On the predictive/fitting capabilities of the advanced differential constitutive equations for branched LDPE melts

Predictive/fitting capabilities of the XPP, PTT–XPP and modified Leonov constitutive equations are compared in both steady as well as transient shear and uniaxial elongational flows using two, highly branched LDPE materials (Escorene LD165BW1 and Bralen RB0323). It has been found that even if all three tested models exhibit very good fitting capability for steady uniaxial extensional viscosity curve, their predicting capabilities may differ significantly for shear viscosity as well as first and second normal stress coefficients.     

Bagley correction: the effect of contraction angle and its prediction

The excess pressure losses due to end effects (mainly entrance) in the capillary flow of a branched polypropylene melt were studied both experimentally and theoretically. These losses were first determined experimentally as a function of the contraction angle ranging from 10° to 150°. It was found that the excess pressure loss function decreases for the same apparent shear rate with increasing contraction angle from 10° to about 45°, and consequently slightly increases from 45° up to contraction angles of 150°. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model were used to predict the end pressures. It was found that the numerical predictions do agree well with the experimental results for small contraction angles up to 30°. However, the numerical simulations under-predict the end pressure for larger contraction angles. The effects of viscoelasticity, shear, and elongation on the numerical predictions are also assessed in detail. Shear is the dominant factor controlling the overall pressure drop in flows through small contraction angles. Elongation becomes important at higher contraction angles (greater than 45°). It is demonstrated in abrupt contractions (angle of 180°) that both the entrance pressure loss and the vortex size are strongly dependent on the extensional viscosity for this branched polymer. It is suggested that such an experiment (visualisation of entrance flow) can be useful in evaluating the validity of constitutive equations and it can also be used to fitting parameters of rheological models that control the elongational viscosity.     

Sensitivity analysis of the Bagley correction to shear and extensional rheology

The excess pressure losses due to end effects in the capillary flow of two linear low-density polyethylene resins (LLDPE) were studied. These losses were first determined experimentally by using two methods: 1) by extrapolating experimental data of pressure drop versus length-to-radius ratios (L/R) to zero capillary length and 2) by means of using orifice dies (L/R≅0). Both methods resulted in practically the same end corrections. Numerical simulation was also used to model this important aspect of experimental rheology. The constitutive equations used in the simulations are a multimode K-BKZ equation, a multimode Phan-Thien/Tanner, and finally a purely viscous Carreau equation. It was found that the numerical predictions agreed qualitatively but underestimated the experimental data for the various geometries used to determine the end effects. Furthermore, it is demonstrated that the entrance pressure loss is also insensitive to extensional rheology, while it depends more strongly on the shear rheology. This finding raises doubts as to the usefulness of end pressure (known also as Bagley correction) as a method of determining the extensional viscosity of polymer melts at high rates. Received: 18 December 1997 Accepted: 21 May 1998     

Extensional rheology of concentrated poly(ethylene oxide) solutions

The shear and extensional rheology of three concentrated poly(ethylene oxide) solutions is examined. Shear theology including steady shear viscosity, normal stress difference and linear viscoelastic material functions all collapse onto master curves independent of concentration and temperature. Extensional flow experiments are performed in fiber spinning and opposed nozzles geometries. The concentration dependence of extensional behavior measured using both techniques is presented. The zero-shear viscosity and apparent extensional viscosities measured with both extensional rheometers exhibit a power law dependence with polymer concentration. Strain hardening in the fiber spinning device is found to be of similar magnitude for all test fluids, irrespective of strain rate. The opposed nozzle device measures an apparent extensional viscosity which is one order of magnitude smaller than the value determined with the fiber spinline device. This could be attributed to errors caused by shear, dynamic pressure, and the relatively small strains developed in the opposed nozzle device. This instrument cannot measure local kinematics or stresses, but averages these values over the non-homogenous flow field. These results show that it is not possible to measure the extensional viscosity of non-Newtonian and shear thinning fluids with this device. Fiber spin-line experiments are coupled with a momentum balance and constitutive model to predict stress growth and diameter profiles. A one-mode Giesekus model accurately captures the plateau values of steady and dynamic shear properties, but fails to capture the gradual shear thinning of viscosity. Giesekus model parameters determined from shear rheology are not capable of quantitatively predicting fiber spinline kinematics. However, model parameters fit to a single spinline experiment accurately predict stress growth behavior for different applied spinline tensions.     

Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

Purely elastic flow asymmetries in flow-focusing devices

The flow of a viscoelastic fluid through a microfluidic flow-focusing device is investigated numerically with a finite-volume code using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models. The conceived device is shaped much like a conventional planar “cross-slot” except for comprising three inlets and one exit arm. Strong viscoelastic effects are observed as a consequence of the high deformation rates. In fact, purely elastic instabilities that are entirely absent in the corresponding Newtonian fluid flow are seen to occur as the Deborah number (De) is increased above a critical threshold. From two-dimensional numerical simulations we are able to distinguish two types of instability, one in which the flow becomes asymmetric but remains steady, and a subsequent instability at higher De in which the flow becomes unsteady, oscillating in time. For the UCM model, the effects of the geometric parameters of the device (e.g. the relative width of the entrance branches, WR) and of the ratio of inlet average velocities (VR) on the onset of asymmetry are systematically examined. We observe that for high velocity ratios, the critical Deborah number is independent of VR (e.g. De_c ≈ 0.33 for WR = 1), but depends non-monotonically on the relative width of the entrance branches. Using the PTT model we are able to demonstrate that the extensional viscosity and the corresponding very large stresses are decisive for the onset of the steady-flow asymmetry.     

An approximate analysis for contraction and converging flows

An approximate analysis is presented for the flow of fluids through planar and axisymmetric contractions. Energy principles are employed to relate the entry pressure drop to flow rate and fundamental rheometric properties. One of the aims of the analysis is to investigate the influence of extensional viscosity on such flows, particularly with regard to the occurrence and enhancement of vortex motion in the entry corners.For the sake of mathematical simplicity, independent power-law models are used to represent the shear and extensional viscosity functions. The analysis indicates that, once significant vortex motion is present, enhancement occurs whenever the Trouton ratio is an increasing function of shearrate (or stretch-rate). It is readily seen how the occurrence of vortices serves as a stress relief mechanism. Indeed, for highly stretch-thickening materials, the entry pressure drop is seen to be dominated by shear properties.The power-law parameters of the extensional viscosity function may be obtained in a straight-forward way from entry pressure drop versus flow rate data.Finally, the extension and application of the analysis to other similar flows, such as through converging nozzles, is briefly discussed.     

The motion of long bubbles through viscoelastic fluids in capillary tubes

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

Low-Reynolds-number motion of a deformable drop between two parallel plane walls

The motion of a three-dimensional deformable drop between two parallel plane walls in a low-Reynolds-number Poiseuille flow is examined using a boundary-integral algorithm that employs the Green’s function for the domain between two infinite plane walls, which incorporates the wall effects without discretization of the walls. We have developed an economical calculation scheme that allows long-time dynamical simulations, so that both transient and steady-state shapes and velocities are obtained. Results are presented for neutrally buoyant drops having various viscosity, size, deformability, and channel position. For nearly spherical drops, the decrease in translational velocity relative to the undisturbed fluid velocity at the drop center increases with drop size, proximity of the drop to one or both walls, and drop-to-medium viscosity ratio. When deformable drops are initially placed off the centerline of flow, lateral migration towards the channel center is observed, where the drops obtain steady shapes and translational velocities for subcritical capillary numbers. With increasing capillary number, the drops become more deformed and have larger steady velocities due to larger drop-to-wall clearances. Non-monotonic behavior for the lateral migration velocities with increasing viscosity ratio is observed. Simulation results for large drops with non-deformed spherical diameters exceeding the channel height are also presented.     

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.     

Birefringence measurements in extensional flow of polyisobutylene around an expanding bubble

Birefringence in liquid polymers offers the possibility of obtaining information about stress in complex flows. In this work, this is done for extensional flows of polyisobutylene in a “breathing bubble” rheometer. In this type of rheometer, a bubble consisting of an incompressible, low-viscosity fluid (usually water) is injected into the sample with a nozzle. Expanding or collapsing the bubble by adding or removing water induces biaxial or uniaxial extension in the surrounding sample. The pressure difference between the bubble and the surroundings can be measured and compared to the predictions of constitutive equations. This measurement only gives one integral value for a complex flow history. In this paper, the birefringence around the bubble is measured in order to learn more about the flow. This is done by comparing pressure and birefringence results to those of standard constitutive equations for a polyisobutylene sample. A good agreement between the pressure and optical measurements and the theory is found with a single value of the stress-optical constant. Received: 25 June 1997 Accepted: 12 November 1997     

Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems

This study extends our previous analysis on pressure-drops for strain-hardening Boger-type fluids in contraction flow settings, into those fluids that manifest mild shear-thinning properties. Numerical simulations are compared and contrasted for a variety of constitutive equations, categorised through their differences in viscometric functional response, considering application on 4:1:4 contraction-expansion flow and 2:1 flow past a sphere. Here, prior results on pressure-drop enhancement for constant shear-viscosity fluids have revealed the counter-influences of first normal stress differences and extensional viscosity. The present comparative work advances this study by selectively including the effects of shear-thinning. Suitable models to accomplish this are chosen from the class of Phan-Thien/Tanner (PTT) models, with cross-reference to FENE-models and Oldroyd-B. Furthermore, the work explores the falling sphere problem with comparison of the drag coefficient factor for various implementations. The numerical computations are performed by appealing to a well-founded hybrid finite element/finite volume algorithm, using structured triangular meshing, semi-implicit time-stepping and subcell technology. The cell-vertex finite volume scheme is particularly suited to the solution of the stress subsystem, and invokes fluctuation-distribution for upwinding and median-dual-cells for source-term representation.     

A note on the extensional viscosity of elastic liquids under strong flows

In this article a parametric study based on a balance between viscous drag and restoring Brownian forces is used in order to construct a nonlinear dumbbell model with a finite spring and a drag correction for a dilute polymer solution. The constitutive equations used are reasonable approximation for describing flows of very dilute polymer solutions such as those used in turbulent drag reduction. We investigate the response of an elastic liquid under extensional flows in order to explore the roles of a stress anisotropy and of elasticity in strong flows. It is found that for low Reynolds numbers, the extensional viscosity of a dilute polymer solution is governed by two parameters: a Deborah number representing the importance of the elasticity on the flow and the macromolecule extensibility that accounts for the viscous anisotropic effects caused by the macromolecule orientation. Two different asymptotic regimes are described.The first corresponds to an elastic limit in which the extensional viscosity is a function of the Deborah number and the particle volume fraction. The second is an anisotropic regime with the extensional viscosity independent of Deborah number but strongly dependent on macromolecule aspect ratio. The analysis may explain from a phenomenological point of view why few ppms of macromolecules of high molecule weight or a small volume fraction of long fibres produce important attenuation of the pressure drop in turbulent flows. On the basis of our analysis it is seen that the anisotropic limit of the extensional viscosity caused by extended polymers under strong flows should play a key role in the attenuation of flow instability and in the mechanism of drag reduction by polymer additives.     

The Giesekus fluid in ω–D form for steady two-dimensional flows

Using the fact that for simple fluids the most general constitutive equation in constant stretch history flows for the extra stress tensor τ is known in an explicit form, the Giesekus fluid model is cast into this (ω–D) form for two-dimensional flows. The three material functions needed to characterize τ are listed. The explicit results for simple shear and planar elongation reveal that the parameter α should be restricted to values less than 0.5. It is demonstrated that in this explicit form the constitutive equation is free from thermodynamic objections and can thus be used as a starting point for numerical calculations of general, but steady, two-dimensional flows. Received: 9 November 1998 Accepted: 20 May 1999     

Dynamics of worm-like micelles: the Cox–Merz rule

The viscoelastic behaviour of worm-like micelles in small-amplitude oscillatory, steady simple shear and uniaxial extensional flows are analyzed with a model that couples the Oldroyd-B constitutive equation with a kinetic equation that accounts for the structural changes induced by the flow. In some cases, the constitutive equation predicts a viscoelastic behaviour that is consistent with the Cox–Merz rule. Departures from this rule are also predicted. Experimental data obtained for two worm-like micellar systems indicate that in these solutions, the Cox–Merz rule is not usually followed, in agreement with the predictions of our model. In uniaxial extensional flow, the model predicts a strain hardening in the extensional viscosity at low extensional rates and a strain-thinning at high extensional rates.     

Numerical simulations of granular free-surface flows using smoothed particle hydrodynamics

This paper presents a two-dimensional SPH model designed to simulate free-surface flows of dense granular materials. Smoothed particle hydrodynamics (SPH) is a mesh-free numerical method based on a Lagrangian discretization of the continuum mass and momentum conservation equations. The rheology of dense granular materials is modelled using a new local constitutive law recently proposed by Jop et al. (Nature, 2006). Of the viscoplastic class, this law is characterized by an apparent viscosity depending both on the local strain rate and local pressure. Validation test cases of the model in steady and unsteady configurations are presented. For steady cases (vertical chute flow and uniform free-surface layers on inclines), excellent agreement with analytical predictions is obtained. In the unsteady case, the simulations satisfactorily capture the dynamics of gravity-driven surges observed in experiments, including behaviours that are very specific to granular materials. Among the various parameters involved in the computations, the influence of SPH particle configuration within the flow and of the threshold viscosity used in the regularization of the constitutive yield criterion are particularly discussed.     

Approximate elongation flow properties utilising the opposed orifice technique – Correction for shear and inertia

Mackay et al. (1995) have presented an approximate technique to determine the elongation viscosity from pressure drop measurements in a simple stagnation flow device. In the present paper we describe experiments using a high viscosity Newtonian oil, aimed at probing some of the assumptions made by Mackay et al. We find that Trouton ratios calculated using the original analysis are well above the value of three expected for Newtonian fluids. Finite element simulations of the flow field show this is due to the net pressure drop having a substantial shear contribution, which should be corrected for before the Trouton ratios are evaluated. Interestingly, most of the shear correction is due to shear on the inside of the orifice near the exit from the central flow region. The shear contribution to the pressure drop occurs for all flow rates, however, at large flow rates there is also an inertial correction to the pressure drop. In this paper we describe an approximate method that corrects for both shear and inertial effects. With these effects recognised and corrected for, the measured Trouton ratios are reduced to around three. Received: 15 December 1997 Accepted: 16 March 1998     

Experimental and numerical study of entry flow of low-density polyethylene melts

The present work deals with experimental and numerical features of entry flows of two polyethylene melts, namely a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE) in an axisymmetric converging geometry. The study also involves rheological characterization of the polymers and determination of flow parameters, at 160°C. For both fluids, the data are fed into a viscoelastic integral Wagner constitutive equation. The numerical flow simulations are performed by using a stream-tube mapping analysis. Consideration of a sub-domain of the total flow domain, the peripheral stream tube, close to the wall of the converging duct permits to relate the results of the numerical simulation to experimental flow characteristics as total and entrance pressure drops. The agreement is good for the total pressure losses, but, concerning LDPE, a lack of consistency remains for the entrance pressure drop.