The study further shows that peristaltic pumping, flow velocity and wall shear stress are significantly altered due to the non-uniformity of the cross-sectional radius of blood vessels of the micro-circulatory system. Moreover, the magnitude of the amplitude ratio and the value of the fluid index are important parameters that affect the flow behaviour. Novel features of SSD wave propagation that affect the flow behaviour of blood have also been discussed.     

环空管内粘弹性流体不定常旋转流的解及流动特性分析

本文用Hankel积分变换法分别求得二阶流体和Maxwell流体在环管内不定常旋转流运动方程的解析解,据此可以分析环管内旋转速度和切应力的分布与变化特征;流体物性参数、管道环隙大小等参量在解析公式中有明确反映,便于定性分析和讨论,本解可以为钻探工程和高分子加工工艺的设计提供理论依据,另外还可用来分析双筒粘度计的流动状态和应力特征,拟合曲线,确定材料的粘弹性参数,在对这种流体进行特性分析时,我们发现,Maxwell流体的旋转流动在起动初期表现为方波振荡,振动的幅度和周期随Ha(物质常数)的增大而增大,此种现象还是首次发现,可能对实际应用有一定的意义。     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

The 3D motion of a solid with large deformations

We study in dimension 3 the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, shells, etc. Therefore, it is wise to choose a third gradient theory for the body. It is known that the stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint, which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time for it may be interrupted by a crushing, which entails a discontinuity of velocity with respect to time, i.e., an internal collision.