Oliver G. Harlen

Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations

A recent constitutive equation derived from molecular considerations on a model architecture containing two branch points a “pom-pom” captures the qualitative rheological behavior of low density polyethylene (LDPE) in shear and extension for the first time [, J. Rheol. 42, 82 (1998)]. We use a hypothetical melt of pom-poms with different numbers of arms to model the behavior of LDPE. The linear relaxation spectra for various LDPE samples are mapped to the backbone relaxation times of the pom-pom modes. Data from start-up flow in uniaxial extension fixes the nonlinear parameters of each mode giving predictions for shear and planar extension with no free parameters. This process was carried out for data in the literature and for our own measurements. We find that multimode versions of the pom-pom equation, with physically reasonable distributions of branching, are able to account quantitatively for LDPE rheology over four decades in the deformation rate in three different geometries of flows. The method sug...

Molecular drag–strain coupling in branched polymer melts

The “pom-pom” model of McLeish and Larson [J. Rheol. 42, 81–110 (1998)] provides a simple molecular theory for the nonlinear rheology of long chain branched polymer melts. A feature of this model is a maximum stretch for the branched molecules. Sharp transitions were predicted in the extensional viscosity at this maximum stretch. We introduce a simple treatment of the coupling between relaxed and unrelaxed polymer segments at branch points. The branch point is allowed to move in a quadratic localizing potential of unknown strength. Taking account of this effect smoothes the sharp transitions of the model and accounts for the extensional viscosity of “pom-pom” model polymers at their maximum stretch. The result is an improved multimode pom-pom fit for low-density polyethylene rheology. By fitting the nonlinear extensional viscosity, quantitative predictions are made for the nonlinear steady shear viscosity and transient first normal stress difference in shear.

A split Lagrangian-Eulerian method for simulating transient viscoelastic flows

A novel numerical method for simulating time-dependent flow of viscoelastic fluids derived from dumbbell models is described. The constitutive equation is solved in a co-deforming frame, where the natural time-derivative is the upper-convected derivative. Mesh reconnection is achieved using a variant of Delaunay triangulation. The velocity and pressure are found via a finite element solution of the momentum equations. The method is tested by applying it to the benchmark problem of a sphere falling along the axis of a cylindrical tube.

Experimental observation and numerical simulation of transient "stress fangs" within flowing molten polyethylene

We report experimental observations and matching numerical simulation for the time-dependent start-up flow of two molten polyethylenes (PEs) within a slit entry and exit geometry. For the case of a low density polyethylene (LDPE), an unexpected transient, birefringence “stress fang” was observed downstream of the slit exit. The stress fang consisted of a localized region of stress concentration. The stress fang, however, was not observed for a linear low density polyethylene (LLDPE) sample subjected to the same processing condition. A matching time-dependent numerical simulation of the flow is also presented. Using a split Lagrangian–Eulerian method for simulating transient viscoelastic flow with the multimode pom–pom constitutive equation, the general features of the stress fangs were predicted for the LDPE. In addition, the simulation did not predict stress fangs for the LLDPE. The paper demonstrates that for this particular case the pom–pom model can successfully discriminate the complex flow behavior ...

The negative wake behind a sphere sedimenting through a viscoelastic fluid

Abstract The phenomenon of the negative wake behind a sphere sedimenting through a viscoelastic fluid is investigated. It is shown that there are two competing viscoelastic forces at work in this flow. The relaxation downstream of shear stresses generated near the side of the sphere drives a flow directed away from the sphere, giving rise to a negative wake. This force is opposed by the extensional stresses generated in the extensional flow at the rear of the sphere, which drives a flow towards the sphere producing an extended wake. The parameter controlling the balance between these forces is the extensibility of the polymer, which limits the extensional viscosity, with high extensibility producing an extended wake and smaller values giving negative wakes.

Constriction flows of monodisperse linear entangled polymers: Multiscale modeling and flow visualization

We explore both the rheology and complex flow behavior of monodisperse polymer melts. Adequate quantities of monodisperse polymer were synthesized in order that both the materials rheology and microprocessing behavior could be established. In parallel, we employ a molecular theory for the polymer rheology that is suitable for comparison with experimental rheometric data and numerical simulation for microprocessing flows. The model is capable of matching both shearand extensional data with minimal parameter fitting. Experimental data for the processing behavior of monodisperse polymers are presented for the first time as flow birefringence and pressure difference data obtained using a Multipass Rheometer with an 11:1 constriction entry and exit flow. Matching of experimental processing data was obtained using the constitutive equation with the Lagrangian numerical solver, FLOWSOLVE. The results show the direct coupling between molecular constitutive response and macroscopic processing behavior, and differentiate flow effects that arise separately from orientation and stretch.

High Deborah Number Flows of Dilute Polymer Solutions

A boundary layer approximation is developed for steady flows of dilute polymer solutions at high Deborah numbers. In these flows polymer molecules are most highly extended within thin elongated regions at or downstream of stagnation points, as only those molecules that pass close to a stagnation point reside in the flow long enough to experience a large strain. This structure can been seen in both optical birefringence experiments (e.g. Cressely & Hocquart 1980, Keller & Odell 1985) where the regions of extended polymer appear as bright birefringent lines, and in numerical calculations using a F.E.N.E. dumbbell model (Chilcott & Rallison 1988). For dilute solutions high elastic stresses occur only within these regions of high molecular strain (here called ‘birefringent strands’) and outside the fluid behaves as a Newtonian fluid of approximately solvent viscosity. This structure is used to devise an asymptotic (‘birefringent strand’) technique for flows with stagnation points in which the birefringent strands are treated as line distributions of forces within an otherwise Newtonian fluid. A number of different flow geometries which contain one or more stagnation points are analysed using this technique. These flows contain a variety of different types of stagnation points: isolated stagnation points and separation points on solid boundaries and free surfaces in both two-dimensional and axisymmetric geometries. This method produces results which are in good agreement with both experiment and full numerical solutions, but without requiring large numerical calculations.

Numerical simulations of a sphere settling through a suspension of neutrally buoyant fibres

The sedimentation of a small dense sphere through a suspension of neutrally buoyant fibres is investigated via a numerical simulation technique that includes both fibre–fibre contact forces and long-range hydrodynamic interactions. In situations where the diameter of the sphere is smaller than the length of the fibres, calculations that exclude the effect of contacts between fibres severely underestimate the drag force on the sphere measured in experiments. By including fibre–fibre contacts in our simulations we are to able to account for this discrepancy, and also the strong dependence of the drag on the initial orientation of the fibres. At low and moderate values of nL 3 , where n is the number of fibres per unit volume and L the fibre length, hydrodynamic interactions are found to be important in moderating the effect of contacts between fibres. An asymptotic solution is presented for the limit when the sphere diameter is much smaller than both the fibre length and inter-fibre spacing, but large compared to the fibre thickness. This is found to be in good agreement with the simulations. Results of calculations on sedimentation through a monolayer of fibres are also presented, as a model of a semi-concentrated suspension. Collisions between fibres are much more frequent, due to the geometric confinement.

High-Deborah-number flow of a dilute polymer solution past a sphere falling along the axis of a cylindrical tube

Abstract The birefringent strand technique (O.G. Harlen, J.M. Rallison and M.D. Chilcott, J. Non-Newtonian Fluid Mech., 34 (1990) 319-349) is used to analyse steady flow past a sphere falling along the axis of a cylinder at high Deborah number but low Reynolds number. The presence of a thin region of highly extended polymer downstream of the sphere causes the velocity in the wake to decay much more slowly than for a Newtonian fluid, and produces a small increase in the drag coefficient. Asymptotic results are presented for a class of models (FENE dumbbels, Phan-Thien—Tanner and Giesekus fluids) that display a high Trouton ratio. The modification of the drag coefficient as the sphere approaches the bottom of the tube is also calculated.

Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams

Abstract The effects of fluid viscoelasticity on the expansion of gas bubbles in polymer foams for the cases of reactive and non-reactive polymers are investigated. For non-reactive polymers, bubble expansion is controlled by a combination of gas diffusion and fluid rheology. In the diffusion limited case, the initial growth rate is slow due to small surface area, whereas at high diffusivity initial growth is rapid and resisted only by background solvent viscosity. In this high Deborah number (De) limit, we see a two stage expansion in which there is an initial rapid expansion up to the size at which the elastic stresses balance the pressure difference. Beyond this time, the bubble expansion is controlled by the relaxation of the polymer. In the model for reactive polymer systems, the polymer molecules begin as a mono-disperse distribution of a single reacting species. As the reaction progresses molecules bond to form increasingly large, branched, structures each with a spectrum of relaxation modes, which gel to form a viscoelastic solid. Throughout this process gas is produced as a by-product of the reaction. The linear spectrum for this fluid model is calculated from Rubinstein et al. [Dynamic scaling for polymer gelation, in: F. Tanaka, M. Doi, T. Ohta (Eds.), Space–Time Organisation in Macromolecular Fluids, Springer, Berlin, 1989, pp. 66–74], where the relaxation spectrum of a molecule is obtained from percolation theory and Rouse dynamics. We discretise this linear spectrum and, by treating each mode as a mode in a multimode Oldroyd-B fluid obtain a model for the non-linear rheology. Using this model, we describe how the production of gas, diffusion of gas through the liquid, and evolution of the largest molecule are coupled to bubble expansion and stress evolution. Thus, we illustrate how the rate of gas production, coupled to the rate of gas diffusion, affects the bubble size within a foam.