Influence of wall slip on extrudate swell: a boundary element investigation

This paper reports a numerical study of the extrudate swell by a transient Boundary Element Method (BEM). Furthermore the fluid, which is modeled by a differential constitutive equation, is allowed to slip at the wall, where the slip velocity is a prescribed function of the wall shear stress. This function is a curve fit of the extensive data of Ramamurthy on linear low density polyethylene which incorporates two parameters: a critical wall shear stress above which slip occurs, and another parameter which governs the shape of the slip velocity versus shear stress curve. The results show that wall slip reduces both the amount of extrudate swell and the critical Weissenberg number above which our numerical scheme no longer converges.     

Numerical simulation of the extrusion of strongly compressible Newtonian liquids

The axisymmetric and plane extrusion flows of a liquid foam are simulated assuming that the foam is a homogeneous compressible Newtonian fluid that slips along the walls. Compressibility effects are investigated using both a linear and an exponential equation of state. The numerical results confirm previous reports that the swelling of the extrudate decreases initially as the compressibility of the fluid is increased and then increases considerably. The latter increase is sharper in the case of the exponential equation of state. In the case of non-zero inertia, high compressibility was found to lead to a contraction of the extrudate after the initial expansion, similar to that observed experimentally with liquid foams and to decaying oscillations of the extrudate surface. The time-dependent calculations show that the oscillatory steady-state solutions are stable. These steady-state oscillatory solutions are not affected by the length of the extrudate region nor by the boundary condition along the wall.     

A simple model to predict extrudate swell of polystyrene and linear polyethylenes

In a previous paper (Guillet and Seriai, 1991) we derived a simple analytical expression which allows the prediction of extrudate swell of polystyrene in a wide range of residence time. This was done using the rubber-like elasticity theory and calculation of the elongational strain recovery of a Lodge fluid. The theoretical extrudate swell ratio mainly depends on the relaxation modulus, the extension ratio and the recoverable shear strain. The main advantage of this model is to provide good accuracy with short dies in a wide range of shear rate. In this paper, we examine the validity of the proposed equation with different contraction ratios at the die entrance (ECR) and its ability to predict extrudate swell of other commodity polymers such as polyethylenes.     

Cessation of Newtonian circular and plane Couette flows with wall slip and non-zero slip yield stress

We solve analytically the cessation flows of a Newtonian fluid in circular and plane Couette geometries assuming that wall slip occurs provided that the wall shear stress exceeds a critical threshold, the slip yield stress. In steady-state, slip occurs only beyond a critical value of the angular velocity of the rotating inner cylinder in circular Couette flow or of the speed of the moving upper plate in plane Couette flow. Hence, in cessation, the classical no-slip solution holds if the corresponding wall speed is below the critical value. Otherwise, slip occurs only initially along both walls. Beyond a first critical time, slip along the fixed wall ceases, and beyond a second critical time slip ceases also along the initially moving wall. Beyond this second critical time no slip is observed and the decay of the velocity is faster. The velocity decays exponentially in all regimes and the decay is reduced with slip. The effects of slip and the slip yield stress are discussed.     

Laminar,transitional and turbulent annular flow of drag-reducing polymer solutions

Mean and rms axial velocity-profile data obtained using laser Doppler anemometry are presented together with pressure-drop data for the flow through a concentric annulus (radius ratio κ = 0.506) of a Newtonian (a glycerine–water mixture) and non-Newtonian fluids—a semi-rigid shear-thinning polymer (a xanthan gum) and a polymer known to exhibit a yield stress (carbopol). A wider range of Reynolds numbers for the transitional flow regime is observed for the more shear-thinning fluids. In marked contrast to the Newtonian fluid, the higher shear stress on the inner wall compared to the outer wall does not lead to earlier transition for the non-Newtonian fluids where more turbulent activity is observed in the outer wall region. The mean axial velocity profiles show a slight shift (～5%) of the location of the maximum velocity towards the outer pipe wall within the transitional regime only for the Newtonian fluid.     

Annular extrudate swell of pseudoplastic and viscoplastic fluids

Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The finite element method (FEM) is used to provide numerical results for different inner/outer diameter ratios κ under steady-state conditions. The Herschel-Bulkley model of viscoplasticity is used with the Papanastasiou regularization, which reduces with appropriate parameter choices to the Bingham–Papanastasiou, power-law and Newtonian models. The present results provide the shape of the extrudate, and in particular the thickness and diameter swells, as a function of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress (in the case of viscoplasticity). The pressures from the simulations have been used to compute the excess pressure losses in the system (exit correction). While shear-thinning leads to reduced swelling relative to the Newtonian values for all κ-values, the opposite is true for shear-thickening fluids, which exhibit considerable swelling. Viscoplasticity leads to decreased extrudate swell as the dimensionless yield stress goes from zero (Newtonian behaviour) to an asymptotic value of fully plastic behaviour. The exit correction decreases to zero with a decrease in the power-law index to zero and an increase in the dimensionless yield stress to its asymptotic limit. However, the decrease is not monotonic: for power-law fluids it has maxima in the range of power-law indices between 0.8 and 0.6, while for viscoplastic fluids it has maxima around Bingham number values of 5.     

Three-dimensional streamlined finite elements: Design of extrusion dies

A method to determine three-dimensional die shapes from extrudate swell and vice versa is presented using a three-dimensional Galerkin finite element method based on a streamlined formulation with the fluid velocities and pressures represented by triquadratic and trilinear basis functions respectively. The three-dimensional streamlined method, an extension of the two-dimensional formulation, uses successive streamsurfaces to form a boundary-conforming co-ordinate system. This produces a fixd, computational domain leaving the spatial location of the elements as unknowns to be determined with the standard primary variables (u, v, w, p). The extrudate produced by a die of a given shape is considered for moderate Reynolds numbers. Finally, the method is extended to address the problem of die design, where a die profile is sought to produce a target extrudate shape.     

Slip yield stress effects in start-up Newtonian Poiseuille flows

Analytical solutions are derived for various start-up Newtonian Poiseuille flows assuming that slip at the wall occurs when the wall shear stress exceeds a critical value, known as the slip yield stress. Two distinct regimes characterise the steady axisymmetric and planar flows, which are defined by a critical value of the pressure gradient. If the imposed pressure gradient is below this critical value, the classical no-slip, start-up solution holds. Otherwise, no-slip flow occurs only initially, for a finite time interval determined by a critical time, after which slip does occur. For the annular case, there is an additional intermediate (steady) flow regime where slip occurs only at the inner wall, and hence, there exist two critical values of the pressure gradient. If the applied pressure gradient exceeds both critical values, the velocity evolves initially with no-slip at both walls up to the first critical time, then with slip only along the inner wall up to the second critical time and finally with slip at both walls.     

Extrusion of a compressible Newtonian fluid with periodic inflow and slip at the wall

We explore a mechanism of extrusion instability, based on the combination of nonlinear slip and compressibility. We consider the time-dependent compressible Newtonian extrudate swell problem with slip at the wall. Steady-state solutions are unstable in regimes where the shear stress is a decreasing function of the velocity at the wall. Compressibility provides the means for the alternate storage and release of elastic energy, and, consequently, gives rise to periodic solutions. The added novelty in the present work is the assumption of periodic volumetric flow rate at the inlet of the die. This leads to more involved periodic responses and to free surface oscillations similar to those observed experimentally with the stick-slip instability. To numerically simulate the flow, we use finite elements in space and a fully-implicit scheme in time.Dedicated to the memory of Prof. Tasos Papanastasiou     

Newtonian flow in a triangular duct with slip at the wall

We consider the Newtonian Poiseuille flow in a tube whose cross-section is an equilateral triangle. It is assumed that boundary slip occurs only above a critical value of the wall shear stress, namely the slip yield stress. It turns out that there are three flow regimes defined by two critical values of the pressure gradient. Below the first critical value, the fluid sticks everywhere and the classical no-slip solution is recovered. In an intermediate regime the fluid slips only around the middle of each boundary side and the flow problem is not amenable to analytical solution. Above the second critical pressure gradient non-uniform slip occurs everywhere at the wall. An analytical solution is derived for this case and the results are discussed.     

Polypropylene-polyethylene blends

Summary Die swell behaviour and morphology of melt blends of isotactic polypropylene (PP) and high density polyethylene for pure polymers and blends with 25, 50 and 75 weight % PP are described in the present study. A light interference contrast microscopy technique was used for the morphological characterization of melt blends and extrudate samples of the blends obtained with an Instron capillary rheometer. The results indicate that the domains from blends where the dispersed phase has higher viscosity than the continuous phase remain as continuous domains in the extrudate whereas domain destruction takes place when blends where the continuous phase has the higher viscosity are extruded.The die swell behaviour as well as the fiber forming properties of extrudates of melts having unstable domains extruded at high shear stresses resemble the behaviour of homopolymers, whereas samples with stable domains are significantly different, die swell increases with temperature at constant shear stress and stable fibers cannot be obtained after necking.With 10 figures and 1 table     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Extrudate swell of Boger fluids

Boger fluids are dilute polymer solutions exhibiting high elasticity at low apparent shear rates, which leads to high extrudate swell. Numerical simulations have been undertaken for the flow of three Boger fluids (including benchmark Fluid M1), obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of dilute polymer solutions. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries without or with gravity. The results provide the extrudate swell and the excess pressure losses (exit correction), as well as the shape and extent of the free surface. All these quantities increase rapidly and monotonically with increasing elasticity level measured by the stress ratio, S_R. It was found that the main reason for the high extrudate swelling is high normal stresses exhibited in shear flow (namely, the first normal-stress difference, N₁). Surprisingly, the elongational parameter of the model or a second normal-stress difference N₂ do not affect the results appreciably. Gravity serves to lower the swelling considerably, and makes the simulations easier and in overall agreement with previous experiments.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

聚合物熔体三维挤出胀大的数值模拟

采用有限元方法分析K-BKZ本构方程描述的聚合物熔体的三维挤出胀大.对于本构方程中偏应力张量的计算,首先给出质点的运动轨迹,分段求出局部的变形梯度张量,再求出整体的变形梯度、Cauchy-Green应变张量和 Finger应变张量,沿轨迹采用分段高斯积分计算应力.把应力作为方程的右端项,给出迭代方法,求解非线性方程组.并根据自由面处的边界条件,迭代得出出口处自由面的最终位置.对轴对称流道和矩形流道进行分析计算,并把结果与二维分析和实验结果进行了比较,显示方法是可行的.     

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.     

黏弹流体挤出胀大的数值模拟研究进展

主要介绍了黏弹流体挤出胀大的数值模拟研究进展.给出了黏弹流体挤出胀大的数学模型,回顾了近20多年以来挤出胀大的主要数值模拟研究工作,然后对主要模拟方法的计算过程、方法特点和形成的结果进行了一定的总结.最后提出了作者对挤出胀大研究的一些看法,包括目前研究中存在的问题和相关研究的发展趋势.      

Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

滑滚表面间的流体膜壁面滑移和流体动力学

经典雷诺润滑理论建立在无壁面滑移的假设基础之上。近年来许多试验报告了发生在流体膜流动的壁面滑移证据。本文研究了两固体表面间的流体膜流动特性和流体动力学,发现壁面滑移显著影响膜的流体动力学问题,流体动压力不仅受黏度和几何间隙的影响,而且还由壁面滑移和表面运动强力控制,通过控制表面的吸附性质,甚至可以得到零摩擦表面。另一方面,如果两个表面具有相同的滑移特性,存在一个临界滑动速度使得流体动压效应完全消失;但是在纯滚动条件下,即使界面极限剪应力很小,仍然有相当可观的流体动压效应。     

A comparative study of the thermotropic mesomorphic tendencies and rheological characteristics of three cellulose derivates: Ethylene and propylene oxide ethers and an acetate butyrate ester

Samples of cellulose acetate butyrate (CAB) hydroxypropyl cellulose (HPC) and ethyl cellulose (EC) are contrasted with commercial (atactic) polystyrene (PS) and isotactic polypropylene (PP) in studies of (i) differential scanning calorimetry, (ii) quiescent polarized light microscopy (iii) optical retardation variation following an imposed stress field. It is concluced that HPC and EC are thermotropic liquid crystals, while CAB behaves in a manner similar to a vitrifying isotropic melt such as PS. Studies of the shear viscosity and dynamic viscosity indicate HPC and EC exhibit yield values while CAB shows a zero shear viscosity and Vinogradov-Malkin reduced viscosity curve identical to PS. The normal stress and extrudate swell behavior of CAB are also similar to PS. The HPC and EC exhibit substantially reduced extrudate swell. Measurement of the principal normal stress difference behavior of the HPC melt is troubled by the existence of yield values.