Planar Extensional Flow Research Articles

Normal Stress Ratio Predicted by Viscoelastic Constitutive Equations.

The first and second normal stress differences, N 1 and N2 in steady shear flow are calculated using differential constitutive equations proposed by Leonov and Giesekus. At low shear rates, the Leonov model gives -N2/ N1=0.25 for both single and multiple relaxation modes. In the Giesekus model, −N2/ N1 increases with increasing anisotropy mobility parameter α. Both models predict that −N2/N1 is a decreasing function of the shear rate at high shear rates. The shear rate dependence of −N2/N1 becomes weaker with increasing width of relaxation time distribution. The BKZ type integral constitutive equation is employed to investigate the effect of a model parameter b (=N2/N 1) on steady planar, uniaxial and biaxial extensional flows. It is found that the strain rate dependences of planar2 and biaxial extensional viscosities are very sensitive to the parameter b, where 2 in planar extension denotes the direction of constant width.

Visualization and microscopic modeling of phase inversion during compounding

AbstractA detailed description of the sequence of deformation steps leading to phase inversion during compounding in a low‐viscosity‐ratio co‐polyester/polyethylene blend is presented. Visualization using a glass window and sampling of the blend at different mixing times enabled identification of the intermediate morphologies of the major component en route to phase inversion. Based on these observations, a theoretical model is developed to predict the time to phase inversion. The model incorporates a simplified flow‐field approximation and the calculation of strain imparted to the major component domains. A strain‐based criterion for phase inversion is then proposed, which, in conjunction with the model, yields an explicit expression for the time to phase inversion during compounding, tP.I.. The model predictions are seen to be in good agreement with the increase of tP.I., on scaleup between two mixing bowls. The correct functional dependence of tP.I. on the nominal maximum‐shear‐rate is predicted. Using combination of pure drag and planar extensional flow, the model predictions are shown to be consistent with the observed dependence of tP.I. on the volume fraction of the minor component and the blend viscosity ratio.

Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid

This paper is an experimental investigation of the deformation and relaxation of a Newtonian drop suspended in a PIB/PB Boger fluid. The suspending fluid is undergoing a planar extensional flow produced in a four-roll mill. We show that increasing elasticity of the suspending fluid has a pronounced effect on both the deformation and relaxation of a drop. For steady flows, as the strength of viscoelastic effects in the suspending fluid is increased, the drops become more deformed, with ends that are generally more pointed. This leads to a decrease in the maximum (“critical”) capillary number for the existence of a steady, deformed drop shape. In transient startup and step flows, the elasticity of the suspending fluid produces a large deformation shape that is more pointed at its ends and more tubular in its midsection than is observed for a drop in a Newtonian fluid (bulbous ends with necking at the waist). This enables a drop in the PIB/PB suspending fluid to be extended to a longer length without breaking upon flow cessation. However, at smaller deformations, the elasticity of the suspending fluid retards the relaxation of the drop. The observed viscoelastic effects on the steady and transient deformation, as well as the relaxation of drops in the PIB/PB suspending fluid, cannot be explained by viscoelastic modifications of the global, undisturbed flow field. Instead, our results suggest the existence of a non-linear coupling between the drop shape, the local disturbance flow, and the polymer configuration in the vicinity of the drop. This coupling enhances elastic effects, such that a drop can display significant non-Newtonian behavior prior to any changes in the global, undisturbed flow field.

An experimental and simulation study of dilute polymer solutions in exponential shear flow: Comparison to uniaxial and planar extensional flows

The rheological properties of dilute polymer solutions in exponential shear, uniaxial, and planar extensional flows are compared using Brownian dynamics simulations of both freely draining, flexible bead-rod and bead-spring chains. We introduce a novel stress function for exponential shear which uses the extinction angle χ to take into account the orientation of the chain as it aligns in the flow. Comparing this new stress function during startup and relaxation in exponential shear with τ11−τ22 in planar extensional flow and τ11−12(τ22+τ33) in uniaxial extensional flow, we find that for both models there is a quantitative agreement among the three different flows over a large range of Wi, strain, and chain length. Furthermore, the distributions of maximum extension show a microstructural equivalence between ensembles of chains in all three flows up to strains of 3 or 4 at all values of Wi simulated. Finally, we show three comparisons between experiment and simulation of the various flows: (a) simulations ...

Rheological Properties of Dough During Mechanical Dough Development

During mechanical development dough is subjected to both shear and extensional deformations. Thus, it is expected that both flow conditions contribute to the development of dough. In order to monitor rheological changes, occurring during mixing, shear and extensional properties of dough prepared with two flours of different strength and various levels of mixing energy were determined using fundamental rheological methods. Rheological measurements included: small deformation and large deformation (shear test), planar extensional flow and a combined shear/extensional flow test, namely extrusion test. Results obtained in this research showed that, during mixing, dough develops with an increase in both apparent shear and extensional viscosities. For all the tests, plots of the measured rheological properties as a function of the mixing energy resembled typical mixing curves. This indicated that the increase in the power drawn by the mixer motor is due to the increase in both apparent shear and extensional viscosities. After peak dough development these properties decreased synchronously with the mixing curves. Results from small deformation shear tests exhibited large variability, particularly when non-mixed and underdeveloped doughs were tested. This variability was associated with poor water distribution in the sample due to insufficient mixing. Results of large deformation tests, including shear, planar extensional flow and the extrusion test, were less variable and showed that mixing and type of flour affect the rheological properties of dough.

Non-Newtonian Rheology of Liquid Crystalline Polymer Monolayers

A rheological study on a two-dimensional Langmuir monolayer has been conducted by measuring the optical anisotropy and the mechanical surface properties. Experiments were conducted on monolayer mixtures of a liquid crystalline polymer, poly(p-phenylene)sulfonic acid (PPPSH), with an aliphatic fatty acid, stearic acid (C18). It is found that these two molecules mix ideally, as supported by studies of average molecular area, collapse pressure, and Brewster angle microscopy. Both surface moduli and surface viscosity were obtained with an interfacial stress rheometer and showed that PPPSH/stearic acid monolayer mixtures are non-Newtonian. The orientational dynamics of polymers under well-defined planar extensional flow was considered using UV dichroism, which revealed a surface pressure-induced isotropic−nematic transition. This transition is difficult to observe using traditional isotherm analysis; however, it is readily evident in measurements of the rheology of the monolayers.

A large fluid-bridge device to measure the deformation of drops in uniaxial extensional flow fields

Uniaxial extensional flow appears in transport and processing and is able to destroy desired rheological properties of emulsions by tearing drops. During the last few decades, many theoretical approaches used to describe the drop-deformation behaviour of emulsions have involved the assumption that uniaxial extensional flow fields are involved. Recent experiments investigating drop deformation in planar extensional flow fields were restricted to single droplets in a surrounding fluid. The present device considerably improves this situation by allowing one to study the drop-matrix interaction for several drops in uniaxial extensional flow. To determine the deformation of a few drops, a uniaxial flow field with a constant rate of extension with respect to space and time is necessary. This is realized by stretching a large cylindrical liquid bridge under microgravity. The bridge should remain cylindrical throughout the stretching. Therefore, a special mechanical adaptation is used to suppress necking at the ends of the filament. However, bridge deformations caused by viscous stress and by dynamic and capillary pressure gradients still remain. The strain and shear rate distributions in the liquid bridge are evaluated in order to determine the most suitable position for drop-deformation experiments. Examples of several drop-deformation experiments with elliptical and pointed-ends shapes of the stretched bridge are presented.

Experimental trajectories of two drops in planar extensional flow

In this paper we map the experimental trajectories of two deformable drops in planar extensional flow and compare the experimental results with theoretical calculations for spherical drops. We examine the effects that deformation, initial position, and viscosity ratio have on the interaction of two drops and the necessity of incorporating deformation into trajectory calculations, which can be used to estimate the collision rates, the collision efficiencies, and the collision interaction times. For drops which do not come into close contact, the existing theoretical calculations for spherical drops accurately predict the symmetric trajectories and capture the increased hydrodynamic interaction for higher viscosity ratios. For drops which come into close contact, the spherical drop theory accurately predicts the approach and exit trajectories and with a slight empirical modification adequately predicts the interaction times for deformable drops with a Taylor deformation parameter up to 0.22. The experimental results show that for drops with close contact, the collision trajectories are asymmetric and irreversible with a minimum separation between the centers of mass that is less than the minimum separation of two spheres. This minimum separation corresponds to the minor axis of the deformed drop and is not captured by the spherical theory. However, overall, the modified trajectory theory based upon the hydrodynamic mobility for spherical drops does provide a reasonable estimate for the trajectories and the interaction times for two deformable drops in planar extensional flow.

Surfactant and viscoelastic effects on drop deformation in 2‐D extensional flow

AbstractThe deformation and breakup of polymeric drops suspended in an immiscible Newtonian fluid are examined for planar extensional flow in a computer controlled four roll mill are examined. Previous work from this lab with aqueous polymeric drops showed an increase in the degree of deformation compared to Newtonian drops, and two distinct breakup mechanisms, tip streaming, and tip streaming with stretch. This led to the conclusion that breakup for non‐Newtonian drops occurred at a lower capillary number and by a distinctly different mechanism than for a Newtonian drop with a similar ratio of internal to external viscosity. This conclusion was puzzling, because numerical computations with the Chilcott‐Rallison model showed that the influence of viscoelasticity for that model was to decrease the degree of deformation rather than increasing it. In this work, the results for aqueous polymer solutions are compared with results for ethylene glycol polymer solutions. The results suggest that tip streaming and tip streaming with stretch in the original experiments resulted from an undetected surfactant effect, which acts to produce a more pointed drop shape that destabilizes the drop to tip streaming. Viscoelasticity, on the other hand, acts to decrease the deformation of polymeric drops at a given capillary number and increases the critical capillary number, as predicted numerically, when compared to Newtonian drops with an equivalent viscosity ratio.

Flow Light Scattering Studies of Polymer Coil Conformation in Solutions in Extensional Flow

The dynamics of isolated polymer chains in dilute solution under steady, extensional flow were investigated by means of flow light scattering. Both the orientation and deformation of the chains during flow were determined by analyzing the angular dependence of the scattered light intensity. The extensional flow field was imparted on the polymeric fluids by a stagnation point flow in the center of a four-roll mill apparatus. The fluids studied were nearly monodisperse high molecular weight polystyrenes (HMPS) of various molecular weights dissolved in either the viscous solvent dioctyl phthalate (DOP) or a mixed solvent of low molecular weight polystyrene (LMPS) and dioctyl phthalate. The flow field in a Newtonian fluid of seeded glycerin was also examined by flow dynamic light scattering techniques and flow visualization to confirm the extensional flow field. Flow dynamic light scattering and flow visualization results verified that the field generated by the four-roll mill was a reasonable approximation of planar extensional flow, under the conditions investigated. Flow (static) light scattering results confirmed that the polymer chains aligned completely with the stretch direction, as expected for this type of flow. The deformation of the chains, however, was significantly less than predicted by elastic dumbbell models. Also studied were the effects of molecular weight and solvent quality on the deformation of the polymer chains. Comparisons are drawn between polystyrene chains in the strong extensional flow field studied here and a steady shearing flow examined in a previous publication as well as with other studies in extensional flows.

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.

Transient flow-induced crystallization of a polyethylene melt

Extensional, flow-induced crystallization (FIC) of a high-density polyethylene (HDPE) melt has been studied using a four-roll mill flow cell. Simultaneous measurement of the birefringence and scattering dichroism are used to quantify the crystallization process during and following transient flow deformation in planar extensional flow. Suspension of the HDPE phase as a droplet in a linear low-density polyethylene carrier phase prevents die blockage on crystallization and allows measurement of the flow kinematics. Initial crystallization rates following a transient flow deformation show a stress-strain dependence. Crystallization induction times during flow correlate with the extension rate during the transient flow deformation. Measurement of the HDPE melt steady and oscillatory flow rheological properties, along with measurements of time constants following step-shear and extensional strains, allow determination of the viscoelastic properties which enhance FIC. Parameters obtained from these experiments are used in a phenomenological model for FIC which allows qualitative and semiquantitative analysis of the data trends, particularly the relaxation behavior of the birefringence during flow cessation/crystallization. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 64: 2165–2176, 1997

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger. A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.

Observations of fiber orientation in suspensions subjected to planar extensional flows

Fibers in a planar extensional flow tend to align parallel to the axis of extension, with a small orientational dispersion resulting from hydrodynamic, fiber–fiber interactions. We have visualized the orientation of an opaque, tracer fiber in the midst of an index of refraction and density matched fiber suspension in a four-roll mill and determined the mean squares of the components of the fiber orientation in the compressional and neutral directions. The observed mean-square orientations in suspensions in Newtonian fluids with nL3=5–17 are qualitatively consistent with the dilute limit of the theory of Shaqfeh and Koch [Phys. Fluids A 31, 728 (1988)]. Here n is the number of fibers per unit volume and L is the fiber length. It was not possible to perform the experiment for higher fiber concentrations, because of difficulties in keeping the tracer fiber at the stagnation point, which were attributed to translational, hydrodynamic diffusion. Measurements of orientational dispersion in polyacrylamide solutions indicated that the non-Newtonian stresses increase fiber alignment, as predicted by Harlen and Koch [Phys Fluids A 4, 5 (1992)].

Computer simulation of dynamics and morphology of discotic mesophases in extensional flows

Abstract A previously presented model is used to simulate the dynamics and microstructure of spatially invariant uniaxial discotic nematic liquid crystals in isothermal, incompressible, irrotational, extensional (shear-free or elongational) flows. Numerical and analytical solutions of the director n and alignment S are presented for given uniaxial extensional, equi-biaxial extensional and planar extensional start-up flows. The unit sphere description of the director is used to discuss and analyse the sensitivity of the director trajectories and the alignment relaxation to the initial conditions (n o, S o), to the alignment Deborah number (De), and to the type of flow. The numerical results are used to characterize the relaxation of the tensor order parameter Q and to compute the steady flow birefringence. The various flows are classified according to their orienting strength and alignment strength, and according to whether they generate geodesic (shortest path) director orbits. Equi-biaxial extensional and planar extensional flows are found to be strongly orienting and strongly aligning flows, while uniaxial extensional flow is a weakly orienting and weakly aligning flow. The number of strain units required to achieve steady state are shown to depend on whether the flow is geodesic (uniaxial extensional and equi-biaxial extensional flows) or not (planar extensional flow).

Strain-coupling effects in steady planar extension

This study is concerned with an evaluation of the capability of the strain-coupling model to predict steady shear and steady planar extensional flows. A comparison is made between the predictions of the K-BKZ and strain-coupling theories.

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.

The extensional viscosity and effective thermal conductivity of a dispersion of aligned disks

This paper discusses the properties of a semidilute suspension of disks, for which nl3≫1 and φ≪1, where n is the number of disks per unit volume, l is their large dimension, and φ is their volume fraction. The effective conductivity of a dispersion of aligned, highly conducting disks is shown to be O[k(nl3)2], where k is the conductivity of the matrix. The extensional viscosity is shown to be O[μ(nl3)2], where μ is the viscosity of the fluid. In addition, similar scaling results are shown to hold for the case of a semidilute suspension of aligned, two-dimensional slabs which are of infinite extent in the direction perpendicular to their plane of cross section. Specifically, for nl2≫1, the effective conductivity and the extensional viscosity are shown to be O[k(nl2)2] and O[μ(nl2)2] respectively, where n is now the number of slabs per unit area and l is the width of the slab. Planar extensional flow simulations of a periodic array of aligned slabs confirm the quadratic scaling for stress in the semidilute regime. The simulations also show the crossover from a linear dependence of the stress on particle concentration in the dilute regime to the quadratic, semidilute scaling.

Assessment of nonlinear strain measures for extensional and shearing flows of polymer melts

From stress-strain experiments in extensional and shearing flows, nonlinear strain measures and effective damping functions are derived for a polyisobutylene melt. The strain measures determined in planar extensional flow and in simple shear flow coincide. Experimental results are compared with predictions of two molecular theories, the Doi-Edwards model and the molecular stress function approach of Wagner and Schaeffer. Discrepancies between theories and experiment lead to a reconsideration of the classification of extensional flows. The symmetry of the flow field is identified and quantified as an important parameter influencing the strain measure, and a unifying strain measure for general extensional and shearing flows of polymer melts is presented.

On the use of stress growth data to determine strain-dependent material functions for factorable K-BKZ equations

The procedure of using experimental stress growth data to determine strain-dependent, or damping, functions for factorable K-BKZ constitutive equations is examined. Damping functions obtained using stress growth data from simple shear and planar extensional flows in conjunction with integrated forms of factored K-BKZ models are compared with the damping function obtained from shear step-strain experiments. Results for the low-density polyethylene (LDPE) known as IUPAC X are presented that demonstrate the difficulties encountered with this method, which can be quite serious for planar extensional flow. These results appear to be consistent with previous studies on several polymer melts where uniaxial extension stress growth data were used to determine damping functions. This body of evidence raises important questions about the use of this procedure and the strain-dependent and, apparently, test-dependent functions that are obtained. An explanation for these results is proposed which suggests that these observations may be due to an intrinsic shortcoming of K-BKZ theories.