A finite difference analysis of the extrudate swell problem

The extrudate swell phenomenon of a purely viscous fluid is analysed by solving simultaneously the Cauchy momentum equations along with the continuity equation by means of a finite difference method. The circular and planar jet flows of Newtonian and power-law fluids are simulated using a control volume finite difference method suggested by Patankar called SIMPLER (semi-implicit method for pressure-linked equations). This method uses the velocity components and pressure as the primitive variables and employs a staggered grid and control volume for each separate variable. The numerical results show good agreement with the analytical solution of the axisymmetric stick-slip problem and exhibit a Newtonian swelling ratio of 13.2% or 19.2% for a capillary or slit die respectively in accordance with previously reported experimental and numerical results. Shear thinning results in a decrease in swelling ratio, as does the introduction of gravity and surface tension.     

Stability of a viscous liquid microjet in DC and AC electric fields

The problem of the stability of a liquid electrolyte jet under the action of a tangential electric field is considered. The radii of these jets, usually observable in experiments, vary from nanoscales to microscales. In this study, we consider microjets with the characteristic thickness of the double ion layer near the interface much less than the jet radius. The stability problem is analytically solved with account for the presence of this small parameter. The assumption on the electric neutrality of the jet as a whole leads to an explicit expression for the surface electric charge induced by the external field. The solution of the hydrodynamic problem in the external domain closes the solution and gives the dependence of the disturbance growth rate on the wavenumber. The cases of DC and AC electric fields are qualitatively compared. The distinctive features of jet stabilization by an AC high-frequency electric field are discussed.     

Boundary conformed co-ordinate systems for selected two-dimensional fluid flow problems. Part II: Application of the BFG method

This paper gives the results of an application of the SWEs (shallow water equations) to a part of the Hamburg harbour area, which is a complex flow domain, using the BFG approach, outlined in Part I. The results of a grid doubling procedure generating the desired computational grid from a coarse initial mesh are also presented. A second class of problems which is addressed, demands time-dependent co-ordinate systems. The problems which are solved are the free surface problem for a moving wave which eventually breaks and for a wave which is reflected by the solid walls of a rectangular basin.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Extrudate swell for a round jet with large surface tension

The problem of extrudate swell of a viscoelastic fluid from a round pipe is studied by the method of domain perturbations. The perturbation problems are solved by a finite-element method through second-order in the flow rate parameter ∈ for small flow rates. The analysis extends the work of Sturges on swelling in two-dimensional channels to round capillary tubes. In perturbation studies for small ∈, the rheology of the fluid may be expressed by three parameters, the viscosity and the two constants α₁ and α₂ appearing at order two in the expansion of the extra stress around zero shear. Surface tension has an important influence on the shape of the jet at low speeds. The shape of the surface on a round jet depends on α₁ and α₂, in the plane jet only on α₁. The analysis predicts that no matter what the constitutive equation may be, the jet will first contract if the radius of the pipe is sufficiently small. The contraction takes place in a length less than $\text{1}\text{10}$ the diameter of the jet and is followed by a swell. The contraction is usually small and may be hard to observe. There are five different contributions to the jet shape at second-order but only the viscoelastic ones persist as the pipe radius goes to zero.     

A case of jet flow of a gravity fluid

The problem of plane steady ideal heavy fluid flow bounded by an impermeable polygonal section, a curvilinear arc section, and a finite section of free surface is investigated in an exact nonlinear formulation. Hydrodynamic singularities may exist in the stream. A large class of captation problems of jet theory reduces to studying this kind of flow. The unique solvability of the problem under investigation is proved for sufficiently large Froude numbers and small arc curvature. A method of solution is given and an example is computed. Such problems have been solved earlier by numerical methods [1–3]. Some problems about jet flows of a gravity fluid with polygonal solid boundaries have been investigated by an analogous method in [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–143, May–June, 1975.     

A family of new solutions on the wall jet

The fluid flow problem within the wall jet created by fluid hitting on a solid surface at the right angle is solved based on the homotopy analysis method (HAM). A new family of solutions for jet with injection/suction which has been overlooked so far are obtained. Numerical evidence seems to suggest that these solutions decay algebraically far away from the wall.     

气流作用下同轴带电射流的不稳定性研究

通过对气体驱动同轴电流动聚焦的实验模型进行简化,开展了电场力和惯性力共同作用下同轴带电射流的不稳定性理论研究.在流动为无黏、不可压缩、无旋的假设下,建立了三层流体带电射流物理模型并得到了扰动在时间域内发展演化的解析形式色散关系,利用正则模方法求解色散方程发现了流动的不稳定模态,进而分析了主要控制参数对不稳定模态的影响.结果表明,在参考状态下轴对称模态的最不稳定增长率最大,因此轴对称扰动控制整个流场.外层气流速度越高,气体惯性力越大,射流的界面越容易失稳.内外层液-液同轴射流之间的速度差越大,射流越不稳定.表面张力对射流不稳定性起到促进作用.轴向电场对射流不稳定性具有双重影响:当加载电场强度较小时,射流不稳定性被抑制;当施加电压大于某一临界值时,轴向电场会促进射流失稳.临界电压的大小与界面上自由电荷密度和射流表面扰动发展关系密切.这些结果与已有的实验现象吻合,能够对实验的过程控制提供理论指导.     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media II: Numerical Results and Applications

In Part I of this study, generalized mathematical models were developed to describe the motion of fluids in porous media. The second part of this study solved the problem of fluid flow in small channels of a periodic elastic solid matrix at the pore scale numerically, and applied the volume-averaging technique to predict the macroscopic behavior of reservoirs. The numerical results demonstrated different macroscopic behavior of a porous medium due to cyclic excitation at various frequencies corresponding to the five separate characteristic macroscopic models identified in Part I. The results emphasize the need to use an appropriate model to interpret the corresponding responses of a saturated porous medium.     

Self-propelling,coalescing droplets

This work proposes and explores a new propulsion mechanism for sessile droplets which could be of interest for microfluidic applications. This mechanism relies on the Marangoni stresses resulting from the surface tension gradient arising when two droplets of different surface tensions coalesce. We argue that the tendency of the fluid to flow towards regions of higher surface tension is sufficient to displace the droplet. The coalescence of two miscible, partially wetting droplets with different surface tensions is investigated theoretically in this paper and modeled in the lubrication approximation framework. The problem is described by a set of three highly non-linear, coupled partial differential equations which is solved with a commercial finite element code. The analysis reveals two important dimensionless numbers which govern the flow characteristics, one related to the strength of the surface tension gradient and the other to the diffusion time scale. The numerical results confirm the occurrence of the self-propulsion behavior and a parametric study is performed to explore the role of the two dimensionless numbers on the propulsion speed and the total displacement. Unsurprisingly, self-propulsion is enhanced for larger values of the surface tension contrast between the two droplets and smaller values of the diffusion time scale which results in more time for the surface tension gradient to act.     

Stress intensity factors for two offset parallel circular cracks: Part III — Elastic layer

Part III of this work is concerned with the interaction of two penny-shaped cracks in the mid-plane of an elastic layer. Two cases, namely the stress free boundary case and the fixed boundary case, are considered. It is shown that these two cases are mathematically equivalent. As in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [1])_for the problems of an infinite solid and Part II (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 227–237 (1994) [2]) of a semi-infinite solid, the problem is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved when the crack size is small compared to the distance between them and the cracks are far away from the boundaries. It is also shown that the problem decouples when the cracks are subjected to normal loading and shear loading. Asymptotic solutions are presented for these two loadings.     

Flow of a conducting gas jet beyond the nozzle outlet of an electromagnetic accelerator

A solution is given for the problem of the motion of a conducting gas beyond the outlet of an accelerator. The form of the jet is found as well as the distribution of all jet parameters. The problem is solved assuming that the flow is plane, that there are no Hall currents, and that the velocity increase in the jet is small compared with the magnitude of the velocity at the exit of the accelerator channel.     

Thermal modulation and breakup of liquid jets

In this paper, we study the breakup behavior of Newtonian liquid and non‐Newtonian liquid jets with an arbitrary variation surface tension imposed along its length. The effect of duty cycle, fluid properties, and the various profiles of the surface tension is investigated. It is shown that the breakup behavior of a jet can be constructed by using the Fourier expansion of the surface tension profile. When the dimensionless wavenumber k is larger than 0.5, the jet breakup behavior is determined by the lowest frequency of the Fourier series expansion of the surface tension profile. As k decreases, higher frequency Fourier modes come to play. In general, for k between, 1∕(n+ 1) and 1∕n,n Fourier modes are needed to determine the jet breakup behavior. The current nonlinear model differs from the existing linear slender jet model in the literature in several ways. While the principle of superposition is valid for the linear model, it is not generally valid for the current nonlinear model. For the linear model, the jet will never break up when the wavenumber is larger than 1. The current model, however, shows clearly that the jet can indeed break up when the wavenumber is larger than 1. Furthermore, the current nonlinear model predicts a breakup time substantially higher than that from the linear model.Copyright © 2011 John Wiley & Sons, Ltd.     

自由液面晃动对旋转充液腔体运动稳定性的影响

针对旋转充液腔体的运动稳定性问题,本文考虑自旋速率范围较宽的情况。引入表示离心力与液体表面张力之比的常数C_f,以Stewartson—Wedemeyer理论和Murphy方法为基础,研究腔内液体晃动对共振不稳定频带的影响。结果表明只有当自旋速率较小,液体表面张力较大时,自由液面晃动对腔体运动稳定性才有明显影响;反之则可忽略自由液面晃动对腔体运动稳定性的影响。     

高温流体与变物性固体耦合系统温度场及热变形的数值计算

在腔体内部有高温流体,其外壳等固体材料的物性温度而为化的瞬态非线性问题中,其边界条件的确定十分困难。本文采用流一固“混合流”模式进行温度场的计算,由于方程计及了瞬态、复杂形状、浮动外边界条件等因系,采用变步长的控制容积法及时域有限元进行计算。对某个高温燃气配气阀的温度场及热变形进行了数值计算,得到与实测相符的结果。     

Two-dimensional Hertzian contact problem with surface tension

In the present paper, we consider a two-dimensional contact problem of a rigid cylinder indenting on an elastic half space with surface tension. Based on the solution of a point force acting on a substrate with surface tension, we derive the singular integral equation of this problem. By using the Guass–Chebyshev quadrature formula, the integral equation is solved numerically to illuminate the influence of surface tension on the contact response. It is found that when the contact width is comparable with the ratio of surface tension to elastic modulus, surface tension significantly alters the pressure distribution in the contact region and the contact width. Compared to that of the classical Hertzian contact, the existence of surface tension decreases the displacements on the half plane and yields a continuous slope of normal stress and displacements across the contact fringe. In addition, it predicts the increase of hardness as the radius of indent cylinder decreasing. The obtained results are useful for the measurement of mechanical properties of materials based on the indentation technique.     

基于SPH方法的表面张力修正算法及其应用

针对传统SPEI方法中基于CSF模型的表面张力算法,在计算边界、尖角等粒子缺失部位的曲率时存在偏差较大,且粒子秩序较差,对大变形问题表面张力计算精度较低的问题,在Morris提出的表面张力SPH方法基础上,通过引入CSPM方法对边界法向的计算和曲率的计算进行修正,得到了表面张力修正方程组.应用本文方法模拟了水溶液中初始...     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

A regularised one-dimensional drop formation and coalescence model using a total variation diminishing (TVD) scheme on a single Eulerian grid

The breakup of an axisymmetric viscous jet is considered in the lubrication approximation. The discretised equations are solved on a fixed equidistant one-dimensional Eulerian grid. The governing equations are implemented in a conservative second order accurate total variation diminishing (TVD) scheme, preventing the numerical diffusivity. Singularities that occur at pinchoff and coalescence are regularised by a small modification on the surface tension. The modification is of the order of the spatial step Δx. This regularisation ensures that the solution of the presented numerical model converges to the exact solution of the breakup of a jet in the lubrication approximation. The results of the presented numerical model agree quantitatively with the analytical solution of the Rayleigh–Plateau instability, and with experimental results on the final stage of the Rayleigh–Plateau instability.