Peter Wapperom

The backward-tracking Lagrangian particle method for transient viscoelastic flows

A new Lagrangian particle method for solving transient viscoelastic flow for both macroscopic and microscopic stress equations is proposed. In this method, referred to as the backward-tracking Lagrangian particle method (BLPM), we specify the particle locations and calculate the trajectories leading to these locations. This backward tracking process is stopped after a specified time (possibly only a single time step), and the initial configuration for the Lagrangian integration of the stress is obtained by interpolating a stored Eulerian field at that time. In order to demonstrate the accuracy, efficiency and stability of the method, we consider two benchmark problems in the context of the FENE dumbbell kinetic theory of dilute polymer solutions and its FENE-P approximate constitutive equation: the high eccentricity journal bearing flow and the 4:1 contraction flow. With the help of these examples, we show in which manner accurate and stable results can be obtained, for transients of both polymer stress and stream function, with a minimum number of particles and a minimum particle path length

Simulation for viscoelastic flow by a finite volume/element method

Stability of a second-order finite element/finite volume (FE/FV) hybrid scheme is investigated on the basis of flows with increasing Weissenberg number. FEs are used to discretise the balances of mass and momentum. For the stress equation a FV method is used, based on the recent development with fluctuation distribution schemes for pure convection problems. Examples considered include a start-up channel flow, flow past a cylinder and the non-smooth 4:1 contraction flow for an Oldroyd-B fluid. A considerable gain in efficiency per time step can be obtained compared to an alternative pure FE implementation. A distribution based on the flux terms is unstable for higher Weissenberg numbers, and this is also true for a distribution based on source terms alone. The instability is identified as being caused by the interaction of the balance equations and stress equation. A combination of distribution schemes based on flux and source terms, however, gives a considerable improvement to the hybrid FE/FV implementation. With respect to limiting Weissenberg number attenuation, the hybrid scheme is more stable than the pure FE alternative for the smooth flow past a cylinder, but less so for the non-smooth contraction flow. The influence of additional strain-rate stabilisation techniques is also analysed and found to be beneficial.

Thermodynamics of viscoelastic fluids: The temperature equation

From the thermodynamics with internal variables we will derive the temperature equation for viscoelastic fluids. We consider the type of storage of mechanical energy, the dissipation of mechanical energy, the compressibility of the fluid, the nonequilibrium heat capacity and thermal expansion, and deformation induced anisotropy of the heat conduction. The well-known stress differential models that fit into the thermodynamic theory will be treated as an example. Adapting a power-law scaling of the shear moduli on temperature and density, as is usual in rubber elasticity, we will derive an approximation of the temperature equation in measurable quantities. This equation will be compared with experimental results.

Numerical simulation of branched polymer melts in transient complex flow using pom-pom models

In recent years, a number of constitutive equations have been derived from reptation theory to describe the rheology of both linear and branched polymer melts. While their predictions in rheometrical flows have been discussed in detail, not much is known of their behaviour in complex flows. In the present paper, we study by way of numerical simulation the transient, start-up how of branched polymers through a planar contraction/expansion geometry. The constitutive equation is the so-called pom-pom model introduced by McLeish and Larson [J. Rheol. 42 (1998) 81], and later modified by Blackwell et al. [J. Rheol. 44 (2000) 121]. By combining the backward-tracking Lagrangian particle [J. Non-Newtonian Fluid Mech. 91 (2000) 273] and deformation field [J. Non-Newtonian Fluid Mech. 89 (2000) 209] methods, we obtain results for the original, integral pom-pom model which makes use of the Doi-Edwards orientation tensor. Two simplified versions of the pom-pom model are also considered, namely one based on the Currie approximation for the orientation tensor, and a differential constitutive equation proposed in [J. Rheol. 42 (1998) 81]. Finally, the simulation results are compared to those obtained with the so-called MGI model proposed recently by Marrucci et al. [Rheol. Acta, submitted for publication] for describing linear polymer melts

Using transient shear rheology to determine material parameters in fiber suspension theory

Fiber suspension theory model parameters for use in the simulation of fiber orientation in complex flows are, in general, either calculated from theory or fit to experimentally determined fiber orientation generated in processing flows. Transient stress growth measurements in startup of shear flow and flow reversal in the shear rate range, γ=1–10 s−1, were performed on a commercially available short glass fiber-filled polybutylene terephthalate using a novel “donut-shaped” sample in a cone-and-plate geometry. Predictions using the Folgar–Tucker model for fiber orientation, with a “slip” factor, combined with the Lipscomb model for stress were fit to the transient stresses at the startup of shear flow. Model parameters determined by fitting at γ=6 s−1 allowed for reasonable predictions of the transient stresses in flow reversal experiments at all the shear rates tested. Furthermore, fiber orientation model parameters determined by fitting the transient stresses were compared to the experimentally determi...

Simulation of linear polymer melts in transient complex flow

Recently, much progress has been made in improving the modelling of linear polymer melts with the aid of reptation theory. In simple shear flows, this has resulted in a much better prediction of the shear viscosity and normal stress ratio. Here, we evaluate in complex flow the transient and steady-state behaviour of a recently proposed reptation model, the Marrucci-Greco-Ianniruberto model [G. Marrucci, F Greco, G. Ianniruberto, Rheol. Acta, 2000, submitted for publication], that includes convective constraint release and a force balance on the entanglement nodes. To incorporate integral type models into the numerical framework of Lagrangian particle methods, developed previously to simulate dilute polymer solutions, we have included the so-called deformation field method. For the contraction/expansion flow that we consider, we find that a correction of the convective constraint release contribution to the relaxation time is necessary to avoid the unphysical situation of negative relaxation times. With this correction, we could obtain mesh and time convergence for high Weissenberg numbers without adding any solvent viscosity. We find that in complex flow also, both the steady-state and transient response of the integral model can be very well approximated by a constitutive equation of differential type. Due to the dominance of the strong thinning in both shear and elongational flows for the model, however, the inelastic Carreau-Yasuda model reproduces the steady-state kinematics and pressure drop as well

Prediction of rheometrical and complex flows of entangled linear polymers using the double-convection reptation model with chain stretch

We study the rheometrical and complex flow response of the double-convection-reptation (DCR) model with chain stretch proposed recently by Ianniruberto and Marrucci (2002) for entangled linear polymers. The single- and two-mode differential versions of the model are used, with the parameter values identified by Ianniruberto and Marrucci (2002) for a nearly monodisperse polybutadiene solution. These authors found that the DCR model with stretch predicts the rheometrical shear behavior of the fluid well in the modest experimental range of deformation rates. Our calculations for the higher shear rates reached in simulations of complex flow reveal anomalous or questionable behavior, namely, shear thickening over an intermediate range of shear rates and large chain stretch in fast shear flows. This behavior is shown to be shared by the original integro-differential DCR theory, of which the differential DCR model is actually a mathematical approximation. We also show that the original DCR theory with stretch predicts excessive shear thinning at high shear rates, while its differential approximation remains stable for all shear rates. Using the backward-tracking Lagrangian particle method [Wapperom et al. (2000)], we investigate the response of the differential DCR model in start-up flow through an axisymmetric contraction/expansion geometry. We compare the single- and two-mode model predictions (in terms of the steady-state vortex structure, chain stretch, and overall pressure drop), and correlate these with the steady and start-up rheometrical responses in shear and extension. Significant chain stretch is predicted in the vicinity of the axis of symmetry and in thin boundary layers located at the constriction wall. As a result, the DCR predictions significantly depart from the stress-optical rule in these flow regions. Chain stretch also affects the flow kinematics, with the appearance of a large upstream steady-state vortex. Surprisingly, however, the predicted pressure drop is not affected much by these kinematical changes, and is, qualitatively described by a simple inelastic, shear-thinning model

Using startup of steady shear flow in a sliding plate rheometer to determine material parameters for the purpose of predicting long fiber orientation

The properties of long glass fiber reinforced parts, such as those manufactured by means of injection molding and compression molding, are highly dependent on the fiber orientation generated during processing. A sliding plate rheometer was used to understand the transient stress and orientation development of concentrated long glass fibers during the startup of steady shear flow. An orientation model and stress tensor combination, based on semiflexible fibers, was assessed in its ability to predict fiber orientation when using model parameters obtained from the fits of the stress responses. Specifically, samples of different initial fiber orientations was subjected to the startup of steady shear flow, and an orientation model based on bead and rod theory was coupled with a derived stress tensor that accounts for the semiflexibility of the fibers to obtain the corresponding model parameters. The results showed the semiflexible orientation model and stress tensor combination, overall, provided improved rheo...

Sample preparation and image acquisition using optical-reflective microscopy in the measurement of fiber orientation in thermoplastic composites

A complete sample preparation procedure used to determine three‐dimensional fiber orientation from optical micrographs of glass fiber‐reinforced thermoplastic composites is presented. Considerations for elimination of irregularities in the elliptical footprints, contrast enhancement between fibers and surrounding polymer matrix, controlled‐etching that allows the identification of small shadows where fiber recedes into the matrix, and topographical reconstruction of the elliptical footprint are described in the procedure. This procedure has produced high‐quality optical micrographs employed to obtain accurate fiber orientation data for thermoplastic composites using the method of ellipses. The optimal definition of the nonelliptical footprints’ borders allows an accurate measurement of orientation in small sampling areas.

Obtaining reliable transient rheological data on concentrated short fiber suspensions using a rotational rheometer

The conventional method for obtaining transient rheological data on short glass fiber-filled polymeric fluids is to use the parallel disk (PP) geometry in a rotational rheometer. Using the PP geometry large transient stress overshoot behavior was observed during the startup of flow measurements on a 30 wt % short glass fiber-filled polybutylene terephthalate. A contributing factor to this behavior is believed to be induced fiber collisions caused by the inhomogeneous velocity field (radial varying velocity gradient). A novel approach was taken in which a “donut” shaped sample was used in a cone-and-plate device (CP-D) to maintain a sufficient gap to fiber length ratio. The magnitude of the first normal stress difference was reduced by 70%, and the time to reach steady state was reduced by 100 strain units. The Lipscomb model coupled with the Folgar–Tucker model for the evolution of fiber orientation was fit to the stress growth behavior measured using both the PP geometry and CP-D resulting in different p...