Rheological predictions of a transversely isotropic fluid model with extensible microstructure

A continuum extensible director theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike semiflexible liquid crystalline polymers. The model is strictly restricted to material that flow-align in shear, and that, in the absence of flow, are sufficiently far from the nematic-isotropic phase transition. The microstructure of the continuum is described by a variable length director, but the extensibility is finite. The model is an extension of the TIF (Transversely Isotropic Fluid) model of Ericksen (1960). The thermodynamic restrictions on the model parameters are found using the non-negative definiteness of the entropy production. The rheological material functions predicted by the model are calculated for steady simple shear and steady uniaxial extensional flows. In the rigid rod limit the model predictions agree with those of the TIF model, and for the finite extensibility case the model predictions are in agreement with those associated with flexible isotropic polymers: strong non-Newtonian shear viscosity, positive first normal stress differences, recoverable shear of order one, negative second normal stress differences, and a maximum in the steady uniaxial extensional viscosity.     

Lattice modelling of nematodynamics

The microstructures of textured nematics under shear are investigated by means of a director lattice model incorporating linear shear response as well as elastic interactions between neighbouring directors. The model can be understood as a lattice implementation of the so-called nematodynamics equation for a constant uniaxial order parameter. The dimensionless number governing the model is found to be a mesoscale Ericksen number, which scales with the square of the lattice cell size. It is shown that the predicted microstructure depends strongly on the scale of that number. In particular, disclination loops are found to grow for a range of mesoscale Ericksen numbers, while below or above that they disappear. We apply the model to investigate the director profiles of tumbling nematics. If the orientations are restricted to lying in the vorticity plane, we reproduce the director wind-up layers and distortion saturation predicted theoretically. In the full three-dimensional case an initially polydomain director field evolves to a vorticity-aligned state up to a critical Ericksen number, above which in-plane orientations with distortion saturation are found. The simulations hence reproduce the transition from log-rolling to flow aligning with increasing shear rate observed experimentally. Received: 9 March 1999 /Accepted: 26 July 1999     

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.     

Linear viscoelastic models: Part IV. From molecular dynamics to temperature and viscoelastic relations using control theory

Viscoelasticity and temperature dependences are explained using molecular dynamics and control theory. We have previously (Borg and Pääkkönen, 2009 [1], [2], [3]) applied control theory to model the relationship between the relaxation modulus, dynamic and shear viscosity, transient flow effects, power law and Cox–Merz rule related to the molecular weight distribution (MWD), and here these topics are discussed more generally. In this paper we show the direct simple relation to molecular dynamics using structural models comprising dumb-bells (Bird et al., 1987 [4]) with internal viscosity and elasticity in a statistical tube. The dumb-bell model is used to obtain the linear relation to the elasticity P′ value of function P′(ω) and the relation to the viscosity P″ value of function P″(ω) from chain friction. The applied principle is also valid for the relaxation modulus or shear viscosity. A new principle is presented for obtaining absolute values such as zero viscosity by modelling, which is first used to obtain absolute values for a target point at a high rate for unentangled chains (since close relaxed states of chain topology are much more complicated). An analytical model for the temperature dependency of viscoelastic flows is presented, which is many times more accurate than WLF or Arrhenius equations. Control theory and variations of tube diameter as a function of temperature gives linear relation between chain dynamics and viscoelastic properties. New compact formulas are presented to simultaneously model different polymer flows and temperatures. We have also found that the MWDs computed from the relaxation modulus or complex and the shear viscosity are not temperature sensitive, in contrast to what time–temperature superposition (TTS) suggests, although absolute viscoelastic values make them appear very temperature-dependent. TTS is verified for thermorheologically simple materials, and the reasons for it not holding are explained.     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

Some superposition theorems for second-order fluids

Theorems are derived, within the framework of the second-order Rivlin-Ericksen theory, relating to the superposition of longitudinal flows on plane flows in an incompressible isotropic viscoelastic fluid. Also, some theorems are obtained concerning the effects of inertia on the solution of plane flow problems in such viscoelastic fluids.     

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．     

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.     

粘弹性Timoshenko梁的变分原理和静动力学行为分析

从线性，各向同性，均匀粘弹性材料的Boltzmann本构定律出发，通过Laplace变换及其反变换，由三维积分型本构关系给出了Timoshenko梁的本构关系，并由此建立了小挠度情况下粘弹性Timoshenko梁的静动力学行为分析的数学模型，一个积分－偏微分方程组的初边值问题。同时，采用卷积，建立了相应的简化Gurtin型变分原理。给出了两个算例，考查了梁的厚度h与梁的长度l之比β对梁的力学行为的影响。     

Oscillatory squeezing flow of a biological material

Large-amplitude oscillatory squeezing flow data are reported for a complex biological material, which is highly shear-thinning in oscillatory shear flow. This soft tissue has a linear viscoelastic limit at a strain of approximately 0.2%. The oscillatory squeezing flow data at large strain are analyzed using two constitutive models: a bi-viscosity Newtonian model, and a non-linear Maxwell model. It is found that although both models may have the same response in shape, the later matches with our non-linear experimental data better. It is also concluded that the non-linear response of the material in large amplitude oscillatory flow is mainly due to the shear thinning of the material. Received: 9 February 2000/Accepted: 22 February 2000     

Dynamic rheology of the immiscible blends of liquid crystalline polymers and flexible chain polymers

Immiscible blends containing liquid crystalline polymers (LCP) as dispersed phases show different dynamic rheological properties than those composed of flexible polymers. The widely used Palierne’s model was shown by many authors to be insufficient to describe the frequency dependence of dynamic modulus of such blends. A new model was presented to describe the dynamic rheology of the immiscible blend containing LCP as a dispersed phase. The flexible chain polymer matrix was assumed to be a linear viscoelastic material under small amplitude oscillatory shear flow, and the LCP was assumed to be an Ericksen’s transversely isotropic fluid. The Rapini-Papoular equation of anisotropic interfacial energy was used to account for the effect of nematic orientation on the interfacial tension. It was found that the orientation of the director and the anchoring energy greatly influenced the storage modulus at the “shoulder” regime. The overall dynamic modulus of the blend can be well described by the model with suitable choice of the orientation of the director and anchoring energy of LCP.     

The effect of parallel combined steady and oscillatory shear flows on blood and polymer solutions

Human blood at physiological volume concentration exhibits non-Newtonian and thixotropic properties. The blood flow in the microcirculation is pulsatile, initiated from the heart pulse and can be considered as superposition of two partial flows: a) a steady shear, and b) an oscillatory shear. Until now steady and viscoelastic behavior were separately investigated. Here we present the response to the combination of steady and oscillatory shear for human blood, a high molecular weight aqueous polymer solution (polyacrylamide AP 273E) and an aqueous xanthan gum solution. The polyacrylamide and xanthan solutions are fluids that model the rheological properties of human blood. In general, parameters describing blood viscoelasticity became less pronounced as superimposed steady shear increased, especially at low shear region and by elasticity, associated with reduction in RBC aggregation. The response of polymer solutions to superposition shows qualitative similarities with blood by elasticity, but their quantitative response differed from that of blood. By viscosity another behavior was observed. The superposition effect on viscous component was described by a modified Carreau equation and for the elastic component by an exponential equation.Paper in part presented at the Symposium on Rheology and Computational Fluid Mechanics dedicated to the memory of Prof. A. C. Papanastasiou, University of Cyprus, Nicosia, July 4–5, 1996     

树脂基复合材料粘弹性损伤本构及其实验测定

粘弹性是复合材料最重要的力学性能之一。基于Ｂｏｌｔｓｍａｎ迭加原理和Ｓｃｈａｐｅｒｙ非线性粘弹性理论，本文提出了一变量分离式粘弹性本构关系，且由Ｋａｃｈａｎｏｖ“连续性”概念，将损伤变量引入本构关系式中，按不同的加载速度和加载方向对正交玻璃布－环氧材料进行了加、卸载实验研究。根据Ｌｅｍａｉｔｒｅ和Ｃｈａｂｏｃｈｅ由宏观弹性常数变化确定损伤量的方法，测得材料的损伤并用一二次函数表示。通过对实验数据的一系列处理方法，确定了材料常数，验证了本文理论的有效性。实验表明，当纤维做为主要承力者时，无粘性产生，材料的损伤和总应变相关而与应变率无关，并得到强度值随应变率增加而升高，极限应变不随应变率变化的结论。     

Brownian dynamics simulations for suspension of ellipsoids in liquid crystalline phase under simple shear flows

Brownian dynamics simulations of shear flows are carried out for various suspensions of ellipsoids interacting via the Gay-Berne potential. In this simulation all the systems of the suspension are in a liquid crystalline phase at rest. In a shear flow they exhibit various motions of the director depending on the shear rate: the continuous rotation, the intermittent rotation, the wagging-like oscillation, and the aligning. The director is almost always out of the vorticity plane when it rotates, that is the kayaking. The number density of the system and the inter-particle potential intensity significantly affect the shear rate dependence of orientation. In particular, the continuous rotation of director is maintained to higher shear rates for the system with a stronger potential. Furthermore, the rheological properties are examined. The shear-thinning in viscosity is observed, but the negative first normal difference is not obtained.     

The prediction of time-dependent non-linear stresses in viscoelastic materials: II. Prediction of shear and uniaxial elongation behaviour

Stress predictions have been made for two materials using the new strain measure developed in Part I. For the startup of steady-shear flow, the predictions show increasing overshoot of shear and normal stresses at increasing shear rates, and both qualitatively and quantitatively there is good agreement with the experimental results for a polymer solution and a melt. The information contained in Part I on the strain measure is found to be insufficient for making exact predictions for flows other than shear flows, but it is shown here that uniaxial elongational flow is sufficiently similar to shear flow for predictions to be made with only a low degree of inaccuracy. Data on the elongational behaviour of the melt are available from the literature and the predictions made here are found to be in satisfactory agreement with the reported behaviour. All of the predictions require as input information only the linear viscoelastic behaviour of the material.     

Rheology of highly concentrated anionic surfactants

Surfactant solution flow behavior is of great importance to both the chemical and consumer product industries. Most studies on the flow behavior of surfactant solutions, however, have focused on the dilute regime. Seldom reported is rheology in the highly concentrated regime where typically these surfactants are processed and delivered. First, we present here the phase diagram for the ternary system: water and two anionic surfactants (sodium salt of lauric and oleic acid) at different temperatures. Then, we present both linear viscoelastic and steady shear flow results in the high (70 to 90%) surfactant regime. We find that high values of the shear modulus are directly dependent on the quantity of surfactant crystals and that the formation of a lamellar liquid crystal phase at 45°C affects both modulus and flow of the system. Lamellar crystals create a stiff network resulting in wall slip at large shear strain. Using serrated plates removes slip at the wall and we find a shear rate where microfractures localize in a preferential plane and the material flows. This behavior is reversible.     

Effect of flow history on the structure of a non-polar polymer/clay nanocomposite model system

The effect of flow history on the linear and non-linear viscoelastic properties of non-polar polymer nanocomposites (PNCs) has been investigated by means of a suitable model system based on a Newtonian matrix. The structural recovery of this model suspension after cessation of different pre-shear rates was monitored by measuring its linear viscoelastic properties while its structural evolution under shear flow was followed by using stepwise changes in shear rate including flow reversal measurements. To assess the kinetics of the structural evolution at rest and under flow, empirical relations of stretched exponential form were used. It is shown that for different pre-shear rates, different equilibrium structures were reached at rest but with a similar kinetics of recovery. As a result, the low frequency behaviour was typical of solid-like or weak gel material, strongly dependent on the flow history. After any given shear rate under steady state, only one reversible equilibrium structure was reached after a kinetics that was dependent on the pre-shear history. Finally, typical flow reversal responses as observed for PNCs are reported and interpreted in light of the microstructure evolution under flow. This paper was presented at the Annual Meeting of the European Society of Rheology, Hersonisos, Greece April 2006.     

The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 10³–10⁶). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow. Received: 19 January 1999 Accepted: 30 June 1999     

Thermo-rheological behavior of model protein–polysaccharide mixtures

The use of mixtures of pea protein isolate (PPI) and κ-carrageenan (κ-C) is increasing rapidly with the aim of increasing the stability and viscosity of food products. Recent works have studied their textural and thermal properties but few have studied the influence of the temperature and concentration on the rheological behavior of model systems. In the present work, we study the thermo-rheological properties in the linear and non-linear viscoelastic regimes, in both shear and extensional flows, of mixtures of PPI and κ-C with the aim of obtaining a model for the temperature-induced gelation of complex mixtures of globular vegetable proteins under linear and non-linear deformations. We analyzed the influence of temperature and protein-to-polysaccharide ratios and showed that there are strong changes in the mechanical properties. In shear flows, small-amplitude oscillatory shear was used to study the linear regime and large-amplitude oscillatory shear was used for the non-linear regime. In extensional flows, studies were carried out via the analysis of the dynamics of capillary thinning and breakup process in a filament-thinning rheometer.