Planar Extensional Viscosity Research Articles

Elastic and shear-thinning effects in contraction flows: a comparison

The flow through a 4:1 planar contraction has been investigated using different rheological models having the same shear viscosity, namely, the inelastic Carreau-Yasuda model (CY), the enhanced Bautista-Manero-Puig (eBMP), and the exponential version of the Phan-Thien/Tanner (PTT). Noticeable discrepancies were observed with the CY model and the eBMP in terms of the velocity profiles along the centerline and in the exit channel (near the end of the geometry) normal to the flow direction. Transient planar extensional viscosity shows a large effect on vortex dynamics although the effect of transient and steady elongation on pressure drop seems negligible. Simulation results allowed gathering that pressure drop is largely influenced by the shear-thinning behavior of the fluid, noticeably affected by elasticity, and less by extensional viscosity.Graphical

Planar extensional viscosity from Oldroyd 8-constant framework

The Oldroyd 8-constant framework established a vision for constitutive equations for polymeric liquids past, present, and future. Though most canonical material functions for this framework are available, the expression for the steady planar extensional viscosity is not. In this work, we solve exactly for any steady homogeneous extensional (shearfree) flow field. Our results recover the well-known solution for the steady uniaxial extension, as they must. We next derive the steady planar extensional viscosity. Though this work has mainly been driven by curiosity, the planar extensional viscosity governs drawability in the film and sheet casting.

A hierarchical multimode molecular stress function model for linear polymer melts in extensional flows

A novel hierarchical multimode molecular stress function (HMMSF) model for linear polymer melts is proposed, which implements the basic ideas of (i) hierarchical relaxation, (ii) dynamic dilution, and (iii) interchain tube pressure. The capability of this approach is demonstrated in modeling the extensional viscosity data of monodisperse, bidisperse, and polydisperse linear polymer melts. Predictions of the HMMSF model, which are solely based on the linear-viscoelastic relaxation modulus and a single free model parameter, the segmental equilibration time, are compared to elongational viscosity data of monodisperse polystyrene melts and solutions as well as to the elongational viscosity data of a bidisperse blend of two monodisperse polystyrenes, and good agreement between model and experimental data is observed. By using a simplified relation between the Rouse stretch-relaxation times and the relaxation times of the melts, the modeling is extended to the uniaxial, equibiaxial, and planar extensional viscosity data of a high-density polyethylene, the uniaxial and equibiaxial extensional viscosity data of a polydisperse polystyrene, the elongational viscosity data of three high-density polyethylenes, and a linear low-density polyethylene. For polydisperse melts, the modeling is again based exclusively on the linear-viscoelastic relaxation modulus with only one material parameter, the dilution modulus, which quantifies the onset of dynamic dilution.

Measurements and modeling of temperature-strain rate dependent uniaxial and planar extensional viscosities for branched LDPE polymer melt

Abstract In this work, novel rectangle and circular orifice (zero-length) dies have been utilized for temperature-strain rate dependent planar and uniaxial extensional viscosity measurements for the LDPE polymer melt by using standard twin bore capillary rheometer and Cogswell model and the capability of five different constitutive equations (novel generalized Newtonian model, original Yao model, extended Yao model, modified White-Metzner model, modified Leonov model) to describe the measured experimental data has been tested. It has been shown that chain branching causes the strain hardening occurrence in both uniaxial and planar extensional viscosities and its maximum is shifted to the higher strain rates if the temperature is increased. The level of uniaxial extensional strain hardening for the branched LDPE sample has been found to be higher in comparison with the planar extensional viscosity within wide range of temperatures.

On using planar microcontractions for extensional rheometry

Planar contractions within a microfluidic environment have shown some potential to be used as a simple, cheap extensional rheometer. In this article, flow within a planar contraction is analyzed using an energy minimization calculation with a measure of slip included. The effect of slip on the calculated flow-field is used to show how the apparent extensional viscosity compares with the true extensional viscosity. In particular, it was found that the apparent planar extensional viscosity would be difficult to determine reliably at either low or high Weissenberg numbers but could be distinguished to some extent at moderate Weissenberg numbers between 0.25 and 1. It was also shown, however, that an unknown extent of slip at the vortex boundary would cause an uncertainty of a factor of 2 in any apparent measure of rheological properties.

Response of an entangled polymer solution to uniaxial and planar deformation

Experimental data on homogeneous planar extension on well-characterized polymeric fluids are rare. As a result, the relationship between planar and uniaxial extensional viscosity is also unclear. We present measurements of the steady–state planar extensional viscosity of a well-characterized polymer solution with about 30 entanglements per chain. The measurements are made by modifying the filament stretching rheometer to enable the stretching of the sample in the shape of a cylindrical tube. When a constant strain rate is imposed on the sample, a well-defined steady-state in the planar extensional viscosity is observed in the experiments. Our experiments show that, within the range of strain rates used, the response of the entangled polymer solution in planar extensional flow is identical to that observed in uniaxial extensional flow. Such a finding, if found to be generally valid, facilitates the comparison between simulations and experimental data, since planar flows are easier to simulate than uniaxial flows. We also compare the predictions of a toy-version of the tube model that describes the flow properties of entangled polymer solutions with the experimental results. We find that the predictions of the model are in qualitative agreement with the experimental observations within the range of strain rates used in this study.

Optimized Cross-Slot Flow Geometry for Microfluidic Extensional Rheometry

A precision-machined cross-slot flow geometry with a shape that has been optimized by numerical simulation of the fluid kinematics is fabricated and used to measure the extensional viscosity of a dilute polymer solution. Full-field birefringence microscopy is used to monitor the evolution and growth of macromolecular anisotropy along the stagnation point streamline, and we observe the formation of a strong and uniform birefringent strand when the dimensionless flow strength exceeds a critical Weissenberg number Wi(crit) ≈ 0.5. Birefringence and bulk pressure drop measurements provide self-consistent estimates of the planar extensional viscosity of the fluid over a wide range of deformation rates (26 s(-1) ≤ ε ≤ 435 s(-1)) and are also in close agreement with numerical simulations performed by using a finitely extensible nonlinear elastic dumbbell model.

Planar extensional flow resistance of a foaming plastic

The planar extensional flow resistance of a foaming-grade polystyrene, with and without a blowing agent, was characterized using two dies, each one consisting of a high-aspect-ratio straight rectangular channel followed by a hyperbolic convergent rectangular channel. With the hyperbolic geometry, the fluid near the centerline of the convergent channel was subjected to a constant rate of extension. The shear viscosity was found from the pressure drop along the straight channel, and the planar extensional viscosity was determined from the pressure drop in the convergent channel, taking into account the pressure drop due to shearing. Values of the planar extensional viscosity are compared with values of the uniaxial extensional viscosity measured with a shear-free fixture. Tests with a polystyrene and with a solution of 5% carbon dioxide in the polystyrene revealed that the gas caused significant reductions in both the shear and extensional viscosities.

Numerical simulation of contraction and expansion flows of Langmuir monolayers

Langmuir monolayers consist of amphiphilic molecules at the air–water interface and can be modeled as two-dimensional fluids. Earlier experiments [D.J. Olson, G.G. Fuller, J. Non-Newtonian Fluid Mech. 89 (2000) 187–207] on 4:1 contraction and 4:1 expansion flows have been simulated using an integral constitutive equation of the K-BKZ type, suitably modified to account for strain-thickening in the planar extensional viscosity. The model has been used to fit linear viscoelastic data ( G′ and G″) and the shear viscosity ( η S ), while the amount of strain-hardening is assumed, due to lack of experimental data. The simulations are in good agreement with the experiments on Newtonian monolayers, which show no vortices in the contraction but large inertial vortices in the expansion. For the viscoelastic monolayer (a poly-octadecyl methacrylate or PODMA), the opposite is true. The contraction flow shows vortices, while in the expansion flow the vortex activity is substantially reduced compared with the Newtonian one. The viscoelastic behavior is well captured by the model, provided that substantial strain-thickening is exhibited by the monolayer in planar extension. The latter behavior is very much like that for a branched LDPE melt, which also shows big vortices due to strain-hardening in planar as well as in uniaxial extension.

ヘルショウセルによる高分子溶液の伸張粘度測定法の開発(S23-1 複雑流体の流れとその応用(1),S23 複雑流体の流れとその応用)

The Hele-Shaw flows of a Newtonian fluid have been used to reproduce the flow patterns of two-dimensional potential flow. In the earlier papers, the present authors showed effects of elasticity on Hele-Shaw flows around a circular and a square cylinder, and a flow passing through a slit, and presented a new method to estimate the planar extensional viscosity from the slit flow pattern. In this study, the method is improved and tested using four polymer solutions. A new photochromic visualization technique is developed and applied to investigate the velocity field to confirm the validity of the improved method.

On the two-dimensional splashing experiment for Newtonian and slightly elastic liquids

When a solid sphere is dropped from some distance above the free surface of a liquid, a rich sequence of events can occur, the most notable being a vertical jet, which can rise some distance above the free surface. The height of this so-called Worthington jet can be dramatically reduced if a very small concentration of a high molecular weight polymer is added to a Newtonian solvent. This reduction has been tentatively linked to the high extensional viscosity of very dilute polymer solutions [J. Non-Newtonian Fluid Mech. 86 (1999)]. If this interpretation is correct, it is clearly of interest to study the corresponding planar situation, when the sphere is replaced by a long rod. In this case, a study of the height of the jet should provide information concerning the planar extensional viscosity. The experiments conducted in the present study seem to indicate that planar jets are not significantly affected by the addition of polymer additives, in contrast to the axisymmetric situation. This would indicate that the dependence of the planar extensional viscosity on polymer additives is not nearly as strong as is the case of the uniaxial extensional viscosity, although more theoretical work is required to confirm this conjecture. In the course of the experiments carried out in the present study, jets of unusual and provocative shapes were obtained when short rods were employed.

Numerical simulation of entry flow of the IUPAC-LDPE melt

Numerical simulations have been undertaken for the creeping entry flow of a well-characterized polymer melt (IUPAC-LDPE) in a 4:1 axisymmetric and a 14:1 planar contraction. The fluid has been modeled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times (Papanastasiou–Scriven–Macosko or PSM model). Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear and elongational data from uniaxial elongation experiments. The numerical solutions show that in the axisymmetric contraction the vortex in the reservoir first increases with increasing flow rate (or apparent shear rate), goes through a maximum and then decreases following the behavior of the uniaxial elongational viscosity. For the planar contraction, the vortex diminishes monotonically with increasing flow rate following the planar extensional viscosity. This kinematic behavior is not in agreement with recent experiments. The PSM strain-memory function of the model is then modified to account for strain-hardening in planar extension. Then the vortex pattern shows an increase in both axisymmetric and planar flows. The results for planar flow are compared with recent experiments showing the correct trend.

Numerical simulation of confined flow of polyethylene melts around a cylinder in a planar channel

Numerical simulations have been undertaken for the flow of three polyethylene melts, two low-density (LDPE) and a linear low-density (LLDPE), around a cylinder placed between parallel plates. Polyethylene melts have been the subject of previous extensive rheological characterizations. The flow geometry corresponds to a 2:1 and a 1.6:1 gap/diameter ratio used in experimental studies for flow-induced birefringence (FIB) measurements. The constitutive equation of this work is an integral-type K-BKZ model with a relaxation spectrum, which fits well experimental data for the shear viscosities and the normal stresses as measured in shear flow. The model has been properly modified to account for strain-thickening in planar extensional flows, such as those studied here. Stable numerical solutions have been obtained for a wide range of flow rates and used to compare the K-BKZ model predictions with the experimental observations for FIB. Good agreement is obtained for the birefringence predictions in all cases, provided that the right amount of strain-thickening has been assumed for the planar extensional viscosity. The simulations are also in agreement with previous simulations for the same problem using multi-mode differential constitutive models. The drag force exerted by the fluid on the cylinder has been calculated and reported to be a decreasing function of the flow rate.

Effect of ingredients on the rheological properties of extruded corn meal

A slit rheometer, mounted on a square cross section barrel extension, was attached to the exit of a single screw extruder to facilitate the simultaneous measurement of steady shear viscosity (ηsh), apparent planar extensional viscosity (ηpa), and the first normal stress differences (N1). The hole pressure method was used to determine N1 and the entrance pressure drop method with Cogswell’s analysis for abrupt planar contraction flow was used to determine ηpa. The effect of slot dimensions on N1 was investigated using low density polyethylene. Increasing the depth and width of the slot by 50% produced no significant change in the hole pressure. N1 predicted using the Higashitani–Pritchard–Baird (HPB) equation compared favorably with extrapolated N1 values obtained on a cone and plate rheometer. Addition of salt and sugar to corn grits affected the rheological properties of the melt, as observed by the changes in ηsh, N1, and ηpa. Of the three material functions, N1 and ηpa were more sensitive to the addition of ingredients. Corn grits exhibited thinning behavior in shear and planar extensional flows.

In-line measurement of rheological properties of polymer melts

Shear viscosity (η), first normal stress difference (N1), and extensional viscosity (ηE) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure η, N1, and the planar extensional viscosity (ηp) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure η and the uniaxial extensional viscosity (ηu) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N1, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate ηE.

SLIT RHEOMETER STUDIES OF WHEAT FLOUR DOUGH1

ABSTRACTRheological properties of bread dough having different levels of added gluten (0, 2.5, 5.0 and 7.5%) were measured. The dough was forced through a slit die attached to a cylindrical container. The material functions of interest included steady shear viscosity, first normal stress coefficient in steady shear flow derived from hole pressure measurements and planar extensional viscosity as estimated from entrance pressure drop measurements. These rheological properties were correlated to the loaf volume of the baked bread. Results indicate that the first normal stress coefficient and planar extensional viscosity were affected by the gluten content of the dough and correlated well with the loaf volume. The entrance pressure drop increased with increasing apparent shear rate but entrance correction was found to decrease with apparent shear rate. Water had a greater effect on the rheological properties than gluten content. The Trouton ratio was rather large and ranged between 850 to 2000 at a strain rate of 1s‐1 and decreased with increasing extension rate for most cases.

Planar extensional viscosity of corn meal dough

The application of the entrance pressure drop method to evaluate the extensional viscosity of corn meal during extrusion-cooking was investigated. An approximate planar extensional flow was generated by forcing the dough from a circular channel barrel into a slit die. The pressure drop along the contraction was calculated to obtain the entrance pressure drop ΔP en. Both Cogswell's and Binding's analyses were applied to estimate the apparent planar extensional viscosity η pa from ΔP en. The magnitude of η pa from Cogswell's analysis was found to be much larger than that predicted by Binding's analysis. In addition, the severe extension-thinning behavior exhibited by the fluid introduced problems in the calculations of Binding's analysis for some cases. Calculation and interpretation of the Trouton ratios have to be done carefully when extrapolation of the viscosities is required. Moderate to large entrance corrections were obtained, with lower values obtained for the lower moisture contents and temperatures of extrusion. In all cases the entrance correction decreased with increasing flow rate.

Some asymptotic results for planar extension

Approximations to planar and uniaxial extensional viscosities are obtained in the limiting case of large extension rate. Model fluids for which results are obtained are the Phan-Thien-Tanner model, the Acierno-Marrucci model, the Giesekus model, and a FENE-dumbbell model. Detailed microstructural models for a dilute solution of rigid rodlike molecules, following Doi and Edwards, and a dilute suspension of slender axisymmetric particles, following Brenner, are also discussed. The effect of using the decoupling approximation (and avoiding solving the equation for the orientational distribution function) is examined, and the relation between results of Brenner and of Doi and Edwards for extensional flows of solutions and suspensions is exhibited. In all these cases, apart from any solvent contribution to the viscosity, it is shown that the elongational viscosity and the planar extensional viscosity are equal at high rates of extension. Under the same conditions the cross-viscosity in planar extension becomes vanishingly small in comparison with the planar extensional viscosity.

On the planar extensional viscosity of mobile liquids

An instrument is described in which steady planar orthogonal stagnation flows of the type proposed by Winter [1] are achieved with lubrication. From pressure readings on steady results acceptable Trouton ratios are achieved for a Newtonian sample and results are obtained for a ‘Boger’ fluid showing high levels of extensional viscosity with Trouton ratios approaching 4 in the limit of zero stretch rate.

A Method for the Measurement of the Planar Extensional Viscosity

A Method for the Measurement of the Planar Extensional Viscosity