A new branch of solutions of boundary-layer flows over a permeable stretching plate

The steady-state boundary-layer flows over a permeable stretching sheet are investigated by an analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM). Two branches of solutions are obtained. One of them agrees well with the known numerical solutions. The other is new and has never been reported in general cases. The entrainment velocity of the new branch of solutions is always smaller than that of the known ones. For permeable stretching sheet with sufficiently large suction of mass flux, the difference between the shear stresses and velocity profiles of two branches of solutions is obvious: the shear stress of the new branch of solutions is considerably larger than that of the known ones. However, for impermeable sheet and permeable sheet with injection or small suction of mass flux, the shear stress and the velocity profile of two branches of solutions are rather close: in some cases the difference is so small that the new branch of solutions might be neglected even by numerical techniques. This reveals the reason why the new branch of solutions has not been reported. This work also illustrates that, for some non-linear problems having multiple solutions, analytic techniques are sometimes more effective than numerical methods.     

FLEXURAL WAVE DISPERSION IN MULTI-WALLED CARBON NANOTUBES CONVEYING FLUIDS

Motivated by the great potential of carbon nanotubes for developing nanofluidic devices, this paper presents a nonlocal elastic, Timoshenko multi-beam model with the second order of strain gradient taken into consideration and derives the corresponding dispersion relation of flexural wave in multi-walled carbon nanotubes conveying fuids. The study shows that the moving flow reduces the phase velocity of flexural wave of the lowest branch in carbon nanotubes. The phase velocity of flexural wave of the lowest branch decreases with an increase of flow velocity. However, the effects of flow velocity on the other branches of the wave dispersion are not obvious. The effect of microstructure characterized by nonlocal elasticity on the dispersion of flexural wave becomes more and more remarkable with an increase in wave number.     

Time-dependent stagnation-point flow over rotating disk impinging oncoming flow

The unsteady stagnation point flow of an incompressible viscous fluid over a rotating disk is investigated numerically in the present study.The disk impinges the oncoming flow with a time-dependent axial velocity.The three-dimensional axisymmetric boundary-layer flow is described by the Navier-Stokes equations.The governing equations are solved numerically,and two distinct similarity solution branches are obtained.Both solution branches exhibit different flow patterns.The upper branch solution exists for all values of the impinging parameter β and the rotating parameter.However,the lower branch solution breaks down at some moderate values of β.The involvement of the rotation at disk allows the similarity solution to be transpired for all the decreasing values of β.The results of the velocity profile,the skin friction,and the stream lines are demonstrated through graphs and tables for both solution branches.The results show that the impinging velocity depreciates the forward flow and accelerates the flow in the tangential direction.     

Particle segregation in slurry flow through vertical tees

An experimental study has been conducted of particle segregation in slurry flow through vertical tees. Water-sand slurries with solids concentrations to 25% by volume were used with tees of various sizes and angles. The experiments showed that the branch concentration is less than the upstream value for all the branches studied at velocity ratios less than unity. The separation ratio was found to be a function of upstream conditions. velocity ratio. branch size and angle. For lateral branches. the inertia effect is dominant at low velocity ratios. whereas the gravity effect becomes important at high velocity ratios. A two-dimensional model explains the results qualitatively.     

理论力学研究性教学新探索: 刚体运动基点法公式与连续介质速度场分解

基础力学课程体系是现代工程教育的重要基础。理论力学作为力学课程体系的起点,提供了力学基本概念和原理。一方面,以离散体系(质点/刚体)为研究对象的理论力学与以连续体系(介质/变形微元体)为对象的连续介质力学,它们之间既存在密切联系又差异明显;另一方面,源于理论/经典力学的动力系统理论和非线性科学研究对应用力学学科产生了广泛影响。围绕这两个侧面,作者将在《理论力学研究性教学新探索》系列教研论文中阐释它们之间的联系。本篇探讨刚体运动基点法公式和连续介质速度场分解之间的关系,它们分别给出刚体模型和连续介质模型的速度分布规律,前者依赖刚体转动角速度矢量,而后者由速度梯度张量所刻画。本文将说明,两者存在对应理论关系,且刚体基点法公式是连续介质速度场分解的退化形式,即忽略变形效应。

     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

Shock waves in diatomic chains—I. linear analysis

Solutions are presented for the response of semi-infinite diatomic chains to a step jump in velocity. Integral transform theory and contour integration are used to express the solutions as definite integrals for the acoustic and optical branch contributions to the response. The contribution of an end mode is indicated for the case of particles that are unequally spaced within a unit cell. Asymptotic approximations are obtained for the contribution of the optical and acoustic branches to the wave solution when the wave propagates far into the lattice. Asymptotic estimates are obtained, also, for the discontinuous speeds at which the head of the pulse travels in a general diatomic chain. Shock profiles for the special case of a simple diatomic chain and for a general diatomic chain are discussed, and upper bounds are obtained for the maximum shock responses that are possible in such chains with the given shock condition.     

Low-frequency dispersion of fundamental waves in anisotropic piezoelectric plates

Low-frequency onset of the fundamental branches in piezoplates is studied with a view to identify the impact of piezoelectric coupling. General analytical expressions for the zero- and leading-order terms of the velocity versus wavenumber expansion in an anisotropic homogeneous piezoplate are obtained. On this ground, it is shown what types of anisotropy and electric boundary conditions enable the onset parameters of fundamental branches to be piezoactive. Particular attention is given to the linear dispersion at the origin of two upper fundamental branches. This property is entirely caused by the piezoeffect, being ruled out for elastic plates. An invariant hierarchy is established between the zero-order velocities of the fundamental waves under different electric boundary conditions in homogeneous and functionally graded plates. It is shown that some of these velocities in a metallized plate become piezoactive specifically if the piezoplate is functionally graded.     

On constitutive equations for branching of response with selectivity

Materials, such as elastic-plastic, which exhibit distinct regimes of response are usually modeled by different constitutive equations in each regime. The present paper explores a method for the construction of a unified constitutive equation from these separate relations. The main idea is to write this unified equation in an implicit form which contains these separate solutions as non-unique solutions. The form is chosen in order to utilize the notions of branch points and branches. Different solutions, corresponding to constitutive equations for different regimes of response, are then regarded as bifurcations at branch points from the fundamental response. The choice of the appropriate branch at a branch point is governed by a selectivity condition which depends on the nature of the response under consideration. A detailed example is provided for elastic-plastic response, with and without the effect of strain rate dependence.     

Experimental investigation of coaxial side branch resonators

Digital particle image velocimetry is employed to investigate acoustically-coupled flow past a coaxial deep cavity (side branch) resonator mounted in a duct. The emphasis is on the effect of the separation between the coaxial side branches on the interaction between separated shear layers that form across the side branch openings. Various resonator geometries are characterized in terms of patterns of instantaneous and time-averaged flow velocity, vorticity, and streamline topology at several phases of the acoustic cycle. In addition, phase-averaged images of the flow in conjunction with unsteady pressure measurements are evaluated in order to provide insight into the mechanisms of acoustic power generation. Generally speaking, the acoustic source undergoes a significant transformation as the distance between the coaxial side branches changes. As the distance between the side branches decreases, interaction between the associated oscillating shear layers results in substantial changes in the spatial structure of the acoustic source region.     

Experimental slowing of flexural waves in dielectric elastomer films by voltage

Shape and physical properties of dielectric elastomers are changeable by voltage. Theoretical works show that these changes can be harnessed to tune the propagation of superposed elastic waves. We experimentally demonstrate this concept by manipulating waves in a dielectric elastomer film, focusing on the flexural mode at low frequencies. To this end, we design an experimental apparatus to pre-stretch, actuate, excite waves at low frequencies in a VHB™ 4910 film, and measure the velocity of the fundamental flexural mode. Our results show that the excited wave velocity is slowed down by the applied voltage, and provide experimental proof of concept for the application of deformable dielectrics as tunable waveguides.     

Phase dynamics of effective drag and lift components in vortex-induced vibration at low mass–damping

In this work, we investigate the dynamics of vortex-induced vibration of an elastically mounted cylinder with very low values of mass and damping. We use two methods to investigate this canonical problem: first we calculate the instantaneous phase between the cylinder motion and the fluid forcing; second we decompose the total hydrodynamic force into drag and lift components that act along and normal to, respectively, the instantaneous effective angle of attack. We focus on the phase dynamics in the large-amplitude–response range, consisting of the initial, upper and lower “branches” of response. The instantaneous phase between the transverse force and displacement shows repeated phase slips separating periods of constant, or continuous-drifting, phase in the second half of the upper branch. The phase between the lift component and displacement shows strong phase locking throughout the large-amplitude range – the average phase varies linearly with the primary frequency – however the modulation of this phase is largest in the second half of the upper branch. These observations suggest that the large-amplitude–response dynamics is driven by two distinct limit cycles – one that is stable over a very small range of reduced velocity at the beginning of the upper branch, and another that consists of the lower branch. The chaotic oscillation between them – the majority of the upper branch – occurs when neither limit cycle is stable. The transition between the upper and lower branches is marked by intermittent switching with epochs of time where different states exist at a constant reduced velocity. These different states are clearly apparent in the phase between the lift and displacement, illustrating the utility of the force decomposition employed. The decomposed force measurements also show that the drag component acts as a damping factor whereas the lift component provides the necessary fluid excitation for free vibration to be sustained.     

Approximation of the flexural velocity branch in plates

The fitting of the flexural velocity branch by means of the Padé approximation is examined. Its use for free isotropic plates is elaborated and compared to other approaches. The flexural-velocity expansion into power series, which underlies the Padé implementation, is detailed. The generalization for fluid-loaded plates and inhomogeneous plates is outlined, and the impact of anisotropy is discussed in some detail.     

圆柱壳的非线性自由振动分析

本文分析了各向同性封闭圆柱壳的非线性自由振动。文中采用经典的非线性弹性力学方法推导了圆柱壳的大振幅运动方程,这些方程的静态形式与冯·卡门的板理论方程具有同样的精度。文中讨论了四种基本振动模态,并且还以数学公式的形式给出了一般的最终结果,一些例子以曲线给出结果,并进行了比较。结果还表明线性振动可以作为非线性振动的一种特例。     

The onset of gas entrainment from a flowing stratified gas–liquid regime in dual discharging branches: Part II: Critical conditions at low to moderate branch Froude numbers

This is the second of a two part investigation. Experiments were performed in a 50.8 mm diameter horizontal pipe with three 6.35 mm diameter branches located at the test section mid-span. The inlet length was 1.8 m, and three branch orientations were tested at 0° (side), 45° (inclined), and 90° (bottom) from horizontal. Water and air, operating at 206 kPa, were used and both fluids flowed co-currently within the inlet in the smooth-stratified regime. The inlet superficial velocity of the liquid phase ranged between 0.04 and 0.15 m/s while in the gas phase values of 0.3, 0.4, and 1 m/s were tested. Three different dual discharge combinations were tested and included side-inclined, side-bottom, and inclined-bottom. The tested branch flow Froude numbers were limited between low to moderate values which ranged between 1 and 23. Extensive experimental data are reported for the critical conditions at the onset of gas entrainment during dual discharge. A novel map was developed for the inclined-bottom branch configuration showing the relationship between the inlet superficial liquid velocity and branch Froude numbers. This map was used to quantify the three observed modes of gas entrainment during dual discharge. These modes were classified as onset of gas entrainment in the inclined branch only, in the bottom branch only, or both the inclined and bottom branches simultaneously. The critical height at the onset of gas entrainment results are compared to published models and data sets and poor agreement was found with studies conducted in stratified gas–liquid reservoirs.     

A new branch of the temperature distribution of boundary-layer flows over an impermeable stretching plate

In 2005 it was reported that the boundary-layer flows over an impermeable stretching plate have a new branch of solutions. In this short communication, the corresponding heat transfer problem is considered, and a new branch of temperature distribution is obtained. It is found that the new branch of temperature distributions is mostly rather close to the known branch of solutions, except in case of small Prandtl number. Thus, it is practically rather hard to distinguish the two branches of temperature distributions.     

Finite-amplitude waves in a homogeneous fluid with a floating elastic plate

Equations for three nonlinear approximations of a wave perturbation in a homogeneous ideal incompressible fluid covered by a thin elastic plate are obtained using the method of multiple scales and taking into account that the acceleration of vertical flexural displacements of the plate is nonlinear. Based on the obtained equations, asymptotic expansions up third-order terms are constructed for the fluid velocity potential and the perturbations of the plate-fluid interface (plate bending) caused by a traveling periodic wave of finite amplitude. The wave characteristics are analyzed as functions of the elastic modulus and thickness of the plate and the length and tilt of the initial fundamental harmonic wave.     

含双侧分支受限空间油气爆炸火焰行为与超压特性大涡模拟

为研究含分支结构狭长受限空间油气爆炸特性规律,基于大涡模拟WALE模型和Zimont预混火焰模型,对横截面为100 mm×100 mm的含双侧分支管道受限空间油气泄压爆炸特性进行了数值模拟。通过对火焰形态、火焰传播速度和动态超压3个物理量的对比,验证了所建立模型对于含分支结构受限空间油气爆炸计算的适用性。基于数值模拟结果,对爆炸过程中的流场结构、火焰形态和超压变化规律进行了分析,指出了"浪花状"火焰的形成原因。结果表明:(1)火焰传播进入分支管道前,在主管道和分支管道交界处会产生旋转方向相反的对称涡旋结构,并随着火焰传播不断向分支管道内部发展;(2)当火焰传播进入分支管道后,分支管道内部前期已建立流场决定了火焰的形态,火焰锋面在涡旋结构作用下呈"浪花状",此后火焰和流场相互影响,流场向湍流转捩,火焰锋面褶皱变形;(3)爆炸超压升压过程可划分为4个阶段,受到火焰锋面面积和分支管道泄压共同作用,表明爆炸流场、火焰行为和动态超压呈现出显著耦合性。     

A study of the recirculating flow in planar,symmetrical branching channels

A numerical method is employed to examine the flow in symmetrical, two‐dimensional branches of Y shape and Tee shape. The methodology is based on a pressure‐correction procedure within the frame of unstructured grids. Specified pressures are imposed at the outlets of the two branches. The area ratio of the branch is allowed to vary in the range of 2–3. Separation of the flow in the bifurcating region is inevitable. With equal outlet pressures, symmetrical flow patterns prevail except for the Y type branch under the conditions of high Reynolds numbers and large area ratios. This implies that the Y‐branch flow is more sensitive to small disturbances. It is shown that with a slightly higher pressure imposed on one of the two branches the structure of the recirculating flow for the Y type is greatly affected and the flow rate is reduced dramatically in the high‐pressure branch channel. In contrast, the influence on the Tee type branch is much lower since the flow behaves like a jet impinging on a confined duct. Copyright © 2005 John Wiley & Sons, Ltd.