Inlet instabilities in the capillary flow of polyethylene melts

Inlet instabilities in the capillary flow of polyethylene melts were studied in this work. Extrudate distortions in branched polyethylenes, produced by unstable upstream flow, were found to be accompanied by pressure oscillations that do not have their origin in the slip phenomenon, but on polymer compressibility. The absence of slip was clearly evidenced in the experiments, and the differences between pressure oscillations occurring in linear and branched polymers are shown.Pressure oscillations in the capillary flow of branched polyethylenes were found to be made up of at least two components of different frequency and amplitude. These two components were identified with different bulk defects appearing in the extrudates. Information about the dynamics of vortices upstream of the contraction and extrudate distortions is obtained from the analysis of pressure oscillations.The influence of capillary entrance angle on flow curves was also investigated. From the results, it is concluded that the extensional component of the flow in the contraction is the main factor responsible for the slope change usually found in the log-log flow curves of both linear and branched polyethylenes.     

Modelization of extensional experiments under constant force using memory-integral constitutive equations and stream-tube analysis

In this paper, we propose a numerical simulation of axisymmetric extensional experiments on a viscoelastic polydimethylsiloxane (PDMS) material, using a falling-weight extensional rheometer. The polymer behaviour is represented by a K-BKZ memory-integral constitutive equation, involving a damping function of the Wagner type. Under the assumption of a homogeneous flow zone in the sample, a numerical model is set up, using the stream-tube method and approximating functions. The governing equations of the problem, associated to a limited number of unknowns, are solved by means of the Levenberg-Marquardt optimization algorithm. The numerical results are found to be consistent with the experimental data and reveal the importance of the non-homogeneous flow zone, in relation to the estimation of the extensional strain rate. The calculations involve the sensitivity of the model on the fluid parameters and those concerning the size of the initial column of fluid. The limited computing (CPU) time of the code is also to be underlined.     

A numerical study of the turbulent flow around the stern of ship models

The present work is concerned with the numerical calculation of the turbulent flow field around the stern of ship models. The finite volume approximation is employed to solve the Reynolds equations in the physical domain using a body-fitted, locally orthogonal curvilinear co-ordinate system. The Reynolds stresses are modelled according to the standard k-ε turbulence model. Various numerical schemes (i.e. hybrid, skew upwind and central differencing) are examined and grid dependence tests have been performed to compare calculated with experimental results. Moreover, a direct solution of the momentum equations within the near-wall region is tried to avoid the disadvantages of the wall function approach. Comparisons between calculations and measurements are made for two ship models, i.e. the SSPA and HSVA model.     

Review of the entry flow problem: Experimental and numerical

This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.     

A numerical study of bifurcation in laminar flow in curved ducts

The bifurcation phenomenon whereby multiple-vortex secondary flow occurs in place of the normal two-vortex flow in laminar flow in curved ducts has previously been studied numerically by several researchers. However, the various results have been conflicting on many points. The present paper describes a set of numerical experiments conducted to study the effect of numerical accuracy on the solution. The results show that the transition from two- to four-vortex structure depends strongly on the differencing scheme and to a lesser extent on the grid size. The study also shows that as the Reynolds number of the flow increases, a two-vortex structure is re-established via a path which involves strongly asymmetric secondary flow patterns. These results are in agreement, at least qualitatively, with recent experimental theoretical and numerical results.     

The strain-hardening behaviour of linear and long-chain-branched polyolefin melts in extensional flows

By generalizing the Doi-Edwards model to the Molecular Stress Function theory of Wagner and Schaeffer, the extensional viscosities of polyolefin melts in uniaxial, equibiaxial and planar constant strain-rate experiments starting from the isotropic state can be described quantitatively. While the strain hardening of four linear polymer melts (two high-density polyethylenes, a polystyrene and a polypropylene) can be accounted for by a tube diameter that decreases affinely with the average stretch, the two long-chain-branched polymer melts considered (a low-density polyethylene and a long-chain branched polypropylene) show enhanced strain hardening in extensional flows due to the presence of long-chain branches. This can be quantified by a molecular stress function, the square of which is quadratic in the average stretch and which follows from the junction fluctuation theory of Flory. The ultimate magnitude of the strain-hardening effect is governed by a maximum value of the molecular stress, which is specific to the polymer melt considered and which is the only free non-linear parameter of the theory. Received: 1 June 1999/Accepted: 24 November 1999     

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

The goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan-Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov-like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non-linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re-entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔP_ent) varied from model to model. The UCM and WM models predicted negative values for ΔP_ent.     

Underwater acoustics and cavitating flow of water entry

The fluid mechanics of water entry is studied through investigating the underwater acoustics and the supercavitation. Underwater acoustic signals in water entry are extensively measured at about 30 different positions by using a PVDF needle hydrophone. From the measurements we obtain (1) the primary shock wave caused by the impact of the blunt body on free surface; (2) the vapor pressure inside the cavity; (3) the secondary shock wave caused by pulling away of the cavity from free surface; and so on. The supercavitation induced by the blunt body is observed by using a digital high-speed video camera as well as the single shot photography. The periodic and 3 dimensional motion of the supercavitation is revealed. The experiment is carried out at room temperature. The project supported by the “BaiRen Plan” of Chinese Academy of Sciences     

Imaging and analysis of wave type interfacial instability in the coextrusion of low-density polyethylene melts

This report covers experimental studies and numerical modelling of interfacial instability in the bi-layer coextrusion flow of two low-density polyethylene melts. Melt streams are converged at an angle of 30° to a common die land. Melt stream confluence was observed in two coextrusion die arrangements. In one die design, which we term ‘bifurcated’ the melt stream is split by a divider plate in the die after being delivered from a single extruder. In the other design melt streams are delivered to a die from two separate extruders. In each die design melt flow in the confluent region and die land to the die exit was observed through side windows of a visualization cell. Velocity ratios of the two melt streams were varied and layer thickness ratios producing wave type interfacial instability determined for each melt for a variety of flow conditions. Stress and velocity fields in the coextrusion arrangements were quantified using stress birefringence and particle image velocimetry techniques.     

Shear flow rheological properties of vinylon- and glass-fiber reinforced polyethylene melts

Shear viscosity, shear stress and first normal-stress difference have been investigated for glass- and vinylon-fiber filled polyethylene melts over a wide range of shear rate by means of three kinds of instruments. The influence of fiber content and fiber properties on the rheological properties is discussed. The viscosity increases with increasing aspect ratio and fiber content, and the influence of these parameters on the flow properties is evident at low shear rates. The first normalstress difference increases more rapidly with increasing glass fiber content, especially at low shear stresses. The influence of vinylon fibers on the first normal stress-difference vs. shear-stress relationship is different from that of glass fibers.     

Pressure effects on viscosity and flow stability of polyethylene melts during extrusion

In the present work, the effects of pressure on the viscosity and flow stability of four commercial grade polyethylenes (PEs) have been studied: linear-low-density polyethylene copolymer, high-density polyethylene, metallocene polyethylenes with short-chain branches (mPE-SCB), and metallocene polyethylenes with long chain branching (mPE-LCB). The range of shear rates considered covers both stable and unstable flow regimes. “Enhanced exit-pressure” experiments have been performed attaining pressures of the order of 500×10⁵ Pa at the die exit. The necessary experimental conditions have been clearly defined so that dissipative heating can be neglected and pressure effects isolated. The results obtained show an exponential increase in both shear and entrance-flow pressure drop with mean pressure when shear rate is fixed and as long as flow is stable. These pressure effects are described by two pressure coefficients, β_S under shear and, β_E under elongation, that are calculated using time–pressure superposition and that are independent of mean pressure and flow rate. For three out of four PE, pressure coefficient values can be considered equal under shear and under elongation. However, for the mPE-LCB, the pressure coefficient under elongation is found to be about 30% lower than under shear. Flow instabilities in the form of oscillating flows or of upstream instabilities appear at lower shear rates as mean pressure increases. Nevertheless, the critical shear stress at which they are triggered remains independent of mean pressure. Moreover, it is found that the β_S values obtained for stable flows do not differ much from the values obtained during upstream instability regimes, and differ really from pressure effects observed under oscillating flow and slip conditions.     

Dynamic flow properties of vinylon fibre and glass fiber reinforced polyethylene melts

An experimental study of the dynamic shear flow properties of polyethylene melts filled with glass fibers and vinylon fibers was carried out and comparison with the steady shear flow properties was made. The effects of loading level and the characteristics of the fibers on the rheological properties of the fiber-filled systems is discussed. The rigidity and complex viscosity of the fiber-filled systems is sensitive not only to the quantity of fibers but also to their length, distribution and properties. The Cox-Merz empirical law for complex viscosity and steady shear viscosity, and Roscoe's empirical relation for estimating the normal-stress coefficients are both able to be applied to pure polymer melts but not to fiber-filled systems.     

钝体倾斜和垂直冲击入水时引起的超空泡流动特性实验研究

对4种不同头型的钝体、以不同初始速度在小倾斜角度和垂直状态下入水,所产生的空泡流进行了的实验观察,分析了不同工况下空泡产生和发展的特性。实验结果表明：对于倾斜入水及垂直入水,圆台头和平头(即空化器均为圆盘)实验体均能形成较稳定的入水弹道;初始入水速度较低时,空泡的闭合方式为深闭合;初始入水速度较高时,空泡的闭合方式为表面闭合,且运动速度衰减得更快。测量得知,钝体倾斜入水产生的空泡的前部外形轮廓与Logvinovich的半经验公式给出的结果相吻合。在垂直入水的情况下,调查了物体头部对空泡的起始点位置及其形态的影响。

     

An experimental study of planar entry flow of rubber compound

An automated rheometer based on an injection molding machine is developed for the evaluation of entry flow problems. Several entry flow geometries having different contraction and expansion angles and different channel lengths are tested. Two pressure transducers are flushmounted along the die length and a displacement transducer is installed to measure the screw motion. Signals generated by the pressure transducers and displacement transducer are supplied to an A/D converter and an IBM PC/AT computer. The pressure losses for a rubber compound are measured between two cross-sections along the flow direction. The time evolution of pressure with overshoot during flow before and after the entry region is observed. At low flow rates the pressure drops of the expansion flow are larger than those of the contraction flow. At high flow rates the pressure drops of the contraction flow become higher than those of the expansion flow. A ratio of the pressure drop to absolute pressure before the entry is found to be almost independent of flow rate.     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

A study of time integration schemes for the numerical modelling of free surface flows

A semi‐discrete finite element methodology for the modelling of transient free surface flows in the context of Eulerian interface capturing is proposed. The focus of this study is put on the choice of an appropriate time integration strategy for the accurate modelling of the dynamics of free surfaces and of interfacial physics. It is composed of an adaptive time integration scheme for the Navier–Stokes equations, and of the implicit midpoint rule for the transport equation of the Eulerian marker variable. The adaptive scheme allows the automatic determination of a time‐step size that follows the physics of the problem under study, which facilitates the accurate modelling of stiff free surface flows. It is shown that the implicit midpoint rule reduces mass loss for each fluid. Various free surface flow problems are studied to verify and validate the proposed time integration strategy. Copyright © 2005 John Wiley & Sons, Ltd.     

Multi-scale numerical simulation of fluidized beds: Model applicability assessment

In the past few decades, multi-scale numerical methods have been developed to model dense gas-solid flow in fluidized beds with different resolutions, accuracies, and efficiencies. However, ambiguity needs to be clarified in the multi-scale numerical simulation of fluidized beds: (i) the selection of the sub-models, parameters, and numerical resolution; (ii) the multivariate coupling of operating conditions, bed configurations, polydispersity, and additional forces. Accordingly, a state-of-the-art review is performed to assess the applicability of multi-scale numerical methods in predicting dense gas-solid flow in fluidized beds at specific fluidization regimes (e.g., bubbling fluidization region, fast fluidization regime), with a focus on the inter-particle collision models, inter-phase interaction models, collision parameters, and polydispersity effect. A mutual restriction exists between resolution and efficiency. Higher-resolution methods need more computational resources and thus are suitable for smaller-scale simulations to provide a database for closure development. Lower-resolution methods require fewer computational resources and thus underpin large-scale simulations to explore macro-scale phenomena. Model validations need to be further conducted under multiple flow conditions and comprehensive metrics (e.g., velocity profiles at different heights, bubbles, or cluster characteristics) for further improvement of the applicability of each numerical method.     

Experimental and numerical study of turbulent flow and heat transfer inside hexagonal duct

Flow and heat transfer characteristics in transition and turbulent regions are studied experimentally and numerically in a horizontal smooth regular hexagonal duct under constant wall temperature boundary condition covering a range of Reynolds number from 2.3 × 10³ to 52 × 10³. Two types of k-omega (standard and shear stress transport (SST)) and three types of k-ε (standard, renormalization (RNG), and realizable) turbulence model are employed for transition and turbulent regions, respectively. Both average and fully developed Darcy friction factor and Nusselt number are presented as a function of Reynolds number. It is seen that k-omega SST and k-ε realizable turbulence models gave the best agreement with the experimental data in transition and turbulent regions, respectively. All the experimental results are correlated within an accuracy of ±13 % and ±7 % for Nusselt number and Darcy friction factor, respectively. Results obtained in this study are compared with circular duct results using hydraulic diameter.