Planar Extensional Flow Research Articles

Planar extensional flow resistance of a foaming plastic

The planar extensional flow resistance of a foaming-grade polystyrene, with and without a blowing agent, was characterized using two dies, each one consisting of a high-aspect-ratio straight rectangular channel followed by a hyperbolic convergent rectangular channel. With the hyperbolic geometry, the fluid near the centerline of the convergent channel was subjected to a constant rate of extension. The shear viscosity was found from the pressure drop along the straight channel, and the planar extensional viscosity was determined from the pressure drop in the convergent channel, taking into account the pressure drop due to shearing. Values of the planar extensional viscosity are compared with values of the uniaxial extensional viscosity measured with a shear-free fixture. Tests with a polystyrene and with a solution of 5% carbon dioxide in the polystyrene revealed that the gas caused significant reductions in both the shear and extensional viscosities.

A comparison of model linear chain molecules with constrained and flexible bond lengths under planar Couette and extensional flows

We compare directly under flow two commonly used coarse grained models of linear polymers, namely the flexible finitely extensible nonlinear elastic (FENE) chain, and the freely jointed tangent sphere chain, otherwise known as the freely jointed chain. The comparison is based on viscometric, structural and dynamical properties. We use non-equilibrium molecular dynamics to simulate steady-state systems under planar Couette flow and planar extensional flow. Computed properties include shear and elongational viscosities, normal stresses, radius of gyration and end-to-end distances, order parameters, alignment angles and spin angular velocities. In all computed properties we observe very little difference between the two molecular models. Therefore, the choice of either model is suitable, though there is a computational advantage in the use of the FENE model.

Dynamics of strain-hardening and strain-softening capsules in strong planar extensional flows via an interfacial spectral boundary element algorithm for elastic membranes

In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution.

Diffusion of linear polymer melts in shear and extensional flows

We present results from molecular dynamics simulations for the anisotropic self-diffusion tensor and the velocity autocorrelation functions of monodisperse systems of dense linear chain molecules under flow. Two molecular models are used in these simulations: The finitely extensible nonlinear elastic chain and the freely jointed tangent sphere chain. Nonequilibrium molecular dynamics is used to simulate these systems under planar Couette flow and planar extensional flow. Under planar extensional flow, results presented here are the first, from simulation, for diffusion and velocity autocorrelation functions of molecules, while for planar Couette flow, we compare the broadest range of conditions. An explicit derivation is provided of the Green-Kubo expression for the diffusion tensor. This expression is then used to derive the relation involving the mean-squared displacement-an often used alternative method to calculate diffusion coefficients. Velocity autocorrelation functions have been used, in combination with results on the alignment of molecules from a previous paper, to provide some details of the molecular scale dynamics that influence diffusive transport under flow.

Investigating the stability of viscoelastic stagnation flows in T-shaped microchannels

We investigate the stability of steady planar stagnation flows of a dilute polyethylene oxide (PEO) solution using T-shaped microchannels. The precise flow rate control and well-defined geometries achievable with microfluidic fabrication technologies enable us to make detailed observations of the onset of elastically driven flow asymmetries in steady flows with strong planar elongational characteristics. We consider two different stagnation flow geometries; corresponding to T-shaped microchannels with, and without, a recirculating cavity region. In the former case, the stagnation point is located on a free streamline, whereas in the absence of a recirculating cavity the stagnation point at the separating streamline is pinned at the confining wall of the microchannel. The kinematic differences in these two configurations affect the resulting polymeric stress fields and control the critical conditions and spatiotemporal dynamics of the resulting viscoelastic flow instability. In the free stagnation point flow, a strand of highly oriented polymeric material is formed in the region of strong planar extensional flow. This leads to a symmetry-breaking bifurcation at moderate Weissenberg numbers followed by the onset of three-dimensional flow at high Weissenberg numbers, which can be visualized using streak-imaging and microparticle image velocimetry. When the stagnation point is pinned at the wall this symmetry-breaking transition is suppressed and the flow transitions directly to a three-dimensional time-dependent flow at an intermediate flow rate. The spatial characteristics of these purely elastic flow transitions are compared quantitatively to the predictions of two-dimensional viscoelastic numerical simulations using a single-mode simplified Phan-Thien–Tanner (SPTT) model.

Deformation of a viscoelastic drop in planar extensional flows of a Newtonian fluid

This communication presents experimental data for steady and transient deformations of a purely elastic drop (PIB/PB Boger fluid) suspended in a Newtonian fluid (PDMS) undergoing a planar extensional flow produced by a four-roll mill. The viscoelastic effects alter the steady drop shape from being ellipsoidal to drop shapes with more blunt ends. The results reflect a balance between the direct tensile stress contribution of the viscoelastic fluid to the normal stress balance, and modifications of the viscous (i.e. Newtonian) stress and pressure due to viscoelastic changes in the flow. This is qualitatively consistent with the conclusion from recent simulation results [S. Ramaswamy, L.G. Leal, The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid, Journal of Non-Newtonian Fluid Mechanics 85 (2–3) (1999) 127–163]. There is no overshoot of the drop deformation upon start-up of the flow, and a relative insensitivity to the Deborah number. However, when the external flow is turned off, there is a broad spectrum of relaxation times for the drop to return to a spherical shape, between the characteristic capillary timescale for a Newtonian fluid, aμ/ σ, to the longer polymer relaxation time, τ R .

Drop coalescence in planar extensional flow and gravity

A film drainage model, which allowed for the effect of the ratio of the viscosities of the drop and liquid film on the interfacial mobility, is found to represent reasonably well the published experimental data on the coalescence times of drops in two different flow fields at a constant force. The data of Yang et al. [2001. The coalescence of two equal-sized drops in a two-dimensional linear flow. Physics of Fluids 13, 1087–1106] on the head-on collision of two equisized drops at the center of the inflow axis under planar extensional hyberbolic flow in a four-roll mill, and the data of several authors on the coalescence time of a single drop with its bulk homophase at a deformable interface under gravity are considered. The scaled coalescence time increased linearly with the product of the viscosity ratio and the applied external to interfacial tension forces ratio over three orders of magnitude. Higher applied external forces resulted in larger film areas while more viscous drops reduced the interfacial mobility thereby increasing the coalescence times.

Spindles, Cusps, and Bifurcation for Capsules in Stokes Flow

Interfacial dynamics of membrane-enclosed fluid volumes in viscous flows is complicated due to the coupling of the fluid dynamics with the membrane properties. Based on computational investigation via our interfacial spectral boundary element algorithm, our study shows that a (strain-hardening) Skalak-type capsule in a planar extensional Stokes flow develops steady-state shapes whose edges from spindled become cusped with increasing flow rate owing to a transition of the edge tensions from tensile to compressive. A bifurcation in the steady-state shapes is also found (i.e., existence of both spindled and cusped edges for a range of high flow rates) by implementing different transient processes, owing to the different evolution of the membrane tensions.

A Lagrangian finite element method for simulation of a suspension under planar extensional flow

A numerical simulation of a suspension of two-dimensional solid particles in a Newtonian fluid under planar extensional flow is presented. The method uses a finite element solution of the flow with a unit cell within the self-replicating lattice for planar extensional flow identified by Kraynik and Reinelt [A.M. Kraynik, D.A. Reinelt, Extensional motions of spatially periodic lattices, Int. J. Multiphase Flow 18 (1992) 1045]. This is implemented using a quotient space representation that maps all points space onto points within the unit cell. This mapping is preserved by using fully Lagrangian grid movement, with grid quality preserved by a combination of Delaunay reconnection and grid adaptivity. The no-slip boundary conditions on the particles are enforced weakly via a traction force acting as a Lagrange multiplier. The method allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The method is illustrated for both isotropic and anisotropic two-dimensional particles and can be easily extended to viscoelastic fluids and to non-rigid particles.

Flow perturbation model for filament buckling

This short communication presents a flow perturbation model for filament buckling that is typically observed under smectic A liquid crystal phase ordering from the isotropic phase. In the absence of flow, liquid crystal filaments buckle with a characteristic wave-length under the action of negative tension generated by the decrease in free energy that drives the growth process. This phenomenon is demonstrated numerically using simulations of a previously presented non-driven mesoscopic model. We then investigate how shear and extensional flow affect the filament buckling process. Using a linearized model, the filament shape dynamics are cast into a Cahn–Hilliard equation whose solutions are the buckling modes. The growth rate of the dominant mode is expressed in terms of the factorized product of the elastic contribution and the flow contribution. The flow Deborah number, D e, contributions to the growth of the dominant wave-vector are found to be 1 ± n D e , where + (−) denotes compression (extension), and where n = 2 (1/2) for planar extensional (shear) flow. The model provides insights on how flow-generated stresses affect shape formation, an area of significant interest to biological form and growth.

The dynamics of the coil-stretch transition for long, flexible polymers in planar mixed flows

We describe the dynamics of an isolated, flexible polymer molecule in an extension-dominated planar mixed flow. We find that as vorticity is added to planar extensional flow, the conformational fluctuations of the molecule increase and the coil-stretch hysteresis [e.g., Schroeder et al., Macromolecules 37(24), 9242 (2004); Science 301, 1515 (2003)] vanishes for a fixed Deborah number and Hencky strain. We present Brownian dynamics simulation results and a kinetic theory, valid in the hysteretic regime, which allows us to determine both the rate at which a coiled molecule unravels and the rate at which a stretched molecule relaxes to a coil when subjected to planar mixed flow. We demonstrate that for all values of the added vorticity, the coil-stretch/stretch-coil transition rates increase with increasing vorticity as a consequence of the conformational fluctuations, which can be explained by the molecule sampling a shear flow through Brownian fluctuations off the extensional axis. We demonstrate that if Rg∕(αL)⪡1, where Rg is the radius of gyration, L is the molecular contour length, and α is the so-called “mixedness” parameter, the transition rate increase is explained in terms of an effective conformational diffusivity, which can be evaluated using a convective dispersion analysis.

Migration and orientation of disk-like particles in microcomposite processing

The objective of this study is to well understand the migration and orientation of thin micro-particles, such as talc and mica, in a suspension flow by means of both experiments and numerical simulations as well as to obtain the knowledge of the processing operations of thin micro-particle reinforced composites. A slit channel was mounted on the sidewall of a large reservoir in the experimental setup. For this reason, the thin disk-like particles were subjected by a planar extensional flow in a reservoir, then by a simple shear flow through a slit channel. The evolution of the orientation of thin disk-like particles was studied in both a planar extensional and simple shear flows by numerical calculation of the Jeffery equation: thin disk-like particles aligned in the parallel orientation to the upper- and lower-walls of the slit channel in a planar extensional flow through the reservoir, then entered into the inlet of the slit channel. On the other hand, in a simple shear flow through the slit channel, the disk-like particles kept this parallel orientation except the occurrence of flip-over. The period of the flip-over became longer with a decrease in aspect ratio (thickness/diameter) of the disk-like particles. Furthermore, the measurements of the orientation of the talc particles in a suspension flow through the slit channel clearly showed that almost the same period of the flip-over was found in spite of different particle size. The experimental result arises from complex geometries and no accurate data of the thickness of talc particles.

マイクロコンポジットの成形中におけるディスク状粒子の配向

The objective of this study is to well understand the migration and orientation of thin micro-particles in a suspension flow by means of both experiments and numerical simulations. In this framework, the evolution of the orientation of thin micro-particles such as talc and mica, which were modeled by a disk-like particle, in a flow through a slit channel was analyzed to obtain the knowledge of the processing operations of thin micro-particle reinforced composites.The thin disk-like particles were subjected by a planar extensional flow in a reservoir, then by a simple shear flow through a slit channel in the experimental apparatus used. The evolution of the orientation of thin disk-like particles was, therefore, studied in both a planar extensional and simple shear flows by numerical calculation of the Jeffery equation: thin disk-like particles aligned in a parallel orientation to upper- and lower-walls of the slit channel in a planar extensional flow through the reservoir, then entered into the inlet of the slit channel. On the other hand, in a simple shear flow through the slit channel, the disk-like particles kept this parallel orientation except the occurrence of a flip-over phenomenon. The period of the flip-over became longer with a decrease in the aspect ratio of the disk-like particles.Furthermore, the measurements of the orientation of the talc particles in a suspension flow through the slit channel clearly showed that almost the same period of the flip-over was found although the particle size was different. These experimental results arise from complex geometries and no accurate data of the thickness of the talc particles.

Effect of Hydrodynamic Interactions on DNA Dynamics in Extensional Flow: Simulation and Single Molecule Experiment

Intramolecular hydrodynamic interactions (HI) in flexible polymer chains influence both the equilibrium and nonequilibrium physical properties of macromoecules. In this work, we utilize a combination of single molecule experimental techniques and Brownian dynamics (BD) simulation to investigate the role of HI and excluded-volume (EV) interactions for DNA molecules ranging in contour length from 150 to 1300 μm. Epifluorescence microscopy is used to directly observe the dynamics of DNA molecules in planar extensional flow, and a semiimplicit bead−spring BD algorithm with fluctuating HI and EV interactions is presented. Quantitatitative agreement between ensemble average transient molecular extension in experiment and BD simulation is shown for DNA with 150 μm contour length. Simulations show polymer conformation hysteresis for larger DNA chains (1300 μm in length) when HI and EV parameters are chosen such that simulation results match the experimental polymer relaxation time and polymer stretch at flow stre...

Flow-induced nonequilibrium thermodynamics of lamellar semicrystalline polymers

In this work we present a first attempt to quantify the effect of flow deformation on the microstructure of semicrystalline polymers. This necessitates bridging the macroscopic flow length scale with the microscopic (segment) length scale of the semicrystalline structure. To achieve this connection we developed a hierarchical approach where a thermodynamically consistent macroscopic constitutive equation is interfaced with a microscopic lattice-based Monte Carlo (MC) simulation of the polymer chain conformation. We first illustrate this approach in a two-dimensional (2D) “toy” application where the 2D equivalent of a macroscopic constitutive equation based on reptation theory is applied to describe the chain deformation and extended free energy in the amorphous bulk phase. The values for the derivative of the free energy with respect to the mean segment orientation tensor, calculated for a planar extensional flow, are then used as an extended nonequilibrium thermodynamic forcing term. This is added in a traditional Metropolis Monte Carlo scheme, developed for a 2D lattice representation of a lamellar semicrystalline polymer, to drive the flow-induced microstructure. Significant flow-induced changes are calculated, steadily increasing as the Weissenberg number increases. We subsequently extend these ideas further in a much more realistic three-dimensional (3D) application where the information for the thermodynamics of the bulk amorphous phase under a uniaxial extensional flow is extracted from a macroscopic network model, such as that of Phan-Thien and Tanner (PTT), connecting the free energy to the second moment of the end-to-end distance of a multisegment chain. Through a series of 3D nonequilibrium Monte Carlo simulations of both the amorphous and the semicrystalline microscopic morphologies, it is shown that the interaction of the flow-induced deformation with the semicrystalline microstructure is nonlinear: the amorphous interlamellar structure changes significantly from its corresponding homogeneous bulk amorphous state, even far away from the crystalline interface. Our approach allows for a quantitative estimation of this effect on both thermodynamic quantities, like the extended microscopic free energy, as well as various statistics of the chain conformations.

Monte Carlo simulation of steady extensional flows

An efficient nonequilibrium Monte Carlo method using the Bond fluctuation model is used to simulate uniaxial and planar extension of dilute polymer solutions. The time scale is obtained from the stress relaxation of a fully stretched chain and can be related to the longest relaxation time of a real molecule, while the length scale is taken to be the statistical Kuhn segment length. The method leads to τ1∼N2.16 and D∼N−1.02 for a freely draining chain with an excluded volume constraint, where τ1, N, and D are the longest relaxation time, chain length, and diffusion coefficient, respectively. The finite extensibility of the bond causes extension thinning following the coil-stretch transition. The Monte Carlo predictions for the transient extension of isolated DNA molecules in a planar extensional flow agree reasonably well with published experimental measurements and Brownian dynamics simulations. “Molecular individualism” is observed in the unraveling of the polymer chain.

Observation of polymer conformation hysteresis in extensional flow.

Highly extensible Escherichia coli DNA molecules in planar extensional flow were visualized in dilute solution by fluorescence microscopy. For a narrow range of flow strengths, the molecules were found in either a coiled or highly extended conformation, depending on the deformation history of the polymer. This conformation hysteresis persists for many polymer relaxation times and is due to conformation-dependent hydrodynamic forces. Polymer conformational free-energy landscapes were calculated from computer simulations and show two free-energy minima for flow strengths near the coil-stretch transition. Hysteresis cycles may directly influence bulk-solution stresses and the development of stress-strain relations for dilute polymer flows.

Transient and steady state three-dimensional drop shapes and dimensions under planar extensional flow

Drop shapes and dimensions in transient and steady states have been studied using a four-roll mill flow apparatus with three-dimensional imaging and real-time derivative edge detection. The deformed drops are approximately ellipsoidal for viscosity ratios of λ > 0.1 with about 4% maximum deviation of actual drop volume from ellipsoidal volume. For λ < 0.1, drops retain their ellipsoidal shape up to deformation of ∼0.5, above which they show progressive deviation from the ellipsoidal shape. The measured length, breadth, and width of deformed drops were compared to predictions of various drop deformation theories. The limit of deformation within which theories predict drop dimensions with 5% accuracy has been tested for 10−4 < λ < 102. The ellipsoidal model of Maffettone and Minale (1998) gives the best prediction for λ < 0.3, while the second order theory of Barthes-Biesel and Acrivos (1973) gives the best overall prediction for λ > 0.3. The application of some theories to the determination of interfacial tension and drop viscosity is discussed.

Entry flow of LDPE melts in a planar contraction

Experiments using laser–Doppler velocimetry (LDV) and numerical simulations have been undertaken for the creeping entry flow of two rheologically well-characterised low-density polyethylene (LDPE) melts in a 14:1 planar contraction. LDV measurements of the flow reveal a distinct behaviour between the two melts regarding vortex formation. The flow of the melts has been modelled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times, modified to properly account for strain hardening in the planar extensional flow relevant for the geometry used. Numerical values for the constants appearing in the equation have been obtained from fitting dynamic data ( G′ and G″) as well as data for the viscosities and normal stresses as measured in shear. Elongational data were derived from transient and steady uniaxial elongation experiments. The experiments show that for one LDPE melt the vortex keeps small with increasing flow rate, while for the other there is a vortex growth in the same range of flow rates. The numerical simulations also predict the same trend in agreement with the experiments. The differences in flow behaviour of the two branched melts are found to be attributed to big differences in the average relaxation times, giving rise to different De numbers or dimensionless stress ratios S R, for the same range of flow rates. Equally important is the more pronounced extensional strain-hardening behaviour of the vortex-producing polymer, giving rise to high Trouton ratios T R.

Predictions and Applicability of the BKZ Type Models in Extensional Flows of Polymer Melts

Predictions on shear, planar, uniaxial and biaxial extensional flows are obtained from the BKZ type constitutive equations with the assumption of time-strain separability. Using a single relaxation mode, the effects of adjustable parameters on the stress growth coefficients are investigated in detail. A parameter describing the normal stress ratio governs the biaxial extensional flow behavior. It also controls the planar extensional flow behavior in the direction of constant width (perpendicular to the extension direction). Applicability of two representative models with this parameter is investigated for multiple relaxation times by comparing with a set of data for polyisobutylene (literature data) and polystyrene melts. We propose a method to obtain the model parameters from shear, uniaxial and biaxial damping functions. With the introduction of the parameter describing the normal stress ratio, both the PSMLT model (Papanastasiou-Screven-Macosko as modified by Luo-Tanner) and the WDFO model (Wagner-Demarmels as modified by Fegel-Ottinger) give good agreements with all the experimental data. It is suggested that the PSMLT model is preferable for engineering purposes, because the model gives non-divergent stresses in almost of all cases investigated.