Kathleen Feigl

The flow of a LDPE melt through an axisymmetric contraction: A numerical study and comparison to experimental results

The flow of a LDPE melt in an abrupt 10:1 axisymmetric contraction is simulated using a finite element program, and comparisons are made with experimental results reported by another researcher. The researcher performed his die entry experiment at a temperature of 150 °C, and he used Laser Doppler Anemometry to measure the velocity field at several flow rates. He thus obtained detailed information about the flow field. In our numerical simulation of this experiment, we use a separable Rivlin‐Sawyers integral constitutive equation with a spectrum of nine relaxation times to model the fluid. We assume that the ratio of second normal stress difference to first normal stress difference is a nonzero constant. The material is well‐characterized with both shear and simple elongational data from which we determine the parameters in the constitutive equation. The general performance of our model is determined by comparing the vortex growth and entrance pressure loss for various flow rates with the experimental results reported by the experimentalist. We then repeat the experimentalist’s detailed analysis of the flow field at a single flow rate using particle tracking. Specifically, particles are tracked along several streamlines and we compute the shear and elongational rates, as well as the relative shear strain and stretch ratios close to the die entry. The detailed experimental data used for comparison were obtained from the measuredvelocity field. Comparisons of experimental and numerical results show good qualitative and, in some cases, quantitative agreement. From our numerical particle tracking, we also compute the shear stress, the normal stress differences, and the invariants along streamlines. Finally, the shear and elongational contributions to the energy dissipation and the entrance pressure loss is determined throughout the entire domain and in various regions. We find that the majority of the contribution to the entrance pressure loss comes from regions close to the die entry. In addition, in regions in front of the die entry, elongational effects dominate, although shear effects are not negligible, even at high flow rates.

A numerical study of the flow of a low‐density‐polyethylene melt in a planar contraction and comparison to experiments

The flow of a low‐density polyethylene (LDPE) melt in a 10:1 slit‐die contraction at 150°C is simulated by using a two‐dimensional finite element program, and comparisons are made with the experiments of Kramer (1993) who used laser Doppler velocimetry to measure the velocity field at several flow rates. The LDPE melt is modeled by a Rivlin‐Sawyers integral constitutive equation with a spectrum of relaxation times where the parameters were determined from shear and elongational data. Evaluation of the general performance of our model and simulation at several flow rates shows that the simulation accurately predicts the vortex size but underpredicts the velocity overshoot along the centerline. Repeating Kramer’s particle tracking analysis of the flow field at one flow rate for a given set of streamlines we find good quantitative agreement for the elongation rates and relative stretch ratio, but we find our simulation generally underpredicts the shear rates and relative shear strain.

A numerical procedure for calculating droplet deformation in dispersing flows and experimental verification

Abstract A three-step numerical procedure for studying droplet deformation in mixed, dispersing-type, flow fields is described. Finite element and numerical particle tracking techniques are used to obtain the history of shear and elongation rates along a particle trajectory in the flow field, and from this history, boundary integral techniques are used to determine the deformation a drop would experience along this path. This approach is then used to investigate the effect of a small change in geometry on the breakup behavior of drops in the annular gap flow between two eccentric cylinders. This flow geometry serves as an idealization of a rotor–stator dispersing device used for highly viscous fluid systems. It is found that an increase in eccentricity produces an increase in dispersing capability. Experiments in an eccentric cylinder geometry were performed to verify the simulation procedure. Under the experimental conditions considered, it is found that the simulations perform well, correctly predicting whether or not drop breakup occurs and the qualitative drop evolution behavior. The simulation procedure outlined in this paper can serve as an effective tool to determine drop breakup in dispersing geometries and hence to optimize dispersing procedures.

Steady flows of viscoelastic fluids in axisymmetric abrupt contraction geometry: A comparison of numerical results

Recently two groups of researchers have reported numerical results simulating the steady flow of a KBKZ fluid in torsion free axisymmetric abrupt contraction geometry. The fluid model used in both cases was a constitutive equation chosen to match laboratory behavior of LDPE. In both cases, quadratic finite elements were used. Large corner vortices were observed in the simulations, similar to those observed in laboratory experiments. The agreement between the results in the two papers is good. We repeat the experiment, using linear finite elements. Tracking is performed via an artificial time method, and a novel ‘‘reduced velocity’’ variable is used in our finite element simulation. There is good qualitative and quantitative agreement between the results reported here and the results previously reported by others. Quantitative measures used in the comparison—vortex opening angle, Couette correction, and vortex intensity—are analyzed.

Non-Newtonian flow behaviour in narrow annular gap reactors

Abstract In this paper local flow investigations under isothermal conditions have been established for a narrow annular gap reactor (NAGR, given by a rotor/stator system with a radius ratio of ri/ro=0.8) including two wall scraper blades of different geometry. Two-dimensional laminar flow fields are considered (with Reynolds numbers below Re

A failure of a class of K-BKZ equations based on principal stretches

The invariants in the K-BKZ constitutive equation for an incompressible viscoelastic fluid are usually taken to be the trace of the Finger strain tensor and its inverse. The basis for this choice of invariants is not derived from the K-BKZ theory, but rather is due to the perception that this is the most natural choice. Research into using other sets of invariants in the K-BKZ equation, such as the principal stretches or the eigenvalues of the Finger strain tensor (i.e., the squares of the principal stretches) is relatively new. We attempt here to derive a K-BKZ equation based on the squares of the principal stretches that models the behavior of a low-density polyethylene melt in simple shear and uniaxial elongational deformation. In doing so, two assumptions are made as to the form of the strain-dependent energy function: first, that there is a function f(q) such that the energy function can be written as the sum of f(qi),i = 1, 2, 3, where the qisare the squares of the principal stretches, and second that f is a power law. We find that the K-BKZ equation resulting from these two assumptions is inadequate to describe both the shear and elongational behavior of our material and we conclude that the second of the above assumptions is not valid. Further investigation, including predictions of the second normal stress difference and some finite element calculations reveals that the first assumption is also invalid for our material.

A new class of stochastic simulation models for polymer stress calculation

We introduce a new class of stochastic models for polymer stresses which offers a blending of continuum mechanics, network theory and reptation theory. The stochastic dynamics of the model involve two independent Gaussian stochastic processes, Q1 and Q2. Associated with each random vector, Qi, is a random variable, Si, that describes the vector’s survival time during which it evolves according to a deterministic equation of motion. The expression for the stress tensor is an ensemble average of f1(Q12,Q22)Q1Q1+f2(Q12,Q22)Q2Q2, where the fi are scalar functions of Q12=Q1⋅Q1 and Q22=Q2⋅Q2. The relationship between this new class of models and the class of factorized Rivlin-Sawyers integral models is indicated, and simulation models from this new class are used to predict rheological behavior of three low-density-polyethylene melts. We find that the steady-state shear data of all three melts, and the time-dependent elongational viscosity of one of the melts, can be predicted well by models with the same fi, b...

The equivalence of the class of Rivlin–Sawyers equations and a class of stochastic models for polymer stress

This paper establishes the precise relationship between the macroscopic class of factorized Rivlin–Sawyers equations and a class of microscopic-based stochastic models. The former is a well-established and popular class of rheological models for polymeric fluids, while the latter is a more recently introduced class of rheological models which combines aspects of network and reptation theory with aspects of continuum mechanic models. It is shown that the two models are equivalent in a defined sense under certain unrestrictive assumptions. The first part of the proof gives the functional relationship between the linear viscoelastic memory function of the Rivlin–Sawyers model and the probability density for creation times of random variables in the stochastic model. The main part of the proof establishes the relationship between the strain descriptions in each model by showing that the difference in corresponding strain expressions can be made arbitrarily small using the appropriate weighted norm from spectr...

Numerical Investigation of the Formation and Detachment of Droplets from Pores in a Shear Flow Field

The formation and detachment behavior of droplets from a pore opening into a simple shear field within a channel gap is investigated using numerical simulations. The mathematical model consists of the governing equations for an incompressible two‐phase flow problem with a moving contact line. These equations are numerically solved using the volume‐of‐fluid method implemented in the open source software OpenFOAM. A parameter study was performed to determine the effect of relevant dimensionless parameters on the formation and detachment behavior of the droplets. These dimensionless parameters involve the pore size, pore flow rate, gap shear rate, interfacial tension, and the viscosity and density of the two fluid phases. For the parameter range considered in this study, different degrees of jetting behavior were observed. Also, the sizes of the detached droplets were seen to decrease as the gap shear rate increased, and increase with the pore flow rate, with the gap shear rate having a larger effect.

Calculation of the Die Entry Flow of a Concentrated Polymer Solution Using Micro-Macro Simulations

A micro-macro simulation algorithm for the calculation of polymeric flow is developed and implemented. The algorithm couples standard finite element techniques to compute velocity and pressure fields with stochastic simulation techniques to compute polymer stress from simulated polymer dynamics. The polymer stress is computed using a microscopic-based rheological model that combines aspects of network and reptation theory with aspects of continuum mechanics. The model dynamics include two Gaussian stochastic processes, each of which is destroyed and regenerated according to a survival time randomly generated from the materials relaxation spectrum. The Eulerian form of the evolution equations for the polymer configurations is spatially discretized using the discontinuous Galerkin method. The algorithm is tested on benchmark contraction domains for a polyisobutylene solution. In particular, the flow in the abrupt die entry domain is simulated and the simulation results are compared to experimental data. The results exhibit the correct qualitative behavior of the polymer and agree well with the experimental data.