Effect of constitutive models on the numerical simulation of viscoelastic flow at an entry region

The poor solutions of numerical simulations at a contraction has often been attributed to the numerical scheme used. However, the solution of a numerical simulation is also highly dependent on the constitutive equation. A study of various constitutive equations on the simulation of the 4:1 contraction flow is made in this paper, and their effects on the results analyzed. The constitutive models considered include the Upper Convected Maxwell model, the Oldroyd-B model, the White-Metzner model, the Phan-Thien-Tanner model and the Giesekus-Leonov model. It was found that although the Phan-Thien-Tanner model gave the best results, the solution at higher shear rates were still not satisfactory.     

注塑制品内应力计算的非线性黏弹性本构方程

在聚合物黏弹性理论的基础上,构建了新的注塑制品内应力计算的四元件串联力学模型,并推导了其瞬态黏弹性响应的非线性本构方程,给出了聚合物材料参数弹性模量和黏壶系数的计算公式,并对PS平板注塑制件脱模前的内应力进行了模拟计算。计算结果与固体高聚物的结构和力学性能的相关研究结论相一致,所建计算模型合理可靠。     

Finite element modeling of the flow of a rubber compound through an axisymmetric die using the CEF viscoelastic constitutive equation

This work is devoted to the simulation of the flow of a high viscosity NR/SBR rubber compound through the die of a single screw extruder with axisymmetric geometry. An in-house developed computer code based on the use of continuous penalty finite element method was employed. Three constitutive equations including two generalized Newtonian models namely; power-law and Carreau and an explicit viscoelastic model named CEF (Criminale-Ericksen-Fillbey) were used to reflect the rheological behavior of the material. Using the parameters of the rheological models determined by a slit die rheometry technique, the flow of the compound was simulated through the die and results were compared with experimentally measured mass flow rates. It is shown that for high viscosity rubber compounds the use of generalized Newtonian models which do not take the normal stress in simple shear flow into consideration gives rise to significant errors in prediction of mass flow rates. On the other hand, comparing the simulations results using the CEF equation with experimental data revealed that this model is the best compromise between generalized Newtonian and full viscoelastic models which need high computational costs and effort. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

An analysis of unsteady one-dimensional stretching of a viscoelastic fluid

The time-dependent response of a viscoelastic liquid to unsteady one-dimensional stretching deformations was examined. Oldroyd's three-constant model for a viscoelastic fluid was used. Two cases representing two different stretching histories were analyzed: a sine stretching pulse and a step stretching pulse. The results show that high elongational viscosity may be easily reached in both cases. As the relaxation time of the liquid becomes comparable to the pulse width, elongational viscosity increases with the increase in maximum stretching rates. Conditions to maintain high levels of elongational viscosity at a subsequently reduced stretching rate were given as functions of the relaxation time and initial stretching rates. In view of recent turbulent boundary layer data, the results were used to discuss possible explanations of turbulent drag reduction in polymer solutions. It was concluded that the basic mechanisms for drag reduction in polymer soluations. It was concluded that the basic mechanisms for drag reducation may be related to the effects of high elongational viscosity and local stabilization of small shear disturbances.     

An integral constitutive equation for nonlinear plasto-viscoelastic behavior of high-density polyethylene

In the present paper an effort is made to model the time-dependent behavior of high-density polyethylene (HDPE) with a one-dimensional integral representation. Owing to the plasto-viscoelastic behavior of the material, we assume that the total strain can be decomposed into a recoverable viscoelastic strain and an irrecoverable plastic strain. The viscoelastic deformation is represented by the Schapery thermodynamic theory. The plastic deformation is assumed to be accumulated during the loading history. An effective time concept is introduced for the plastic deformation, so that the response due to complex loading can be accounted for. The present representation gives a very good prediction of the responses of creep and recovery, two-step creep, and constant stress rate loading and unloading. It is also applied successfully to describe the process of preconditioning of semicrystalline polymers.     

Dispersion in gel permeation chromatography. A rapid numerical solution to Tung's integral equation

A rapid iteration method has been developed to correct the molecular weight averages calculated from raw GPC data for dispersion. Though simple in its performance, it covers the general case that the instrumental spreading characteristics (Tung's resolution factor h) depend on the elution volume. Moreover, it is irrelevant whether the calibration curve, being the logarithmic plot of the molecular weight versus the elution volume, is linear or not. The method has been applied to a number of well-characterized polystyrene mixtures and yields molecular weight averages which agree with those predicted theoretically. The effect of asymmetry exerted by the dispersion on both molecular weight averages M?_n and M?_w is also discussed.     

Transient behavior of viscoelastic fluid in an extruder

A simplified model is used for calculating the time-dependent velocity of polymeric fluid in an extruder. The flow properties of the fluid are characterized by a simple constitutive equation based on two parameters: a constant viscosity μ and a constant elasticity modulus G. It was found that the transient velocity fluctuates periodically, and the time t_t needed to restore the steady-state velocity from a disturbance varies with the ratio G/μ and the dimensionless group _ρH²G/μ², where ρ is the density of the fluid and H is the screw depth of the extruder.     

分数导数型粘弹性阻尼器的动力学有限元方程及数值解

采用分数导数Kelvin固体模型建立粘弹性阻尼器在外力作用下的分数阶动力学有限元方程,并利用Newmark数值积分法得到数值解.结果表明,Zhang and Shimizu分数导数数值积分法能够很好地满足精度、收敛性和稳定性等要求,通过减小时间步长能够有效减小因引入Newmark而导致的周期误差,从而提高计算精度.     

High weissenberg number simulation of an annular extrudate swell using the differential type constitutive equation

Annular extrudate swell simulations at high Weissenberg numbers were made using a differential type constitutive equation. The streamline-upwinding method with a sub-element for extra stress components, which is called SU4 × 4, is one of the best mixed finite element methods for computation of viscoelastic flows. Planar and capillary extrudate swell calculations at high Weissenberg numbers (We > 1000) were accomplished by SU4 × 4. However, annular extrudate swell simulations at high We by SU4 × 4 were not successful. The calculated We was less than about 4. A new calculation technique using a Newton-Raphson discretization of the equation of motion was developed. This technique is called a “new under-relaxation method.” The calculated We of annular extrudate swell simulation by the new under-relaxation method with SU4 × 4 was about 6～250 times larger than those by SU4 × 4. Reasonable calculation results were obtained in an annular flow and a capillary extrudate swell by this method, and the reliability and the utility of the new under-relaxation method are shown. It is now possible to consider the swell shapes of annular extrudate under industrially useful conditions. The calculated swelling ratios were also compared with experimental ones.     

黏弹性表面活性剂压裂液的研究与应用进展

黏弹性表面活性剂压裂液是以黏弹性表面活性剂为主剂的清洁压裂液。黏弹性表面活性剂压裂液体系具有破胶后无残渣、携砂性好、滤失控制性能好等特点,但随着对环保问题的日益重视及钻井深度的不断增加,丰富黏弹性表面活性剂压裂液体系迫在眉睫。本文介绍了黏弹性表面活性剂压裂液的发展和应用。根据压裂液配方不同,将其分为常规黏弹性表面活性剂压裂液和非常规新型黏弹性表面活性剂压裂液,总结了不同种类的黏弹性表面活性剂压裂液的组成、耐温耐剪切等性能及应用情况。分析表明,降低成本、研制简单的配制工艺是常规黏弹性表面活性剂压裂液的主要发展方向;在油田进行大规模实际应用及得到更完善的体系是非常规新型黏弹性表面活性剂压裂液的主要发展方向。     

密炼机部分充满流场的二维非稳态数值模拟

以两相流模型为研究对象,利用流体力学软件FLUENT的流体体积函数（VOF）模型和动网格技术,对部分充满密炼机内的丁苯橡胶胶料-空气两相流动进行二维非稳态数值模拟,对胶料空间分布和不同区域的流场进行了详细分析。结果表明,利用VOF模型可以较真实地模拟出胶料自由界面的动态变化,得到胶料-空气两相的瞬时分布;密炼机部分充满时胶料受到转子剪切后在密炼室壁上黏附形成胶料层,转子背后形成空隙,较容易实现物料的传递和交换;适当的填充系数能够使转子棱推进区总是充满物料,且能提供较高的剪切应力使填料聚集体破碎分散;在转子背后形成不被胶料占据的空隙,扰动胶料的层流流动,并促进胶料的运动、反转和交换。     

粘弹性清洁压裂液的性能评价与应用

采用一种长链脂肪酸季铵盐类阳离子表面活性剂配制成粘弹性清洁压裂液,室内性能评价结果表明,该压裂液的最佳使用浓度为1.5%².0%;60℃时静态滤失系数为3.15×10-4m/min1/2;静态悬砂速度为0.25 mm/s,常温下与原油混合可迅速破胶;具有良好的耐温性能(65℃)、稳定性和抗剪切性能,与地层水配伍性良好;岩心伤害率9.9%2.0%;60℃时静态滤失系数为3.15×10-4m/min1/2;静态悬砂速度为0.25 mm/s,常温下与原油混合可迅速破胶;具有良好的耐温性能(65℃)、稳定性和抗剪切性能,与地层水配伍性良好;岩心伤害率9.9%¹0.3%,比瓜胶压裂液下降了65%。在同类井况条件下,粘弹性清洁压裂液和瓜胶压裂液现场应用对比实验结果表明,该清洁压裂液在增油效果上具有一定优势,推广应用前景良好。     

Simulation of Vortex growth in planar entry flow of a viscoelastic fluid

A numerical simulation of entry flow in a slit die has been undertaken for a fluid that is Newtonian in shear but exhibits normal stresses (Boger fluid). Experimentally measured normal stress and viscosity data are included in a simple rheological model. Flow patterns reveal the existence of vortices in the reservoir corners. Vortex size and intensity increase rapidly with elasticity level.     

网状孔板纵向流换热器壳程流体流动及换热的三维数值模拟

利用Fluent软件对网状孔板纵向流换热器壳程流体流动及换热进行了三维数值模拟,得到了壳程流体温度场、速度场及压力场等细观信息。根据模拟的结果,揭示了网状孔板强化壳程流体换热的机理,分析了壳程流体沿轴向流动及换热的性能,总结了近壁区流场及温度场的特点。     

Finite element simulation of three-dimensional viscoelastic planar contraction flow with multi-mode FENE-P constitutive model

Viscoelasticity is a characteristic of many complex fluids like polymer melts, petroleum, blood, etc. The investigation of viscoelastic flow mechanism has practical significance in both scientific and engineering field. Owing to strongly nonlinear, numerical method becomes a practical way to solve viscoelastic flow problem. In the study, the mathematical model of three-dimensional flow of viscoelastic fluids is established. The planar contraction flow as a benchmark problem for the numerical investigation of viscoelastic flow is solved by using the penalty finite element method with a decoupled algorithm. The multi-mode finitely extensible nonlinear elastic dumbbell with a Peterlin closure approximation (FENE-P) constitutive model is used to describe the viscoelastic rheological properties. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids.     

新型静态混合器的三维流场数值模拟

利用专业的计算流体力学(CFD)软件FLUENT的对具有两块稀释孔板的新型静态混合器内的三维可压缩湍流流场进行数值模拟.采用标准,κ-ε双方程湍流模型,得到混合器内流场、湍动能、压降及混合浓度场的变化等.通过计算压力降可满足工艺要求,混合相对均匀.     

Application of a constitutive equation to polymer melts

Yamamoto's integral constitutive equation in which the memory function is dependent on the second invariant of the rate of deformation tensor at past times has been found to be successful in predicting many of the nonlinear viscoelastic functions from the linear viscoelastic data for melts of linear polyethylenes, polypropylenes, and polystryene but not for those of branched polyethylenes with high level of long-chain branching. A specific functional form for the rate-dependent relaxation spectrum is used and is based on the physical meaning resulting from the molecular entanglement theory of Graessley on steady shearing flow. No arbitrary constant is involved in such an interconversion scheme. The data examined are dynamic storage modulus and loss modulus, steady flow viscosity, first normal stress difference, and parallel superimposed small oscillations on steady shear flow. The theory predicts that in such parallel superimposed experiments, storage modulus G′(ω,\documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \dot \gamma } $\end{document}) divided by the square of frequency shows a maximum under finite shear and that G′(ω,\documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \dot \gamma } $\end{document}) would itself become negative at a frequency whose value is about one third the superimposed rate of shear. The experiments are in line with such predictions. Possible reasons for the failure of the theory for branched polyethylenes are considered, and a possible approach is suggested so that the interconversion scheme may be successful for such resins.     

Lateral migration of a rigid sphere in torsional flow of a viscoelastic fluid

A single polystyrene sphere of radius a, between 141 and 275 micron, when in jected into a disk-plate (torsional) flow of polybutene (viscosity 36 poise, M_n ∼680), migrates radially inward at a rate that is dramatically increased by dissolving 1% of a high-molecular-weight polyisobutylene (M_v ∼10⁶) to make the fluid viscoelastic. The torsional flow field was created by rotating a 21-cm-diameter disk at a rate ω of 6–9 rpm with a gap H of 3.7–5.4 mm between this and a ground glass plate with the fluid in the gap. Lateral migration toward a lower shear rate increased with increasing shear rate and with increasing shear rate gradient in the manner predicted by Brunn (1976) or Chan and Leal (1977). Shear rates up to 25 s⁻¹ were investigated. Below a shear rate of 8 s⁻¹ the radial migration velocity is of the form v_r = − L(ωa/H)²r, where L is a positive constant containing the fluid properties, and r is the radial position of the particle.