Numerical simulation of contraction and expansion flows of Langmuir monolayers

Langmuir monolayers consist of amphiphilic molecules at the air–water interface and can be modeled as two-dimensional fluids. Earlier experiments [D.J. Olson, G.G. Fuller, J. Non-Newtonian Fluid Mech. 89 (2000) 187–207] on 4:1 contraction and 4:1 expansion flows have been simulated using an integral constitutive equation of the K-BKZ type, suitably modified to account for strain-thickening in the planar extensional viscosity. The model has been used to fit linear viscoelastic data (G′ and G″) and the shear viscosity (η_S), while the amount of strain-hardening is assumed, due to lack of experimental data. The simulations are in good agreement with the experiments on Newtonian monolayers, which show no vortices in the contraction but large inertial vortices in the expansion. For the viscoelastic monolayer (a poly-octadecyl methacrylate or PODMA), the opposite is true. The contraction flow shows vortices, while in the expansion flow the vortex activity is substantially reduced compared with the Newtonian one. The viscoelastic behavior is well captured by the model, provided that substantial strain-thickening is exhibited by the monolayer in planar extension. The latter behavior is very much like that for a branched LDPE melt, which also shows big vortices due to strain-hardening in planar as well as in uniaxial extension.     

Effects of the side walls on the unsteady flow of a second-grade fluid in a duct of uniform cross-section

In this paper, the effects of the side walls on the unsteady flow of a second-grade fluid in a duct of rectangular cross-section are considered. Two types of unsteady flows are investigated. One of them is the unsteady flow in a duct of rectangular cross-section moving parallel to its length and the other is the unsteady flow due to an applied pressure gradient in a duct of rectangular cross-section whose sides are at rest. It is shown that a Newtonian fluid reaches steady-state earlier than a second-grade fluid and the effect of the side walls on a second-grade fluid is more effective than that on a Newtonian fluid.     

Velocity measurement of start-up flow of polymeric solutions in a planar contraction channel

The start-up flow of polymeric solutions in an abrupt contraction channel with a rectangular cross-section was experimentally studied. Aqueous solutions with 0.2 wt% and 1.0 wt% of polyacrylamide were used as test fluids. Temporal changes in velocity were measured with a laser Doppler velocimeter. The velocity overshot just after the onset of the flow. The changes in velocity caused by the rearrangement of the velocity distribution were observed. A three-dimensional flow structure and the development of a vortex region were found near the entrance to the contraction. A decrease in the axial velocity just upstream from the contraction was observed. The velocity profile is related to the three-dimensional nature of the flow and the elongational rheological properties of the test fluid. The experimental results indicate that three-dimensional analysis of viscoelastic flows is required especially for the flow in rectangular channels. Received: 6 July 1998 Accepted: 1 December 1998     

Effects of the side walls on starting flows in ducts

This article considers the effects of the side walls on the unsteady flow of an incompressible viscous fluid in a duct of uniform cross-section. In order to show the effects of the side walls, three illustrative examples are given. They are: the starting flow in a duct of semicircular cross-section, the starting flow in a duct of rectangular cross-section and the starting flow in a duct of circular cross-section. The velocity distributions and the volume fluxes obtained for these flows are compared and it is shown that the flow in a duct of semicircular cross-section reaches steady state earlier than those for the flow in a duct of circular cross-section and for the flow in a duct of square cross-section. It is found that there are remarkable effects of the side walls of a duct on the required time to attain the asymptotic values of flow properties.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer.We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids.In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.     

Numerical simulation of entry flow of the IUPAC-LDPE melt

Numerical simulations have been undertaken for the creeping entry flow of a well-characterized polymer melt (IUPAC-LDPE) in a 4:1 axisymmetric and a 14:1 planar contraction. The fluid has been modeled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times (Papanastasiou–Scriven–Macosko or PSM model). Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear and elongational data from uniaxial elongation experiments. The numerical solutions show that in the axisymmetric contraction the vortex in the reservoir first increases with increasing flow rate (or apparent shear rate), goes through a maximum and then decreases following the behavior of the uniaxial elongational viscosity. For the planar contraction, the vortex diminishes monotonically with increasing flow rate following the planar extensional viscosity. This kinematic behavior is not in agreement with recent experiments. The PSM strain-memory function of the model is then modified to account for strain-hardening in planar extension. Then the vortex pattern shows an increase in both axisymmetric and planar flows. The results for planar flow are compared with recent experiments showing the correct trend.     

Exact solutions for the unsteady flow of a Burger’s fluid in a duct induced by time-dependent prescribed volume flow rate

In this investigation, some unsteady flows in a circular duct have been studied. The fluid obeys viscoelastic non-Newtonian model with the Burgers’ constitutive equation and all fluid properties are constant. The flows in a duct are due to the prescribed arbitrary time dependent inlet volume flow rates. Four types of flow situations are considered. The governing equations are first developed and then solved using Laplace transform technique. Results indicate the strong effect of Burgers’ fluid parameter on the velocity fields and pressure gradients.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

Influence of elongational properties on the contraction flow of polyisobutylene in a mixed solvent

Over the last decade several international programmes have been developed around different standard fluids, one of which is the so-called S1 fluid. This is a solution of polyisobutylene in a mixed solvent and the aim of the programme has been to study the rheology of polymer solutions from the dilute solution to the melt. The focus of this paper will be on the flow visualisation of contraction flows of S1 through orifice dies and on the estimation of some of its extensional properties. The contraction ratios range from 24.4:1 and 124.3:1. The measured entry pressure drops will be correlated with contraction ratio and apparent wall shear rate. Experimental evidence will show that, when regarded as a function of wall shear rate, the entry pressure drops are independent of the contraction ratios. The flow fields for different contraction ratios, at any constant apparent wall shear rate, however, differ substantially. The evolution of the flow fields is monitored and it is shown that an initial increase in vortex size is followed by a slower decrease, this happening at a constant Weissenberg number. At the same Weissenberg number, small scale instabilities start occurring near the office. As the shear rate is increased further, these instabilities grow in size until, eventually, the flow structure is destroyed. An extensional viscosity is evaluated using a modified form of the Binding analysis for contraction flows and we show that the results are not only in qualitative agreement with those from other groups, but also that the analysis is able to predict exactly the onset of the aforementioned flow instabilities. Received: 20 March 1997 Accepted: 18 September 1997     

粘弹流体轴对称流动的格子Boltzmann方法

基于BGK碰撞模型，通过在迁移方程中引入作用力项，建立了粘弹流体的轴对称格子Boltzmann模型．通过Chapman-Enskog展开，获得了准确的柱坐标下轴对称宏观流动方程．采用双分布函数对运动方程和本构方程进行迭代求解，模拟分析了粘弹流体管道流动，获得了流场中的速度和构型张量的分布，通过与解析解进行比较，验证了模型的准确性．研究了作为粘弹流体流动基准问题的收敛流动，对涡旋位置进行了定量分析，将回转长度的计算结果与有限体积法进行了比较，两种数值结果十分吻合．研究结果表明，模型能够准确表征粘弹流体的轴对称流动，具有较广阔的应用前景．     

Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

Exact solutions of starting flows for second grade fluid in a porous medium

This paper concentrates on the unsteady flows of a magnetohydrodynamic (MHD) second grade fluid filling a porous medium. The flow modeling involves modified Darcy's law. Three problems are considered. They are (i) starting flow due to an oscillating edge, (ii) starting flow in a duct of rectangular cross-section oscillating parallel to its length, and (iii) starting flow due to an oscillating pressure gradient. Analytical expressions of velocity field and corresponding tangential stresses are developed. These expressions are found to be significantly affected by the applied magnetic field, permeability of the porous medium and the material parameter of the fluid. Moreover, the influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn. Finally, a comparison is also made with the existing results in the literature.     

PIV measurements of flow past a confined cylinder

This study investigates the flow past a confined circular cylinder built into a narrow rectangular duct with a Reynolds number range of 1,500 ≤ Re _d ≤ 6,150, by employing the particle image velocimetry technique. In order to better explain the 3-D flow behaviour in the juncture regions of the lower and upper plates and the cylinder, respectively, as well as the dynamics of the horseshoe vortex system, both time-averaged and instantaneous flow data are presented for regions upstream and downstream of the cylinder. The size, intensity and interaction of the vortex systems vary substantially with the Reynolds number. Although the narrow rectangular duct with a single built-in cylinder is a geometrically symmetrical arrrangement, instantaneous flow data have revealed that the flow structures in both the lower and upper plate–cylinder junction regions are not symmetrical with respect to the centreline of the flow passage. The vortical flow structures obtained in side-view planes become dominant sometimes in the lower juncture region and sometimes in the upper juncture region in unsteady mode.     

Frottements et pertes de pression des fluides non newtoniens dans des conduites non circulaires

The study of fluid flow in a duct requires characteristic parameters of the flow and dimensionless numbers to correlate and compare experimental results. For Newtonian fluids in simple configurations, the definition of the Reynolds number is quite standard, but for non-Newtonian fluid flows in ducts with arbitrary shape of cross section, the dependence of the apparent viscosity with the shear rate requires a generalization of this dimensionless number. This note proposes a general method valid for a large class of non-Newtonian fluids and for all duct shapes. An application is developed for a viscoelastic flow through a rectangular duct. Results obtained in the present investigation are in a good agreement with available correlations. To cite this article: M. Mahfoud et al., C. R. Mecanique 333 (2005).     

Dynamics of worm-like micelles: the Cox–Merz rule

The viscoelastic behaviour of worm-like micelles in small-amplitude oscillatory, steady simple shear and uniaxial extensional flows are analyzed with a model that couples the Oldroyd-B constitutive equation with a kinetic equation that accounts for the structural changes induced by the flow. In some cases, the constitutive equation predicts a viscoelastic behaviour that is consistent with the Cox–Merz rule. Departures from this rule are also predicted. Experimental data obtained for two worm-like micellar systems indicate that in these solutions, the Cox–Merz rule is not usually followed, in agreement with the predictions of our model. In uniaxial extensional flow, the model predicts a strain hardening in the extensional viscosity at low extensional rates and a strain-thinning at high extensional rates.     

Numerical simulation for rotating internal weakly viscoelastic flows in rectangular ducts

The present work develops a numerical method for the solution of rotating internal weakly viscoelastic flows in rectangular ducts for dimensionless parameters such as the Reynolds, Rossby and Weissenberg numbers, taken respectively in the intervals between 171 and 12000, 0.047 and 1/12 and up to 1/10000. It is shown that the usual counter‐rotating double‐vortex configuration of secondary flow breaks down with the increase of the Reynolds number (over the threshold of 171). For higher Reynolds numbers such as 7500 and 12000 the secondary flow diffuses to the interior of the duct where it assumes a fully developed configuration and the transition to the turbulence structure is observed. The Sobolev norms increase almost proportionally to the increase of the Reynolds number, and play an essential role for more complex problems involving transition to turbulence modelling. Copyright © 2002 John Wiley & Sons, Ltd.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

A potential-flow,deformable-body model for fluid–structure interactions with compact vorticity: application to animal swimming measurements

This paper presents an approach to quantify the unsteady fluid forces, moments and mass transport generated by swimming animals, based on measurements of the surrounding flow field. These goals are accomplished within a framework that is independent of the vorticity field, making it unnecessary to directly resolve boundary layers on the animal, body–vortex interactions, or interactions among vortex lines in the wake. Instead, the method identifies Lagrangian coherent structures in the flow, whose dynamics in flows with compact vorticity are shown to be well approximated by potential flow concepts, especially the Kirchhoff and deformation potentials from deformable body theory. Examples of the application of these methods are given for pectoral fin locomotion of the bluegill sunfish and undulatory swimming of jellyfish, and the methods are validated by analysis of a canonical starting vortex ring flow. The transition to a Lagrangian approach toward animal swimming measurements suggests the possibility of implementing recently developed particle tracking (vis-à-vis DPIV) techniques for fully three-dimensional measurements of animal swimming.     

黏弹性流体扩展及收缩流动的格子Boltzmann模拟分析

本文应用格子Boltzmann 方法(LBM)并结合Oldroyd-B 模型,讨论了不可压缩的 Navier-Stokes 方程和平流扩散本构方程的解耦及各自求解方法,以及两类问题的边界处理格式,实现了黏弹性流体在二维1:3 扩展流道以及3:1 收缩流道中的流动的数值模拟．获得了不同雷诺数Re 和维森伯格数Wi 以及黏度vs 下流动的流线分布,计算给出了漩涡的涡心位置和大小,并分析了参数Re、Wi 和vs 对流动特点的影响．模拟结果表明本文所采用模型和边界处理方法具有良好的精度和稳定性．