Flow-induced crystallization of high-density polyethylene: the effects of shear and uniaxial extension

The effects of shear, uniaxial extension and temperature on the flow-induced crystallization of two different types of high-density polyethylene (a metallocene and a ZN-HDPE) are examined using rheometry. Shear and uniaxial extension experiments were performed at temperatures below and well above the peak melting point of the polyethylenes in order to characterize their flow-induced crystallization behavior at rates relevant to processing (elongational rates up to 30 s^− 1 and shear rates 1 to 1,000 s^− 1 depending on the application). Generally, strain and strain rate found to enhance crystallization in both shear and elongation. In particular, extensional flow was found to be a much stronger stimulus for polymer crystallization compared to shear. At temperatures well above the melting peak point (up to 25°C), polymer crystallized under elongational flow, while there was no sign of crystallization under simple shear. A modified Kolmogorov crystallization model (Kolmogorov, Bull Akad Sci USSR, Class Sci, Math Nat 1:355–359, 1937) proposed by Tanner and Qi (Chem Eng Sci 64:4576–4579, 2009) was used to describe the crystallization kinetics under both shear and elongational flow at different temperatures.     

Rheo-PIV analysis of the slip flow of a metallocene linear low-density polyethylene melt

The continuous extrusion of a metallocene linear low-density polyethylene through a transparent capillary die with and without slip was analyzed in this work by rheometrical measurements and particle image velocimetry (PIV). For this reason, a comparison was made between the rheological behaviors of the pure polymer and blended with a small amount of fluoropolymer polymer processing additive. Very good agreement was found between rheometrical and PIV measurements. The pure polymer exhibited stick-slip instabilities with nonhomogeneous slip at the die wall, whereas the blend showed stable flow. The slip velocity was measured directly from the velocity profiles and was negligible for the pure polymer before the stick-slip but increased monotonously as a function of the shear stress for the blend. The flow curves and the slip velocity as a function of the shear stress deviated from a power law and were well fitted by continuous “kink” functions. Comparison of PIV data with rheometrical ones permitted a direct proof of the basic assumption of the Mooney theory. Finally, the analysis of the velocity profiles showed that there is a maximum in the contribution of slip to the average fluid velocity, which is interpreted as the impossibility for the velocity profile to become plug like in the presence of shear thinning.     

Fingerprinting the processing behavior of polyethylenes from transient extensional flow and peel experiments in the melt state

The flow curves of linear (linear-low and high density) and branched polyethylenes are known to differ significantly. At increasing shear rates, the linear polymers exhibit a surface melt fracture or sharkskin region that is followed by an unstable oscillating or stick-slip flow regime when a constant piston speed capillary rheometer is used. At even higher shear rates, gross melt fracture appears. Unlike their linear counterparts, branched polyethylenes rarely exhibit sharkskin melt fracture and although gross melt fracture appears at high shear rates there is no discontinuity in their flow curve. The various flow regimes of these two types of polyethylenes are examined by performing experiments in the melt state using a unique extensional rheometer (the SER by Xpansion Instruments) that is capable of performing accurate extensional flow and peel experiments at very high rates not previously realized. The peel strength curves of these linear and branched polyethylenes exhibit all of the distinct flow regimes exhibited in their respective flow curves, thereby providing a fingerprint of their melt flow behavior. Moreover, these extensional flow and peel results in the melt state provide insight into the origins and mechanisms by which these melt flow phenomena may occur with regard to rapid tensile stress growth, melt rupture, and adhesive failure at the polymer wall interface.     

Creep recovery behavior of metallocene linear low-density polyethylenes

The rheological behavior of two metallocene linear low-density polyethylenes (mLLDPE) is investigated in shear creep recovery measurements using a magnetic bearing torsional creep apparatus of high accuracy. The two mLLDPE used are homogeneous with respect to the comonomer distribution. The most interesting feature of the two mLLDPE is that their molecular mass distributions are alike. Therefore, as one of the mLLDPE contains long-chain branches, the influence of long-chain branching on the elastic properties of polyethylene melts could be investigated. It was found that long-chain branches increase the elasticity of the melt characterized by the steady-state recoverable compliance. The long-chain branched mLLDPE has a flow activation energy of 45 kJ/mol which is distinctly higher than that of the other mLLDPE. The shear thinning behavior is much more pronounced for the long-chain branched mLLDPE. A discrepancy between the weight average molecular mass M _w calculated from size exclusion chromatography measurements by the universal calibration method and the zero shear viscosities of the two mLLDPE was observed. These observations are discussed with reference to the molecular architecture of the long-chain branched mLLDPE. The rheological properties of the long-chain branched mLLDPE are compared with those of a classical long-chain branched LDPE. It is surprisingly found that the rheological behavior is very much the same for these two products although their molecular mass distributions and presumedly the branching structures differ largely. Received: 15 February 1999 Accepted: 10 June 1999     

An inclined wall jet: Mean flow characteristics and effects of acoustic excitation

 The mean velocity field of a 30° inclined wall jet has been investigated using both hot-wire and laser Doppler anemometry (LDA). Provided that the nozzle aspect ratio is greater than 30 and the inclined wall angle (β) is less than 50°, LDA measurements for various β show that the reattachment length is independent of the nozzle aspect ratio and the nozzle exit Reynolds number (in the range 6670–13,340). There is general agreement between the reattachment lengths determined by LDA and those determined using wall surface oil film visualisation technique. The role of coherent structures arising from initial instabilities of a 30° wall jet has been explored by hot-wire spectra measurements. Results indicate that the fundamental vortex roll-up frequency in both the inner and outer shear layer corresponds to a Strouhal number (based on nozzle exit momentum thickness and velocity) of 0.012. The spatial development of instabilities in the jet has been studied by introducing acoustic excitation at a frequency corresponding to the shear layer mode. The formation of the fundamental and its first subharmonic has been identified in the outer shear layer. However, the development of the first subharmonic in the inner shear layer has been severely suppressed. Distributions of mean velocities, turbulence intensities and Reynolds shear stress indicate that controlled acoustic excitation enhances the development of instabilities and promotes jet reattachment to the wall, resulting in a substantially reduced recirculation flow region. Received: 24 November 1998/Accepted: 24 August 1999     

Measurement of pressure coefficient of melt viscosity: drag flow versus capillary flow

The pressure coefficient of viscosity of poly(α-methylstyrene-co-acrylonitrile) was measured using a high-pressure sliding plate rheometer (HPSPR) and two types of capillary rheometer: a piston-driven device with a throttle at the exit [piston capillary rheometer with throttle (PCRWT)] operated at a fixed flow rate, and a counter-pressure nitrogen capillary rheometer (CPNCR) operated at a fixed pressure drop. In the HPSPR, the pressure, shear rate, density, and viscosity are all uniform throughout the sample, while the analysis of capillary data is complicated by the axial pressure gradient and the radial shear rate gradient. The polymer was found to be piezorheologically simple, and the HPSPR data indicated that the pressure coefficient of viscosity β ≡ dln(a _P)/dP decreased slightly with increasing pressure at high pressure. While β from PCRWT data from different laboratories and instruments agreed fairly well, the β values were on average about 2/3 of that from the HPSPR. The CPNCR yields β about 18% lower than that of the HPSPR.     

Influence of flow path modification on oscillation of cavity shear layer

This study investigates the influence on the oscillating characteristics of a cavity shear layer by introducing either a sloped bottom or a flow path modifier at the bottom of the cavity. All the experiments are performed in a recirculating water channel. The laser Doppler velocimetry system and the laser sheet technique are employed to perform the quantitative velocity measurements and the qualitative flow visualization, respectively. The Reynolds number, based on the momentum thickness at the upstream edge of the cavity, is kept at about Re _{θ 0}=194 ± 3.4. It is found that, in addition to the feedback effect, the upstream moving part of the recirculating flow inside the cavity also plays an important role in changing the oscillating characteristics of the unstable shear layer. As the bottom of the cavity is either negatively or positively sloped, the oscillating characteristics of the cavity shear layer are modified to different extents. Significant reduction of the oscillating amplitude within the cavity is found while the bottom slope increases up to d/L=± 2/5. As the bottom slope further increases up to d/L=± 1/2, the self-excited oscillation is completely suppressed. In addition, the ability to suppress the self-excited oscillation by the negative bottom slopes is superior to that in the case of a positive bottom slope. Depending upon the fence locations, the upstream moving part of the recirculating flow will perturb the unstable shear layer at different x/L locations, leading to different oscillating amplitudes. The ability to promote the enlarged oscillating amplitude of the unstable shear layer is better for a fence inclined at a positive angle than for one at a negative angle. Received: 31 May 2000/Accepted: 11 January 2001     

Detection and quantification of industrial polyethylene branching topologies via Fourier-transform rheology,NMR and simulation using the Pom-pom model

The significance of sparse long-chain branching in polyolefines towards mechanical properties is well-known. Topology is a very important structural property of polyethylene, as is molecular weight distribution. The method of Fourier-transform rheology (FTR) and melt state nuclear magnetic resonance (NMR) is applied for the detection and quantification of branching topology (number of branches per molecule), for industrial polyethylenes of various molecular weight and molecular weight distributions. FT rheology consists of studying the development of higher harmonics contribution of the stress response to a large amplitude oscillatory shear deformation. In particular, when applying large-amplitude oscillatory shear (LAOS), one observes the development of mechanical higher harmonic contributions at 3ω ₁, 5ω ₁,..., in the shear stress response. We correlate the relative intensity, I _3/1, and phase Φ ₃ of these harmonics with structural properties of industrial polyethylene, i.e. polymer topology and molecular weight distribution. Experiments are complemented by numerical simulations, using a multimode differential Pom-pom constitutive model (DCPP formulation), by fitting to the experimental linear and nonlinear viscoelastic behaviours. Simulation results in the nonlinear regime are correlated with molecular properties of the “pom-pom” macromolecular architecture. Qualitative agreement is found between predicted and experimental FT rheology results.     

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.     

Calculation of oscillatory regimes in couette flow in the neighborhood of the point of intersection of bifurcations initiating taylor vortices and azimuthal waves

Self-oscillating regimes of motion of a fluid close to Couette flow between rigid oppositely-rotating cylinders are investigated in a small neighborhood of the point of intersection of the neutral curves of the monotonic and oscillating instabilities. Bifurcation theory methods, together with computer calculations make it possible to detect transitions associated with the generation of both quasiperiodic oscillations and chaotic attractors. Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–93, July–August, 1998.     

Two-phase flow instabilities in a silicon microchannels heat sink

Two-phase flow instabilities are highly undesirable in microchannels-based heat sinks as they can lead to temperature oscillations with high amplitudes, premature critical heat flux and mechanical vibrations. This work is an experimental study of boiling instabilities in a microchannel silicon heat sink with 40 parallel rectangular microchannels, having a length of 15 mm and a hydraulic diameter of 194 μm. A series of experiments have been carried out to investigate pressure and temperature oscillations during the flow boiling instabilities under uniform heating, using water as a cooling liquid. Thin nickel film thermometers, integrated on the back side of a heat sink with microchannels, were used in order to obtain a better insight related to temperature fluctuations caused by two-phase flow instabilities. Flow regime maps are presented for two inlet water temperatures, showing stable and unstable flow regimes. It was observed that boiling leads to asymmetrical flow distribution within microchannels that result in high temperature non-uniformity and the simultaneously existence of different flow regimes along the transverse direction. Two types of two-phase flow instabilities with appreciable pressure and temperature fluctuations were observed, that depended on the heat to mass flux ratio and inlet water temperature. These were high amplitude/low frequency and low amplitude/high frequency instabilities. High speed camera imaging, performed simultaneously with pressure and temperature measurements, showed that inlet/outlet pressure and the temperature fluctuations existed due to alternation between liquid/two-phase/vapour flows. It was also determined that the inlet water subcooling condition affects the magnitudes of the temperature oscillations in two-phase flow instabilities and flow distribution within the microchannels.     

Instabilities of micellar systems under homogeneous and non-homogeneous flow conditions

The rheological behavior of a cetylpyridinium chloride 100 mmol l^–1/sodium salicylate 60 mmol l^–1 aqueous solution was studied in this work under homogeneous (cone and plate) and non-homogeneous flow conditions (vane-bob and capillary rheometers), respectively. Instabilities consistent with non-monotonic flow curves were observed in all cases and the solution exhibited similar behavior under the different flow conditions. Hysteresis and the sigmoidal flow curve suggested as characteristic of systems that show constitutive instabilities were observed when running cycles of increasing and decreasing stress or shear rate, respectively. This information, together with a detailed determination of steady states at shear stresses close to the onset of the instabilities, allowed one to show unequivocally that "top and bottom jumping" are the mechanisms to trigger the instabilities in this micellar system. It is shown in addition that there is not a true plateau region in between the "top and bottom jumping". Finally, the flow behavior beyond the upturn seemed to be unstable and was found accompanied by an apparent violation of the no-slip boundary condition.     

Prediction of steady-state viscous and elastic properties of polyolefin melts in shear and elongation

The linear and nonlinear steady-state viscosities and elastic compliances were measured in shear and elongational flows for two low-density polyethylenes, a linear polypropylene, and two metallocene catalyzed polyethylenes (one linear and one long-chain branched) by Wolff et al. (Rheol Acta 49:95?C103, 2010) and Resch (dissertation, 2010). Comprehensive data of this type are rarely found in the literature, and comprehensive modeling of both viscous and elastic effects is even rarer. In this contribution, the reliability of a modeling approach proposed by Laun (J Rheol 30(3):459?C501, 1986) and based on the damping function concept is tested. The strain hardening seen for the long-chain branched polymers and its absence in the case of the linear polymer, the stronger decrease of the tensile compliance in comparison to the shear compliance with increasing stress, as well as the extended linear-viscoelastic regime of the shear viscosity in contrast to the shear compliance are correctly modeled. While the modeling of the nonlinear response in shear can be achieved with only one material parameter for most of the polymers considered here, the nonlinear modeling in elongation is achieved with two parameters. The same parameter values are shown to describe viscous as well as elastic properties of the melts, and thus the relations of Laun can be used to test the consistency of viscosity and compliance measurements.     

Experimental investigations of a swirling jet in both stationary and rotating surroundings

The ‘plug’ flow emerging from a long rotating tube into a large stationary reservoir was used in the experimental investigation of swirling jets with Reynolds numbers, Re = 600, 1,000 and 2,000, and swirl numbers, S = ΩR/U, in the range 0–1.1, to cover flow regimes from the non-rotating jet to vortex breakdown. Here Ω is the nozzle rotation rate, R is the radius of the nozzle exit, and U is the mean mass axial velocity. The jet was more turbulent and eddies shed faster at larger Re. However the flow criticality and shear layer morphology remained unchanged with Re. After the introduction of sufficient rotation, co-rotating and counter-winding helical waves replaced vortex rings to become the dominant vortex structure. The winding direction of the vortex lines suggests that Kelvin–Helmholtz and generalized centrifugal instability dominated the shear layer. A quantitative visualization study has been carried out for cases where the reservoir was rotating independently with S _a  = Ω _a R/U = ±0.35, ±0.51 and ±0.70 at Re = 1,000 and 2000, where Ω _a is the rotation rate of the reservoir. The criterion for breakdown was found to be mainly dependent on the absolute swirl number of the jet, S. This critical swirl number was slightly different in stationary and counter-swirl surroundings but obviously smaller when the reservoir co-rotated, i.e. S _c  = 0.88, 0.85 and 0.70, respectively. These results suggest that the flow criticality depends mainly on the velocity distributions of the vortex core, while instabilities resulting from the swirl difference between the jet and its ambient seem to have only a secondary effect.     

Influence of long-chain branches in polyethylenes on linear viscoelastic flow properties in shear

 This contribution presents a survey on the influence of long-chain branching on the linear viscoelastic properties zero shear-rate viscosity and steady-state recoverable compliance of polyethylene melts. The materials chosen are linear and slightly long-chain branched metallocene-catalyzed polyethylenes of narrow molecular mass distribution as well as linear and highly long-chain branched polyethylenes of broad molecular mass distribution. The linear viscoelastic flow properties are determined in shear creep and recovery experiments by means of a magnetic bearing torsional creep apparatus. The analysis of the molecular structure of the polyethylenes is performed by a coupled size exclusion chromatography and multi-angle laser light scattering device. Polyethylenes with a slight degree of long-chain branching exhibit a surprisingly high zero shear-rate viscosity in comparison to linear polyethylenes whereas the highly branched polyethylenes have a much lower viscosity compared to linear samples. Slightly branched polyethylenes have got a higher steady-state compliance in comparison to linear products of similar polydispersity, whereas the highly branched polyethylenes of broad molecular mass distribution exhibit a surprisingly low elasticity in comparison to linear polyethylenes of broad molecular mass distribution. In addition sparse levels of long-chain branching cause a different time dependence in comparison to linear polyethylenes. The experimental findings are interpreted by comparison with rheological results from literature on model branched polymers of different molecular topography and chemical composition. Received: 12 July 2001 Accepted: 30 October 2001     

Laminar forced convection in two-dimensional impacting tee junctions

This study presents numerical predictions of the laminar fluid-flow and heat-transfer characteristics in planar (two-dimensional) impacting tee junctions. The applicable Navier-Stokes equations and the energy equation were solved for airflow at two inlet Reynolds numbers (Re₁) and a wide range of the mass split ratio (β). The results include wall shear stress distributions, streamlines showing the number, location, and size of the re-circulation zones, the pressure loss coefficient, wall heat flux distributions, isotherms, and the overall rate of heat transfer. These results indicate that two re-circulation zones always form on the inside-bend wall of the tee at all values of β and Re₁. Two more re-circulation zones may form on the impacting wall of the tee depending on the values of β and Re₁. It was also found that the pressure loss coefficient reaches a minimum and the overall rate of heat transfer reaches a maximum at even mass split (β = 0.5).     

Numerical investigation of a laminar pulsating flow in a rectangular duct

A pulsating laminar flow of a viscous, incompressible liquid in a rectangular duct has been studied. The motion is induced under an imposed pulsating pressure difference. The problem is solved numerically. Different flow regimes are characterized by a non‐dimensional parameter based on the frequency (ω) of the imposed pressure gradient oscillations and the width of the duct (h). This, in fact, is the Reynolds number of the problem at hand. The induced velocity has a phase lag (shift) with respect to the imposed pressure oscillations, which varies from zero at very slow oscillations, to 90° at fast oscillations. The influence of the aspect ratio of the rectangular duct and the pulsating pressure gradient frequency on the phase lag, the amplitude of the induced oscillating velocity, and the wall shear were analyzed. Copyright © 1999 John Wiley & Sons, Ltd.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Exponential shear flow of branched polymer melts

  The behavior of a low-density polyethylene melt in exponential shear strain histories is examined and compared to its behavior in constant rate planar elongation. A new set of shear stress and first normal stress difference data in exponential shear are presented and used in several different material functions that have been previously proposed. Viscosities composed of principal stress differences for the two flows showed no correspondence suggesting that, contrary to previous assertions, exponential shear and constant rate planar elongation flows are fundamentally different. It is further suggested that the presence of vorticity makes exponential shear a weak, rather than strong, flow. Received: 5 March 1999/Accepted: 1 September 1999     

Effects of drift angle on model ship flow

The effects of drift angle on model ship flow are investigated through towing tank tests for the Series 60 C_B=0.6 cargo/container model ship. Resistance, side force, drift moment, sinkage, trim, and heel data are procured for a range of drift angles β and Froude numbers (Fr) and the model free condition. Detailed free-surface and mean velocity and pressure flow maps are procured for high and low Fr=0.316 and 0.16 and β=5° and 10° (free surface) and β=10° (mean velocity and pressure) for the model fixed condition (i.e. fixed with zero sinkage, trim, and heel). Comparison of results at high and low Fr and previous data for β=0° enables identification of important free-surface and drift effects. Geometry, conditions, data, and uncertainty analysis are documented in sufficient detail so as to be useful as a benchmark for computational fluid dynamics (CFD) validation. The resistance increases linearly with β with same slope for all Fr, whereas the increases in the side force, drift moment, sinkage, trim, and heel with β are quadratic. The wave profile is only affected near the bow, i.e. the bow wave amplitude increases/decreases on the windward/leeward sides, whereas the wave elevations are affected throughout the entire wave field. However, the wave envelope angle on both sides is nearly the same as β=0°, i.e. the near-field wave pattern rotates with the hull and remains within a similar wave envelope as β=0°. The wave amplitudes are significantly increased/decreased on the windward/leeward sides. The wake region is also asymmetric with larger wedge angle on the leeward side. The boundary layer and wake are dominated by the hull vortex system consisting of fore body keel, bilge, and wave-breaking vortices and after body bilge and counter-rotating vortices. The occurrence of a wave-breaking vortex for breaking bow waves has not been previously documented in the literature. The trends for the maximum vorticity, circulation, minimum axial velocity, and trajectories are discussed for each vortex. Received: 16 September 1999/Accepted: 8 November 2001