Rheological material functions of polymer solutions using a generalized Maxwell model with slip functions Part I: Theory

A generalization of the Maxwell model for polymer systems is derived that replaces the velocity gradient in the Eulerian expression for the upper convected derivative by a tensorial kinematic function. Applying the principle of objectivity this tensorial function is reduced to two scalar slip functions. In shear flows, only one of the two occurs. Material functions are calculated in closed form, and asymptotic conditions are formulated that guarantee isotropic behaviour of the material in sudden strains.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada.     

Vibration analysis of foam plates based on cell volume distribution

In this paper, vibration analysis of irregular-closed-cell foam plates is per-formed. A cell volume distribution coefficient is introduced to modify the original Gibson-Ashby equations of effective Young’s modulus of foam materials. A Burr distribution is imported to describe the cell volume distribution situation. Three Burr distribution pa-rameters are obtained and related to the cell volume range and the diversity. Based on the plate theory and the effective modulus theory, the natural frequency of foam plates is calculated with the change of the cell volume distribution parameters. The relationship between the frequencies and the cell volumes are derived. The scale factor of the average cell size is introduced and proved to be an important factor to the performance of the foam plate. The result is shown by the existing theory of size effects. It is determined that the cell volume distribution has an impact on the natural frequency of the plate structure based on the cell volume range, the diversity, and the average size, and the impact can lead to optimization of the synthesis procedure.     

Motion of a bubble in a viscous liquid

The discussion concerns steady-state flow of a viscous fluid around a spherical bubble at small Reynolds number R. Asymptotic matching [1] provides a way of calculating the resistance force, which agrees well with the measured force for R < 5. The rate of growth or dissolution of the bubble is calculated on the assumption that the Péclet number is large.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1971.We are indebted to V. G. Levich for a discussion.     

Characteristics of developing vertical bubbly flow under normal and microgravity conditions

The axial development of the void fraction profile, interfacial area concentration and Sauter mean bubble diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured using stereo image processing under normal and microgravity conditions. The flow measurements were performed at four axial locations (axial distance from the inlet, z normalized by the pipe diameter, D, z/D = 5, 20, 40 and 60) and with various flows: superficial gas velocity of 0.00840-0.0298 m/s, and superficial liquid velocity of 0.138-0.914 m/s. The effect of gravity on radial distribution of bubbles and the axial development of two-phase flow parameters is discussed in detail based on the obtained database and visual observation. Following Serizawa-Kataoka’s phase distribution pattern criteria under normal gravity conditions, the phase distribution pattern map was developed. Similar to normal gravity two-phase flows, wall, core and intermediate void peak patterns are observed under microgravity conditions but a transition void distribution pattern is not observed in the current experimental conditions. The data obtained in the current experiment are expected to contribute to the benchmarking of CFD simulation of phase distribution pattern and interfacial area concentration in forced convective pipe flow under microgravity conditions.     

Bubbling behavior of cohesive particles in a two-dimensional fluidized bed with immersed tubes

Fluidization hydrodynamics are greatly influenced by inter-particle cohesive forces. This paper studies the fluidization of large cohesive particles in a two-dimensional fluidized bed with immersed tubes using “polymer coating” to introduce cohesive force, to gain better understanding of bubbling behavior when particles become cohesive and its effect on chemical processes. The results show that the cohesive force promotes bubble splitting in the tube bank region, thereby causing an increase in the number and a decline in the aspect ratio of the bubbles. As the cohesive force increases within a low level, the bubble number increases and the bubble diameter decreases, while the aspect ratio exhibits different trends at different fluidization gas velocities. The difference in the evolution of bubble size under various cohesive forces mainly takes place in the region without tubes. When the cohesive force is large enough to generate stable agglomerates on the side walls of the bed, the bubble number and the bed expansion sharply decrease. The tubes serve as a framework that promotes the agglomeration, thus accelerating defluidization. Finally, the bubble profile around tubes was studied and found to greatly depend both on the cohesive forces and the location of tubes.     

Size effects in the constrained deformation of metallic foams

The constrained deformation of an aluminium alloy foam sandwiched between steel substrates has been investigated. The sandwich plates are subjected to through-thickness shear and normal loading, and it is found that the face sheets constrain the foam against plastic deformation and result in a size effect: the yield strength increases with diminishing thickness of foam layer. The strain distribution across the foam core has been measured by a visual strain mapping technique, and a boundary layer of reduced straining was observed adjacent to the face sheets. The deformation response of the aluminium foam layer was modelled by the elastic-plastic finite element analysis of regular and irregular two dimensional honeycombs, bonded to rigid face sheets; in the simulations, the rotation of the boundary nodes of the cell-wall beam elements was set to zero to simulate full constraint from the rigid face sheets. It is found that the regular honeycomb under-estimates the size effect whereas the irregular honeycomb provides a faithful representation of both the observed size effect and the observed strain profile through the foam layer. Additionally, a compressible version of the Fleck-Hutchinson strain gradient theory was used to predict the size effect; by identifying the cell edge length as the relevant microstructural length scale the strain gradient model is able to reproduce the observed strain profiles across the layer and the thickness dependence of strength.     

Age dependence of the drag force in an aqueous foam

The drag force on a sphere moving through an aqueous foam is measured as the foam ages. After an initial period, the steady-state drag decreases with age T as T ^{−0.54±0.14}. As the mean bubble size R in the foam coarsens as T ^0.5, this implies that the drag force scales as The transient buildup of the force when the sphere starts to move is described by a single exponential approach to the steady-state drag while its relaxation when the motion stops is described by the sum of three exponential relaxations. This is as for fresh foam, but the coefficients and time constants vary systematically with age. For the most part, these quantities also show a power law scaling with T. The age dependence of the quantities determined in this study is discussed in terms of the mean bubble size.     

Spark-generated bubble near an elastic sphere

The interaction between a spark-generated bubble and an elastic sphere is investigated. A spark-generated bubble is created at various distances horizontally away from a suspended elastic sphere made of silicone rubber or super absorbent polymer (of shear modulus of elasticity G of between 5 and 312 kPa), using a low-voltage spark discharge method. We observe pronounced deformation and elongation of the elastic sphere when the spark-bubble is generated very close to a sphere. This happens when the elastic sphere has a small modulus of elasticity and a small size ratio R’ between the bubble and the elastic sphere (i.e. the bubble and the sphere have similar radii). Numerical simulations are also conducted using a Boundary Element Method (BEM) model coupled with a Finite Element Method (FEM) solver. The simulation results compare well with the experimental data. The numerical model is then extended to study the effects of elasticity and experimental parameters, such as the dimensionless stand-off distance H’, and size ratio R’, on the degree of deformation of the elastic cell and the dynamics of the bubble.     

Lattice gas simulations of two-dimensional liquid foams

Liquid foam is a dense random packing of gas bubbles in a small amount of immiscible liquid containing surfactants. The liquid within the Plateau borders, although small in volume, causes considerable difficulties to investigations of the physical properties of foams, and the situation becomes even more complicated if the flow of the liquid through the foam is considered too. Here we propose a fresh approach to tackling these issues by introducing a discrete two-dimensional hybrid lattice gas model of liquid foams. While lattice gas models have been used to model two-phase liquids in the past, their application to the study of liquid foams is novel and proves promising. We represent bubble surfaces by a finite number of nodes, and model the surrounding liquid as a lattice gas (with a finite number of liquid particles). The gas in the bubbles is treated as an ideal gas at constant temperature. The model is tested by choosing an arbitrarily shaped bubble that evolves into a circular shape in agreement with Laplaces law. The model is then employed to simulate periodic ordered and disordered dry and wet foams. Since our model is specifically designed to handle wet foams up to a critical liquid fraction of 0.16 (void fraction of random packing of disks), we are able to compute the variation in coordination number (average number of neighbours of a bubble) over the whole range of liquid fractions, and we find it to be a linear function of the shear modulus.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, 11–13 September 2003.     

A theory for large deformation and damage of interpenetrating polymer networks

Elastomers and gels can be formed by interpenetrating two polymer networks on a molecular scale. This paper develops a theory to characterize the large deformation and damage of interpenetrating polymer networks. The theory integrates an interpenetrating network model with the network alteration theory. The interpenetration of one network stretches polymer chains in the other network and reduces its chain density, significantly affecting the initial modulus, stiffening and damage properties of the resultant elastomers and gels. Double-network hydrogels, a special type of interpenetrating polymer network, have demonstrated intriguing mechanical properties including high fracture toughness, Mullins effects, and necking instability. These properties have been qualitatively attributed to the damage of polymer networks. Using the theory, we quantitatively illustrate how the interplay between polymer-chain stiffening and damage-induced softening can cause the Mullins effect and necking instability. The theory is further implemented into a finite-element model to simulate the initiation and propagation of necking instability in double-network hydrogels. The theoretical and numerical results are compared with experimental data from multiple cyclic compressive and tensile tests.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.     

On the stability of Taylor bubbles inside a confined highly porous medium

This work focuses on the local hydrodynamics of a multiphase gas–liquid flow forced into an innovative medium of high porosity (96%): an open cell solid foam. The gas (nitrogen) and liquid (ethanol) phases are injected at constant flow-rates in a millichannel to form a well-controlled Taylor flow which enters the porous medium. Based on a fluorescence technique, the apparent liquid holdup in the porous medium is quantified, and its evolution in time and along the porous medium extracted from spatiotemporal diagrams. The analysis of the main frequency, when varying the gas–liquid flow-rate ratio, leads to the identification of two hydrodynamic regimes. A model based on a scaling analysis is proposed to quantify the dimensionless numbers describing the transition between both regimes. It points out that the bubble length fixed by the Taylor flow is the control parameter. The model prediction of the critical bubble length at which the transition occurs is in good agreement with the experimental observations.     

Nucleate pool boiling in microgravity: Recent progress and future prospects

Pool boiling on flat plates in microgravity has been studied for more than 50 years. The results of recent experiments performed in sounding rocket are presented and compared to previous results. At low heat flux, the vertical oscillatory motion of the primary bubble is responsible for the increase in the heat transfer coefficient in microgravity compared to ground experiments. The effect of a non-condensable gas on the stabilisation of the large primary bubble on the heater is pointed out. Experiments on isolated bubbles are also performed on ground and in parabolic flight. The effect of a shear flow on the bubble detachment is highlighted. A force balance model allows determining an expression of the capillary force and of the drag force acting on the bubble.     

3-D measurements of separated flow over rigid plates with spanwise periodic cambering at low Reynolds number

Measurements of the flow field around a flat plate and rigid plates with spanwise periodic cambering were performed using volumetric three-component velocimetry (V3V) at a Reynolds numbers of 28,000 at α=12° where the flow is fully separated. The Reynolds normal and shear stresses, and the streamwise, spanwise and normal components of the vorticity vector are investigated for three-dimensionality. Flow features are discussed in context of the periodic cambering and corresponding aerodynamic force measurements. The periodic cambering results in spanwise variation in the reversed-flow region, Reynolds stresses and spanwise vorticity. These spanwise variations are induced by streamwise and normal vortices of opposite directions of rotation. Moreover, measurements were carried out for the cambered plates at α=8°, where a long separation bubble exists, to further understand the behavior of the streamwise and normal vortices. These vortices become more organized and increase in strength and size at the lower angle of attack. It is also speculated that these vortices contribute to the increase in lift at and beyond the onset of stall angle of attack.     

Caractérisation de la gélification d'une résine thermodurcissable par méthode rhéologique

Résumé Une nouvelle façon de repérer la gélification d'une résine thermodurcissable est proposée, à partir de mesures rhéologiques. Le point de gel est relié à une diminution de la vitesse de croissance du module visqueux observée sur les courbes expérimentales en cours de cinétique à température constante. Les temps de gel obtenus sont du même ordre que ceux donnés par les autres méthodes rhéologiques, mais font cependant apparaître des différences sensibles. Le temps de gel obéit à une loi d'Arrhénius en fonction de la température de cuisson.Le module visqueux au point de gel et, par conséquent la viscosité en ce point, varient avec la température. Il en est de même pour le facteur de perte tan . Par contre, le module élastique au point de gel se conserve quelle que soit la température. Ces résultats ont été obtenus sur deux formulations de résine: DGEBA (n = 0) – mPDA et DGEBA (n = 0) – DDM à la stoechiométrie.

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A new method of characterizing gelation of a thermosetting resin from rheological measurements is proposed. Gelation is associated with a reduction in the speed of growth of the viscous modulus which is observed on the experimental curves during isothermal kinetics. Times of gelation obtained in this way are of the same order of magnitude as those found with other rheological methods, however distinct differences are observed. These times follow the Arrhenius' law as a function of the processing temperature.The viscous modulus at the gelation point and, as a consequence, the viscosity at this point, vary with temperature. This comes also true for the loss factor tan . In contrast, the elastic modulus at the gelation point does not change with the temperature. These results were obtained with two resin formulas: DGEBA (n = 0) – mPDA and DGEBA (n = 0) – DDM in stoechiometric concentration.

     

泡内气体热力学性质对空泡溃灭的影响

数值研究固壁附近轴对称空泡溃灭问题. 忽略泡内气体与周围流体之间的质量和热交换, 假设气体瞬时处于热平衡状态, 通过引入不同的热力学模型, 考察泡内气体在空泡溃灭过程中的作用. 采用原始变量的Navier-Stokes方程作为流场的控制方程, 用流体体积方法跟踪运动空泡壁. 数值结果显示空泡溃灭过程中, 伴随空泡变形, 空泡发出多个高压脉冲和高速射流. 对于不同的热力学模型, 等温, 绝热和准绝热过程, 绝热过程能够最大程度抑制空泡溃灭, 从而减弱空泡溃灭对固壁造成的空蚀破坏. 在绝热及其类似过程中, 出现空泡回弹现象.     

胞体椭球比对泡沫塑料力学性能的影响

本文通过数值法研究了胞体椭球比对材料模量及泊松比的影响；在单向受力情况下，研究了变形对材料孔隙度、椭球比、杨氏模量和泊松比等材料参数的影响。     

Non-Markovian diffusion process in polymers and stretched exponential relaxation

In this contribution, we model the long-time behaviour of the desorption from an LDPE sheet, using non-Markovian random walks. It is shown that the mass of penetrant in the final stage of desorption decays as t ^–m , where m is proportional to the exponent of the probability distribution (t) t ^–(1+u), 0 < v < 1. Furthermore, it is shown that this model may lead to the so-called mechanical stretched exponential relaxation, and that Wagner's memory function can be obtained as a special case.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

A numerical strategy for the direct 3D simulation of the expansion of bubbles into a molten polymer during a foaming process

In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo‐plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity–pressure‐mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first‐order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space–time discontinuous‐Galerkin method, coupled with an r‐adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D. Copyright © 2007 John Wiley & Sons, Ltd.