Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

Biaxial orientation of polymers by solid state extrusion through convergent-divergent dies

Solid-state extrusion of crystalline thermoplastics is a well known method for the production of monoaxially oriented filaments exhibiting very high modulus and strength. This is achieved primarily by forcing the polymer through a converging conical die at temperatures below its melting point.In the present study the polymer is subjected to deformations simultaneously along the longitudinal and transverse directions by using dies featuring converging and diverging walls, perpendicular to each other, to produce flat extrudates exhibiting biaxial orientation. Two geometries are examined to determine the factors controlling the relative magnitude of the orientation in the extrusion and transverse directions respectively, which is being assessed by measuring the birefringence and tensile strength in various sections of the extrudates.Experiments have been carried out over a wide range of temperatures on billets of PTFE and UHMWPE, using dies mounted on a compression testing apparatus to measure the extrusion forces.The mechanics of the converging-diverging flow has been analysed to calculate the extrusion forces as function of the yield strength and coefficient of friction of the polymer and to establish the relationship between type of orientation in extruded products and die geometry.     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Three-dimensional study of extrusion processes by Boundary Element Method.

This paper describes an implementation of a Boundary Element method to solve a general three-dimensional viscoelastic flow problem. The Boundary Element method is formulated in terms of unknown boundary velocity and traction fields. The fluid is incompressible and is modelled by a differential constitutive equation. The steady-state stress field is obtained by a time marching process of integration. For the first time, some results for steady state isothermal creeping flow extrusion of a viscoelastic fluid from triangular and square dies are described. The concept of an axisymmetric-equivalent swell ratio is introduced to compare the present results with the results of axisymmetric extrusion studies reported in the literature. It is shown that reasonable agreement is achieved.     

Die swell measurements of second‐order fluids: numerical experiments

An analysis of the flow of a second‐order fluid is presented. Reference values for some variables are defined, and with these a non‐dimensional formulation of the governing equations. From this formulation, three dimensionless numbers appear; one is the Reynolds number, and two numbers that are called the first‐ and second‐dimensionless normal stress (NSD) coefficients. The equations of motion are solved by a finite element method using a commercially available program (Fidap), and the steady state converged solution was used to measure the die swell. The factors that influence die swell and that are studied in this work include: the die geometry for circular cross sectional dies, including tubular, converging, diverging, half‐converging/half‐tubular shapes; fluid characteristics such as Reynolds number and first‐ and second‐DNS coefficients (both positive and negative values); and flow rates, as determined by the maximum velocity in a parabolic velocity profile at the entrance to the die. The results suggest that shear and deformation histories of the fluid directly influence not only swell characteristics, but also convergence characteristics of the numerical simulation. © 1999 John Wiley & Sons, Ltd.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

Large-scale spanwise periodicity in a turbulent boundary layer induced by highly ordered and directional surface roughness

The effect of converging–diverging riblet-type surface roughness (riblets arranged in a ‘herringbone’ pattern) are investigated experimentally in a zero pressure gradient turbulent boundary layer. For this initial parametric investigation three different parameters of the surface roughness are analysed in detail; the converging–diverging riblet yaw angle α, the streamwise fetch or development length over the rough surface F_x and the viscous-scaled riblet height h⁺. It is observed that this highly directional surface roughness pattern induces a large-scale spanwise periodicity onto the boundary layer, resulting in a pronounced spanwise modification of the boundary layer thickness. Hot-wire measurements reveal that above the diverging region, the local mean velocity increases while the turbulent intensity decreases, resulting in a thinner overall boundary layer thickness in these locations. The opposite situation occurs over the converging region, where the local mean velocity is decreased and the turbulent intensity increases, producing a locally thicker boundary layer. Increasing the converging–diverging angle or the viscous-scaled riblet height results in stronger spanwise perturbations. For the strongest convergent–divergent angle, the spanwise variation of the boundary layer thickness between the diverging and converging region is almost a factor of two. Such a large variation is remarkable considering that the riblet height is only 1% of the unperturbed boundary layer thickness. Increasing the fetch seems to cause the perturbations to grow further from the surface, while the overall strength of the induced high and low speed regions remain relatively unaltered. Further analysis of the pre-multiplied energy spectra suggests that the surface roughness has modified or redistributed the largest scale energetic structures.     

Use of diverging or converging arrangement of plates for the control of chaotic mixing in symmetric sinusoidal plate channels

This paper presents an experimental flow visualization study of the effect that the variation of converging or diverging angles plays in the flow field of a symmetric sinusoidal channel. The experiments were performed in a water tunnel and the visualization technique was laser illumination of seeded particles whose traces were captured using long time exposure photography. Geometrical parameters such as wave amplitude, wavelength and distance between plates were kept constant, while the Reynolds number and divergence or convergence angles were varied. It was found from the experiments that the divergence of the plates is a good way to promote chaotic mixing in channel flows, as the flow becomes more unstable for diverging channels. For the case of converging channels, the flow becomes very stable even for large values of the Reynolds number. These results were compared with those of a channel formed by a pair of sinusoidal parallel plates.     

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Péclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.     

A gas jet superposition model for CFD modeling of group-hole nozzle sprays

In the present study, a jet superposition modeling approach is explored to model group-hole nozzle sprays, in which multiple spray jets interact with each other. An equation to estimate the merged jet velocity from each of the individual jets was derived based on momentum conservation for equivalent gas jets. Diverging and converging group-hole nozzles were also considered. The model was implemented as a sub-grid-scale submodel in a Lagrangian Drop–Eulerian Gas CFD model for spray predictions. Spray tip penetration predicted using the present superposition model was validated against experimental results for parallel, diverging and converging group-hole nozzles as a function of the angle between the two holes at various injection and ambient pressures. The results show that spray tip penetration decreases as the group hole diverging or converging angle increases. However, the spray penetration of the converging group-hole nozzle arrangement is more sensitive to the angle between the two holes compared to diverging nozzle because the radial momentum component is converted to axial momentum during the jet–jet impingement process in the converging group-hole nozzle case. The modeling results also indicate that for converging group-hole nozzles the merged sprays become ellipsoidal in cross-section far downstream of the nozzle exit with larger converging angles, indicating increased air entrainment.     

Stress amplification and plastic flow for spheroidal shells: Applications to myopia

Amplification of effective stress and plastic strain rates after yielding are derived for the anisotropic stress fields of the prolate and oblate spheroidal shells as models of the myopic and hyperopic eyes. Dimensionless closed-form results are presented for arbitrary axis ratio with both constant shell thickness and variable shell thickness, using the constant scleral mass assumption. The results show that the myopic and hyperopic eyes are susceptible to failure by plastic yielding at the equator and pole, respectively, for high intraocular pressures. Experimental data from the equatorial zone of rabbit sclera shows scleral yielding and plastic flow for intraocular pressures greater than 32 mm Hg. Practical applications include glaucoma and pathological myopia.     

Shear and extensional flow of polyacrylamide solutions

As part of an EEC Science Stimulation programme on extensional viscosity two major conferences were organised on the subject. The second of these was devoted to the results obtained on a standard fluid, M 1. The data obtained in shear flow was remarkably consistent from laboratory to laboratory. Extensional flow results presented quite a different picture. Using a series of nonequilibrium techniques such as the spinline rheometer, opposing jet, falling drop and converging flow, extensional viscosity results were obtained which differed by as much as two to three orders of magnitude. Nevertheless, it was apparent that consistancy did exist between similar techniques. It is in the context of this information that the measurements described below have been made.The shear and extensional flow properties of partially ionised polyacrylamide in solution at concentrations ranging from 5 ppm were measured. The method of solution preparation was found to have a profound effect on the behaviour of the solutions in shear flow. The influence of salt concentration and pH was investigated and is discussed in the context of molecular shape in solution.Extensional flow measurements, using the spinline rheometer, show that the solutions are strongly strain thickening even at concentrations as low as 5 ppm. These results are considered in the light of polymer entanglement and association in the strong flow field.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Extensional flow resistance of dilute polyacrylamide and surfactant solutions

The extensional flow behaviour of dilute aqueous solutions of a partiallyhy-drolyzed polyacrylamide and a surfactant were investigated in an extensional flow cell. The cell was designed such that fluids were subjected to steady shear before undergoing extensional motion in a converging channel. Extensional resistance was monitored by measuring the pressure drop through the channel. Such measurements were made over a range of extensional rates at fixed values of the upstream shear rate. Solutions of different concentrations were tested — up to 40 ppm of polyacrylamide and 450 ppm of surfactant — at various temperatures in the case of surfactant and for different types and amounts of salt in the case of polyacrylamide. Of the results, the more notable are that the extensional resistance of polyacrylamide solutions is affected much more by CaCl₂ than by NaCl and that surfactant solutions do not exhibit extensional resistance unless they are pre-sheared.     

Hydrodynamisch induzierte Eigenbewegung von idealisierten Teilchen in newtonschen Suspensionsmedien unter Berücksichtigung begrenzender Wände.

Zusammenfassung Die hydrodynamisch induzierte Bewegung starrer und elastischer Teilchen, die in einem newtonschen Medium suspendiert sind, wird unter Berücksichtigung gekrümmter Begrenzungen untersucht. Die Geometrie eines hyperbolischen Zylinders erlaubt die Simulation einer Strömung durch poröse Medien und in den Grenzfällen durch ein Rohr und ein kreisförmiges Loch. Der Einfluß des inhomogenen Strömungsfeldes verursacht eine Migrationsbewegung der suspendierten Teilchen quer zu den Stromlinien. Es wird gezeigt, daß eine inhomogene Suspension elastischer Teilchen nach Durchlauf eines konvergenten und eines divergenten Bereiches eine Entmischung erfährt. Die Konzentration der Teilchen nimmt sowohl im Kernbereich der Strömung als auch aufgrund des abstoßenden Wandeffektes in unmittelbarer Wandnähe ab.

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The flow-induced migration of rigid and elastic particles suspended in a Newtonian medium is studied. The flow situation investigated consists of pressuredriven flow through a hyperbolic cylinder. This permits the simulation of flow through a porous medium and in the limiting cases through a tube and through a circular hole. The influence of the nonhomogeneous flow field effects a migration of the suspended particles across the streamlines. It is deduced that nonhomogeneous suspensions of elastic particles receive a separation according to size after passing a converging and diverging section. The results imply that the particle concentration decreases in the centre of the flow as well as in the immediate neighbourhood of the walls.

Zweiter Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation     

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Finite difference solutions have been obtained by the perturbation method to investigate the influence of shear thinning and elasticity on the flow around an inclined circular cylinder of finite length in a uniform flow. In this numerical analysis a generalized upper-convected Maxwell model, in which the viscosity changes according to the Cross model, has been used.The local flow over the cylinder is only slightly deflected. However, in the wake flow behind the cylinder the particle path is remarkably influenced by the axial flow and rapidly flows up parallel to the cylinder's axis. Then it gradually rejoins direction of the incoming flow. It is found that viscoelastic fluids are prone to flow axially in the vicinity of the cylinder. The numerical predictions generally agree with the flow visualization results.The numerical solutions also demonstrate that elasticity has a strong effect on the velocity profile especially around both ends of the cylinder; elasticity increases the asymmetric profiles of both circumferential velocity and axial velocity with respect to equal to 90° and decreases a difference in the circumferential velocity between the windward end and the leeward end.For non-Newtonian fluids, the length of the wake flow is influenced by not only the Reynolds number but also the cylinder diameter and it is larger for the cylinder with the smaller diameter at the same Reynolds number.Partly presented at the 9th Australasian Fluid Mechanics Conference, University of Auckland, New Zealand, 8–12 December, 1986     

The motion of rodlike particles in the pressure-driven flow between two flat plates

The motion of freely suspended rodlike particles has been observed in the pressure-driven flow between the two flat plates of a Hele Shaw flow cell at low Reynolds numbers. Data are reported for rodlike particles with aspect ratios of 12.0 suspended in a Newtonian fluid for gap thickness to particle length ratios of 3, 6, and 20; and for rodlike particles with aspect ratios between 5 and 8 in a non-Newtonian fluid (79.25 wt.% water, 20.2 wt.% glycerine, and 0.55 wt.% polyacrylamide). For the Newtonian fluid, the time-dependent orientation of the particles near and far from walls was shown to be in quantitative agreement with Jeffery's theory for ellipsoids suspended in a simple shear flow if an effective aspect ratio is calculated from the experimental period of rotation. Particles aligned with the flow direction and less than a particle half-length from a wall interacted irreversibly with the wall. For the non-Newtonian fluid, the timedependent orientation far from a wall was shown to be in qualitative agreement with Leal's theory for a second-order fluid; however, particles that were aligned with the flow direction and were near walls did not rotate.     

On the cone and plate deformation of a rubber-like material: Instability

In this report it is shown that the place boundary-value problem, for a small deformation superimposed on the large cone-and-plate deformation of a Mooney-Rivlin material, has no unique solution at some displacement angles of the cone. The kinematics of the small deformation are chosen such that the problem is reduced to a set of ordinary differential equations. With respect to this restricted class of kinematics the cone-and-plate deformation is unstable.     

Instabilities in multi-hole converging flow of viscoelastic fluids

Studies of the onset of instabilities were conducted on single hole and multi-hole contractions using laser speckle visualization. A well characterized elastic fluid was used with constant viscosity of 13.1 Pa · s and elasticity characterized by a longest relaxation time constant of 2.233 s. The onset of instabilities was characterized in terms of the Deborah number and the contraction ratio. Three types of instabilities were observed: pulsing vortices, azimuthally rotating vortices, and swirling vortices. For the single hole contractions the critical Deborah number for instability increased from 4.4 to 5.07 to 5.25 as the contraction ratio increased from 4: 1 to 8: 1 to 12: 1. The magnitude of the instabilities was much greater for the 4: 1 contraction than for the other two contraction ratios. For the multi-hole contraction a square array of nine holes was used and the ratio of the hole diameter to hole spacing was varied. The height of the vortices is very similar for the single hole and multi-hole contractions at low Deborah numbers. At high Deborah numbers the effect of adjacent holes is to reduce the height of the vortices by a factor of three. For the 4: 1 spacing no secondary vortex was observed below a Deborah number of De = 3.7. Secondary vortices occurred for the 8:1 and 10:1 spacing at all Deborah numbers. Unstable pulsing vortices appeared for all spacings at a critical Deborah number around 5.5. Adjacent holes decreased the strength of the unsteady vortex motions. The centerline velocities were measured for the multi-hole contraction at shear rates of 5, 30, and 300 s^–1. The elongational strain rates are similar at a low shear rate of 5 s^–1. As shear rate is increased the onset of stretching occurs closer to the plane of the contraction for the smaller contraction ratios.     

Effect of secondary zone dimensions on melt fracture manifestations of high density polyethylene

An experimental study was performed on melt fracture phenomena in extrusion of high density polyethylene. The purpose of the work was to study the sensitivity of melt fracture driven roughness to the size of recirculation zones, viz. secondary zones. The experimental apparatus consists of a right angle die and a hypodermic needle used as a capillary. The position of the needle relative to the die was adjusted using a special fixture. The roughness of the extrudate was studied as a function of penetration depth. A developed procedure provides a comparison between profile lengths of extruded strands. The computed mean, median, and mode values for roughness were presented as a function of capillary position. A qualitative analysis was conducted for the force oscillations during extrusion with a separate set of dies, equipped with the fixed capillaries of identical lengths and different depths of penetration. It was observed that the oscillatory pattern is sensitive to the sizes of the secondary zones. This qualitative observation supports the conclusions from the quantitative analysis that the roughness of the extrudate can be controlled through an adjustment of the secondary zone sizes.Partly presented at the 58th Annual Meeting of the Society of Rheology, Tulsa, Oklahoma, October 20–23, 1986