Showing papers in "Journal of Non-newtonian Fluid Mechanics in 2000"

A thermodynamic frame work for rate type fluid models

Abstract: In this paper, we develop a thermodynamic approach for modeling a class of viscoelastic fluids based on the notion of an `evolving natural configuration'. The material has a family of elastic (or non-dissipative) responses governed by a stored energy function that is parametrized by the `natural configurations'. Changes in the current natural configuration result in dissipative behavior that is determined by a rate of dissipation function. Specifically, we assume that the material possesses an infinity of possible natural (or stress-free) configurations. The way in which the current natural configuration changes is determined by a `maximum rate of dissipation' criterion subject to the constraint that the difference between the stress power and the rate of change of the stored energy is equal to the rate of dissipation. By choosing different forms for the stored energy function ψ and the rate of dissipation function ξ, a whole plethora of energetically consistent rate type models can be developed. We show that the choice of a neo-Hookean type stored energy function and a rate of dissipation function that is quadratic, leads to a Maxwell-like fluid response. By using this procedure with a different choice for the rate of dissipation, we also derive a model that is similar to the Oldroyd-B model. We also discuss several limiting cases, including the limit of small elastic strains, but arbitrarily large total strains, which leads to the classical upper convected Maxwell model as well as the Oldroyd-B model.

Simulation of melt spinning including flow-induced crystallization: Part I. Model development and predictions

Abstract: A mathematical model based on the formalism of Doufas et al. [A.K. Doufas, I.S. Dairanieh, A.J. McHugh, J. Rheol. 43 (1999) 85–109] was developed for the simulation of both low- and high-speed melt spinning including the combined effects of flow-induced crystallization (FIC), viscoelasticity, filament cooling, air drag, inertia, surface tension and gravity. Both an amorphous phase, simulated as a modified Giesekus fluid, and a semi-crystalline phase, approximated as rigid rods that grow and orient in the flow field, are coupled through the stress and momentum balance and the feedback of crystallinity to the system relaxation times. Since the onset of crystallization occurs at the equilibrium melting point, the freeze point arises naturally. The model is robust over a wide range of processing conditions and input parameters and exhibits material behavior consistent with that observed for semi-crystalline polymers under all spinning conditions. The model predicts neck-like deformation and associated strain softening in high-speed spinning, as well as the related velocity-, diameter-, temperature-, tensile stress-, apparent elongational viscosity-, orientation- and crystallinity-profiles. Calculations for the systems studied indicate that extensional softening followed almost immediately by FIC provides the primary mechanism responsible for neck formation, in agreement with experimental observations. The model provides a framework for the simulation and optimization of melt spinning involving FIC.

An experimental determination of the stress–microstructure relationship in semi-concentrated fiber suspensions

Abstract: The relationship between the stress and the fiber orientation distribution in semi-dilute (nL3⪢1,nL2d 1) fiber suspensions was investigated. Here, n is the fiber number density, L the length, and d is the diameter. A highly viscous, index-matched suspension was developed to permit measurements of both the microstructure and rheology using the same suspension. By removing the ambiguity of comparing data taken using different suspending fluids and fibers, a more accurate evaluation of available stress–structure models was made possible. The measured period of rotation and the distribution among Jeffery orbits were compared to the results of a theory for hydrodynamic fiber interactions and a simulation incorporating mechanical contacts. At low concentrations, the period increased above the dilute, Jeffery value. As the fiber loading was increased, the period peaked and decreased to approach the dilute result. The distribution of orbits shifted slightly towards the vorticity axis with increasing concentration. The inclusion of a nematic potential in the hydrodynamic theory provided a possible explanation for the decrease in the period of rotation. Measurements of the viscosity and first normal stress differences of the same suspensions were compared to theoretical predictions based on the orientation results. The measured viscosity was in good agreement with the mechanical contact simulation results but was much larger than predicted by hydrodynamic theories. The high viscosity and the measurement of significant first normal stress differences are suggestive of an enhanced stress resulting from the presence of fiber–fiber contacts.

A model for the nonlinear rheology of associating polymers

Abstract: Theories for transient networks formed by associating telechelic polymers require that the kinetics of chain detachment and reattachment processes are somehow specified. We propose here very simple choices for such kinetics, that are nevertheless based on plausible molecular mechanisms. Specific evolution equations are then formulated for the populations of both elastically-active chains and chains temporarily detached from the network. By using typical closure approximations, the rheological model is finally elaborated into a set of ordinary differential equations. For steady shear flows, these nonlinear equations predict shear thickening at moderate shear rates followed by shear thinning at higher shear rates, in agreement with observations. In the limit of slow flows, analytical expressions for the viscosity and the relaxation times are also obtained.

A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows

Abstract: A distributed Lagrange multiplier/fictitious domain method (DLM) is developed for simulating the motion of rigid particles suspended in the Oldroyd-B fluid. This method is a generalization of the one described in [R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, Int. J. Multiphase Flows 25 (1998) 755–794] where the motion of particles suspended in a Newtonian fluid was simulated. In our implementation of the DLM method, the fluid–particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing equations for the Oldroyd-B liquid are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. We use the Marchuk–Yanenko operator-splitting technique to decouple the difficulties associated with the incompressibility constraint, nonlinear convection and viscoelastic terms. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The nonlinear convection problem is solved using a least square conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The code is verified performing a convergence study to show that the results are independent of the mesh and time step sizes. Our simulations show that, when particles are dropped in a channel, and the viscoelastic Mach number (M) is less than 1 and the elasticity number (E) is greater than 1, the particles chain along the flow direction; this agrees with the results presented in [P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–325]. In our simulations of the fluidization of 102 particles in a two-dimensional bed, we find that the particles near the channel walls form chains that are parallel to the walls, but the distribution of particles away from the walls is relatively random.

Simulation of melt spinning including flow-induced crystallization: Part II. Quantitative comparisons with industrial spinline data

Abstract: The mathematical model for melt spinning of Doufas et al. [A.K. Doufas, A.J. McHugh, C. Miller, J. Non-Newtonian Fluid Mechanics, 1999] coupling fiber microstructure (molecular orientation and crystallinity) with the macroscopic velocity/stress and temperature fields, is tested extensively against industrial spinline data for several nylon melts. Model fits and predictions are shown to be in very good quantitative agreement with spinline data for the fiber velocity and temperature fields at both low and high-speed conditions, and, with birefringence data available for high speeds. The effects of processing parameters: quench air velocity, capillary diameter and mass throughput, as well as material characteristics: molecular weight (RV) and polymer type (i.e., homopolymers with or without additives, and copolymers), on the spinline dynamics are accurately predicted. Under high-speed conditions, strain softening occurs and the tensile stress at the freeze point is predicted to be essentially independent of the processing parameters investigated, in agreement with experimental observations. Birefringence data and model predictions show that crystallization occurs mostly after the freeze point, under the locked-in tensile stress. Under low-speed conditions, the velocity and crystallization profiles (experimental and predicted) are shown to evolve smoothly towards a plateau value and strain hardening behavior is predicted throughout the spinline. The ability to quantitatively describe spinline data over a wide range of conditions and material characteristics, renders the model a useful tool for optimization of melt spinning processes as well as a framework for simulation of other polymer processes involving flow-induced crystallization.

Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows

Abstract: Improved accuracy and enhanced convergence rate are achieved when a finite-volume method (FVM) is used in conjunction with a high-resolution scheme (MINMOD) to represent the stress derivatives in the constitutive equation, because it avoids oscillations of the solution field near sharp stress gradients. Calculations for the benchmark flow of an upper-convected Maxwell fluid through a 4:1 plane contraction were carried out at a constant Reynolds number of 0.01 and varying Deborah numbers in four consistently refined meshes, the finest of which had a normalised cell size of 0.005 in the vicinity of the re-entrant corner. The MINMOD scheme was able to provide converged solutions up to Deborah numbers well beyond those attained by other second-order accurate schemes. The asymptotic behaviour of velocity and stresses near the re-entrant corner was accurately predicted as compared with Hinch’s theory [1]. The simulations improved previous results for the same flow conditions obtained with less accurate schemes, and the present results can be used as benchmark values up to a Deborah value of 3 with quantified numerical uncertainties. © 2000 Elsevier Science B.V. All rights reserved.

A molecular dynamics study of a short-chain polyethylene melt. I. Steady-state shear

Abstract: Utilizing a united atom potential model and reversible reference system propagator algorithm (rRESPA) multitimestep dynamics, we have performed equilibrium and nonequilibrium molecular dynamics simulations of a monodisperse C100H202 polyethylene melt at 448 K and 0.75 g/cm 3 . We report a variety of properties calculated at equilibrium including rotational relaxation time and self-diffusion coefficient as well as shear-enhanced diffusion and rheological properties calculated under steady-state shearing conditions. Shear thinning is observed in the viscosity and normal stress coefficients over the range of strain rates studied. A minimum in the hydrostatic pressure is observed at an intermediate strain rate that is associated with a minimum in the intermolecular Lennard‐Jones potential energy as well as transitions in the strain-rate-dependent behavior of several other viscous and structural properties of the system. The shear field also imposes significant alignment of the chains with the flow direction, approaching a limiting angle of approximately 3 at high strain rate. In addition, the self-diffusion coefficients (calculated in terms of the unconvected positions according to the Cummings‐Wang formalism) are markedly enhanced under shear compared to the equilibrium state (up to two orders of magnitude at the highest shear rate studied). © 2000 Elsevier Science B.V. All rights reserved.

On the shear banding flow of elongated micellar solutions

Abstract: Under steady shear flow, elongated micellar solutions show shear stress saturation above a critical shear rate due to the formation of shear bands that result in non-homogeneous flow. Long transients and oscillations accompany this stress plateau. When measurements are done with a controlled stress rheometer, frequently a metastable branch is observed. At higher shear rates, a second upturn is observed above a second critical shear rate, which indicates that homogeneous flow is recovered. Here, a model consisting of the codeformational Maxwell constitutive equation coupled to a kinetic equation to account for the breaking and reformation of the micelles is presented to reproduce the features described above in steady shear flow. The model also predicts a second metastable branch and long transients at higher shear rates and the existence of an inflexion point in stress-shear rate plots above which no shear banding behavior is detected.

The backward-tracking Lagrangian particle method for transient viscoelastic flows

Abstract: 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. (C) 2000 Elsevier Science B.V. All rights reserved.

Non-linear oscillatory flow of a soft solid-like viscoelastic material

Abstract: Large-amplitude oscillatory shearing flow data are reported for a wheat-flour dough, which is highly shear-thinning in its dynamic data. The large-amplitude data are analysed using a recently proposed constitutive equation for bread dough, in which dough is considered as a solid-like material. It is found that the model predictions agree well with experimental data, giving a certain degree of confidence in the constitutive equation. Furthermore, the markedly non-linear response of the material, even at as low an amplitude as 0.05, is mainly due to the strain softening behaviour of the material; and this non-linearity cannot be predicted by a model with shear rate dependent parameters alone.

Dynamics of weakly strain-hardening fluids in filament stretching devices

Abstract: We investigate the transient viscoelastic response of weakly strain-hardening fluids to imposed elongational deformation in filament-stretching devices. We combine time-dependent finite-element simulations with quantitative experimental measurements on a rheologically well-characterized test fluid to investigate how well the device reproduces the ideal transient uniaxial extensional viscosity that is predicted theoretically. A concentrated polymer solution containing 5.0 wt% monodisperse polystyrene is used as the test fluid and the experiments are conducted using the filament-stretching rheometer, developed by Spiegelberg et al. The axisymmetric numerical simulations incorporate the effects of viscoelasticity, surface tension, fluid inertia and a deformable free surface. Single and multi-mode versions of the Giesekus constitutive equation are used to model the rheology of the shear-thinning test fluid. Excellent agreement between the measured transient Trouton ratio and the numerical predictions over a range of deformation rates is reported. The numerical simulations also reveal some important aspects of the fluid kinematics exhibited by weakly strain-hardening fluids during stretching—including a rapid necking of the filament diameter near the axial mid-plane of the fluid column, and an associated elastic recoil phenomenon near the rigid end-plates. This necking instability of a viscoelastic filament can be understood through a generalized Considere criterion, as recently documented by Hassager et al. As a consequence of this necking, spatial and temporal homogeneity in the extensional deformation of the filament is never achieved, even at large Hencky strains. This is in sharp contrast to the numerical and experimental studies for strongly strain-hardening dilute polymer solutions that have been reported to-date. Nonetheless, the present computational rheology study shows that filament stretching devices can still be used to accurately extract the transient extensional viscosity function for weakly strain-hardening fluids, provided that the evolution history of the tensile force at the end-plate and the filament radius at the mid-plane are carefully measured and that the experimental data are correctly processed.

Measuring the viscosity of large-particle (and flocculated) suspensions — a note on the necessary gap size of rotational viscometers

Abstract: A simple theory is presented to account for the viscometer walls effects encountered when measuring suspensions of large particles (or flocs). The often-recommended minimum ratio of gap-to-particle size ratio of 10:1 is shown to be appropriate for phase volumes up to 25%, but above this value, the ratio needs to be greater. The theory is supported by recent experimental evidence.

Squeeze flow of highly concentrated suspensions of spheres

Abstract: The squeeze flow behaviour of highly concentrated suspensions of spheres in a Newtonian fluid is studied experimentally. Analysing the evolution of the squeeze force as a function of time for different controlled velocities, the suspension is found to present two main flow regimes. The first regime is characterised by the situation in which the force decreases when the velocity decreases, which corresponds to a viscous flow of the suspension. In the second regime, the force increases when the velocity decreases, which is shown to correspond to a filtration of the solvent through the particle skeleton that behaves then as a deformable porous media. By varying the concentration, the sphere diameter and the solvent viscosity, it is found that the transition between the two regimes is a result of a competition between the viscous shear forces due the flow of the suspension and the damping force caused by the filtration of the fluid through the porous media made up by the particles.

Axial annular flow of a nonlinear viscoelastic fluid — an analytical solution

Abstract: An analytical solution is given for the kinematic and stress variations across the radial gap of a concentric annular flow in fully developed conditions. The fluid is viscoelastic and obeys the non-linear rheological constitutive equation proposed by Phan-Thien and Tanner [1]. This constitutive model simulates well the material functions of many polymer melts and solutions and therefore, the present results are useful in a number of practical situations. The ratio of pressure drop to flow rate drop is found to be a complex mathematical function of the radial position of zero shear stress and this, in turn, depends weakly on the elasticity, based on the product of an elongational parameter by a Deborah number defined with an averaged velocity. There is thus a non-linear coupling which could not be solved in an explicit way for the inverse problem of an imposed flow rate, but an iterative procedure gives a ready result. For the direct problem of a given pressure drop the present results represent an exact explicit solution to the axial annular flow problem. Representative profiles of the solution are given and discussed. It is found that, for a given flow rate, the pressure drop scaled with the corresponding Newtonian value is independent of the diameter ratio. © 2000 Elsevier Science B.V. All rights reserved.

Effects of shear thinning on migration of neutrally buoyant particles in pressure driven flow of Newtonian and viscoelastic fluids

Abstract: The pattern of cross stream migration of neutrally buoyant particles in a pressure driven flow depends strongly on the properties of the suspending fluid. These migration effects have been studied by direct numerical simulation in planar flow. Shear thinning has a large effect when the inertia or elasticity is large, but only a small effect when they are small. At moderate Reynolds numbers, shear thinning causes particles to migrate away from the centerline, creating a particle-free zone in the core of the channel, which increases with the amount of shear thinning. In a viscoelastic fluid with shear thinning, particles migrate either toward the centerline or toward the walls, creating an annular particle-free zone at intermediate radii. The simulations also give rise to precise determination of slip velocity distributions in the various cases studied.

A new method for blood viscosity measurement

Abstract: The present study introduces the concept of a newly designed scanning viscometer with a capillary tube for use in measuring whole blood viscosity without the use of anticoagulants in a clinical setting. Both flow rate and pressure drop measurements that are usually required for the operation of a capillary tube viscometer are replaced with a single measurement of liquid-height variation with time. In addition, the present method overcomes the drawbacks of conventional viscometers in the measurement of the whole blood viscosity. First, with the present study, whole blood viscosity can be accurately and consistently measured at 37°C over a wide shear rate range including shear rates as low as 1 s −1 . Second, the present method can measure whole blood viscosity over a range of shear rates in less than 2 min without any anticoagulants so that one can measure the viscosity for unadulterated blood.

Numerical study of heat transfer in a non-Newtonian Carreau-fluid between rotating concentric vertical cylinders

Abstract: Centrifugally forced convection, mixed and natural convection are numerically studied in a short vertical annulus with a heated and rotating inner cylinder. The cooled outer cylinder is at rest and the hortizontal endplates are assumed adiabatic. The effect of a non-Newtonian shear thinning viscosity modeled by the Carreau-shifted constitutive equation is examined. Computations were performed for different values of the flow index and Weissenberg number with the Prandtl number based on the zero-shear-rate viscosity, the radius ratio and the ratio of height to gap spacing are kept fixed. The results show that the shear thinning effect decreases the friction factor at the rotating cylinder and increases the heat transfer through the annular gap. It is also shown that the reduction in apparent viscosity may produce oscillatory flows, especially for centrifugally forced convection.

Wall slip and shear stresses of polymer melts at high shear rates without pressure and viscous heating effects

Abstract: We present experimental data demonstrating wall slip of high temperature polymer melts at high shear rates (up to ~70 s) without pressure and viscous heating effects. Results for polydisperse (linear low density polyethylene) and monodisperse (polystyrene) melts are presented. For the monodisperse case two distinct regimes of slip are seen: (1) a low shear rate regime in which the slip velocity increases slowly with shear rate (weak slip) and (2) a high shear rate regime in which the increase is dramatic (strong slip). Through scaling of our shear stress/shear rate data we demonstrate that the crossover between the two regimes occurs when the bulk polymer chains are effectively disentangled. These findings are in qualitative agreement with molecular models of slip which invoke an entanglement-disentanglement scenario for the transition from weak to strong slip. The polydisperse system shows less critical slip behaviour and the slip velocity increases; yet, the increase is less dramatic than the monodisperse system. (C) 2000 Elsevier Science B.V. All rights reserved.

Matched asymptotic solutions for the steady banded flow of the diffusive Johnson–Segalman model in various geometries

Abstract: We present analytic solutions for steady flow of the Johnson–Segalman (JS) model with a diffusion term in various geometries and under controlled strain rate conditions, using matched asymptotic expansions. The diffusion term represents a singular perturbation that lifts the continuous degeneracy of stable, banded, steady states present in the absence of diffusion. We show that the stable steady flow solutions in Poiseuille and cylindrical Couette geometries always have two bands. For Couette flow and small curvature, two different banded solutions are possible, differing by the spatial sequence of the two bands.

An experimental/theoretical investigation of interfacial instabilities in superposed pressure-driven channel flow of Newtonian and well characterized viscoelastic fluids

Abstract: Purely viscous and purely elastic interfacial instabilities as well as the layer encapsulation phenomena in superposed pressure-driven channel flow of well characterized Newtonian and viscoelastic fluids have been studied. We have characterized both the base and the perturbed flows by measuring velocity profiles and encapsulation rates as well as determining the stability of the interface. Specifically, it has been shown that above a critical depth and viscosity ratio, encapsulation occurs and the rate of encapsulation is a strong function of viscosity ratio at a given depth ratio. In addition, for the first time purely viscous and purely elastic interfacial instabilities in superpose plane Poiseuille flows have been experimentally observed. Specifically, it has been shown that the incipient instability is two-dimensional and well described by theoretical predictions using linear stability theory with a constitutive equation that can accurately describe the rheology of the test fluids. Moreover, it has been demonstrated that in absence of encapsulation and significant second normal stresses in the test fluids the interfacial waves are nearly two-dimensional in the weakly non-linear regime.

Convective orientational renewal in entangled polymers

Abstract: The predictions of tube models in fast flows have recently been improved by incorporating convective constraint release (CCR). In this paper, we revisit CCR showing that, whenever chains are not stretched, an additional effect named convective conformation renewal (CCR2 to distinguish it from CCR) should also be considered. CCR2 consists in the loss of orientation of the unstretched chain due to flow-induced lengthening of tube segments. A simple quantitative model including both CCR and CCR2 is then developed, which reduces to double reptation in the linear limit. Although the model ignores fluctuations, the predicted flow curve is monotonic, and compares favourably with data for nearly monodisperse entangled polymers. For the case of a single relaxation time, typical of polymer-like micelles, a maximum in the flow curve still remains.

Rheology and rheological morphology determination in immiscible two-phase polymer model blends

Abstract: The Maffettone–Minale analysis describes the shape of ellipsoidal droplets in immiscible two-phase polymer model blends during flow. It is used here to calculate the elastic interfacial contribution to the shear stress (σ12,int), and first normal stress difference (N1,int). The ratio N1,int/σ12,int can be linked to the orientation angle of the inclusions. Under steady state flow the Maffettone–Minale model gives an analytical expression between the orientation angle and the capillary number. In this manner the capillary number can be deduced from the orientation angle. From this the droplet size is calculated. Good agreement is found between the results and droplet sizes determined independently from dynamic measurements using the Palierne model. The stresses calculated from the model, with given values for the droplet size, compare quite well with the stresses measured on a model system for relatively small capillary numbers.

Optimal control of non-isothermal viscoelastic fluid flow

Abstract: Optimal control of viscoelastic fluid flow in a 4 to 1 contracting channel is investigated. The control mechanism is based on heating or cooling the fluid along a portion of the boundary of the flow domain. In order to perform the control, a non-isothermal model for viscoelastic fluids is used consisting of the PTT model with relaxation time and elastic viscosity depending on temperature (following an exponential dependency, the WLF model). Moreover, the momentum, mass and constitutive equations are coupled with the heat equation forming the so-called primal system. The goal of the control is to find an optimal temperature on the boundary of the domain such that the large recirculation zones at the corners of the contracting channel are reduced and the fluid behaves like a viscous one. Two different cost functionals are used to reach this goal: one of tracking type, the other penalizing negative contributions of the velocity component in direction of the span of the channel. The minimization of these cost functionals is achieved with a gradient algorithm. An optimality system, derived by the use of the Lagrange multipliers, allows us to deduce this optimal control. The primal and dual systems are solved using the same numerical method: a finite differences discretization on staggered grids, a decoupling approach and the Gauss–Seidel solver with a multigrid algorithm. Numerical simulations are performed with a Weissenberg number equal to 10. We succeed in reducing the recirculation zone by applying the correct boundary temperature.

Mixing of non-Newtonian fluids in time-periodic cavity flows

Abstract: Fluid mixing in two- and three-dimensional time-periodic cavity flows as a function of rheological fluid parameters is studied. Computational methods are applied to obtain accurate descriptions of the velocity fields, which form the basis of the mixing analysis. In addition to some classical techniques, like Poincare maps and the analysis of periodic points, a recently developed mapping method is used to determine mixing efficiency over a wide range of different flow parameters. Within this framework, different mixing protocols can be evaluated with respect to their long term mixing behaviour and compared quantitatively. It is shown that local ‘optimal’ parameter settings for mixing of Newtonian fluids can result in a considerable worse mixing behaviour for non-Newtonian fluids, and vice versa, illustrating the importance of taking the rheology of the fluid into account.

A new method to determine interfacial tension from the retraction of ellipsoidal drops

Abstract: In this study, a new convenient method for the determination of the interfacial tension between two immiscible polymers in melted state is proposed. For an extended thread in a liquid matrix, it is difficult to get a regular sinusoidal disturbance in some systems. But after a liquid thread in matrix breaks up, we can see a process of ellipsoidal drop retraction. By directly using this process, we can measure interfacial tension of polymer melts avoiding the limit of the breaking thread method. The new method is much easier when compared to traditional drop retraction methods. We use a second rank tensor S to describe the shape evolution instead of the deformation parameter D in Taylor’s theory. Analysis of this interfacial tension driven process leads to a theoretical relation between the shape retraction rate, the three semi-axis of ellipsoidal drop and rheological characteristics. The interfacial tensions determined by the method are in good agreement with those data given in the literatures for systems with either high or low viscosity ratio.

Dynamic simulation of freely-draining, flexible bead-rod chains : Start-up of extensional and shear flow

Abstract: We present a study of the rheology and optical properties during the start-up of uniaxial extensional and shear flow for freely-draining, Kramers bead-rod chains using Brownian dynamics simulations. The viscous and elastic contributions to the polymer stress are unambiguously determined via methods developed in our previous publication [1]. The elastic contribution to the polymer stress is much larger than the viscous contribution beyond a time of 5.3l1:N 2 where N is the number of beads in the chain and l1 is the longest relaxation time of the chain. For small Wi (at arbitrary strains) and for small strains (at arbitrary Wi) the stress-optic law is found to be valid. The stress-optic coefficients based on the shear stress and first normal stress difference are equal for all Wi (even when the stress-optic law is not valid) suggesting the stress-optic coefficient is in general a scalar quantity rather than its most general form as a fourth order tensor. We show that a multimode FENE-PM or Rouse model describes the rheology of the bead-rod chains at small strains, while the FENE dumbbell is an accurate model at larger strains. We compare the FENE-PM and FENE model to experimental extensional stress data of dilute polystyrene solutions and find that a multimode FENE-PM with a Zimm relaxation spectrum describes the data well at small strains while a FENE dumbbell with a conformation dependent drag is in quantitative agreement at larger strains. © 1998 Elsevier Science B.V. All rights reserved.

Deformation of a two-dimensional viscoelastic drop at non-zero Reynolds number in time-periodic extensional flows

Abstract: The kinematics of a potential vortex offers an interesting flow history for a rheologically complex material, and earlier work on that subject led us to consider the behavior of a Newtonian drop in three related time dependent flow fields [K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation, J. Fluid Mech., 2000, submitted for publication; K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional viscous drop in time-periodic extensional flows: analytical treatment, J. Fluid Mech., 2000, submitted for publication]. In the work reported here the drop, characterized by an upper-convected Maxwell model (UCM), is suspended in an incompressible Newtonian fluid. Again, three related flows are considered. The first is that of a potential vortex, modeled by an extensional flow field near the drop with rotating axes of stretching. The second is a generalization of the first and is called rotating extensional (RE) flow, in which the frequency of revolution of the flow is varied independently of the shear rate. Finally, we consider oscillating extensional (OE) flow. Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We have developed an analytic elastic-viscous stress splitting scheme obtained by an integration by parts of the constitutive equation. The scheme explicitly separates the diffusive part of the momentum equation for a wide range of differential constitutive relations. An ADI implementation is executed for the diffusive part. We investigate the effects of periodicity, Reynolds number and relaxation time on the drop dynamics. For a vortex and an RE flow, the long-time deformation reaches a steady value, and the drop attains a revolving, steady elliptic shape. The long-time values of deformation show complex non-monotonic behavior with variation in Weissenberg number, an effect of the decreased damping and increased elasticity, as well as the presence of a shear wave triggered by the UCM constitutive relation. The first two effects are modeled successfully by a simple ODE presented in Appendix A. The wave effects are briefly discussed in Appendix B.

Slipping fluids: a unified transient network model

Abstract: Wall slip in polymer solutions and melts play an important role in fluid flow, heat transfer and mass transfer near solid boundaries. Several different physical mechanisms have been suggested for wall slip in entangled systems. We look at the wall slip phenomenon from the point of view of a transient network model, which is suitable for describing both, entangled solutions and melts. We propose a model, which brings about unification of different mechanisms for slip. We assume that the surface is of very high energy and the dynamics of chain entanglement and disentanglement at the wall is different from those in the bulk. We show that severe disentanglement in the annular wall region of one radius of gyration thickness can give rise to non-monotonic flow curve locally in that region. By proposing suitable functions for the chain dynamics so as to capture the right physics, we show that the model can predict all features of wall slip, such as flow enhancement, diameter-dependent flow curves, discontinuous increase in flow rate at a critical stress, hysteresis in flow curves, the possibility of pressure oscillations in extrusion and a second critical wall shear stress at which another jump in flow rate can occur.

Finite element analysis of stability of two-dimensional viscoelastic flows to three-dimensional perturbations

Abstract: We present numerical methods for analysis of the stability of two-dimensional steady viscoelastic flows to small-amplitude, two-dimensional and three-dimensional disturbances based on finite element calculations of the steady base flow and the perturbation. Direct time integration of the linearized equations of motion and iterative calculation of the most dangerous components of the eigenspectrum are tested. Finite element discretizations based on the DEVSS-G finite element discretization with Newton’s method used to compute steady-state solutions. Two different time integration schemes are tested for computing the time evolution of general, random disturbances: a θ -method operator-splitting scheme and a fourth-order Runge–Kutta method. For both time integrators, time stepping is decoupled into a solution of a modified Stokes problem and an evaluation of the time-dependent constitutive equation. The overall efficiency of both methods is extremely high, as is the potential for implementation on parallel computers. An algorithm also is presented for calculating eigenvalues with the largest real part that combines time integration of the linearized equations with a Krylov subspace method to accelerate the calculation of the eigenvalues. Although this method does not dramatically reduce the computational cost over use of time integration alone, it does provide a more complete analysis of the eigenspectrum. For both direct time integration and the hybrid time integration/Krylov calculation, the stability results for cylindrical Couette flow show quantitative agreement with the eigenvalues calculated by using other methods of analysis [M. Avgousti, A.N. Beris, J. Non-Newtonian Fluid Mech. 50 (1993) 225–251]. Contrary to the results in our previous paper [R. Sureshkumar, M.D. Smith, R.C. Armstrong, R.A. Brown, J. Non-Newtonian Fluid Mech. 82 (1999) 57–104], we find that the flow of an Oldroyd-B fluid through a closely-spaced cylinder array is stable to two-dimensional perturbations. However, allowing the perturbations to be three-dimensional and considering an isolated cylinder does not alter the conclusions of our earlier study [R. Sureshkumar, M.D. Smith, R.C. Armstrong, R.A. Brown, J. Non-Newtonian Fluid Mech. 82 (1999) 57–104] of the two-dimensional stability of widely-spaced arrays of cylinders; the flow around an isolated cylinder is computed to be stable for all values of the Weissenberg number obtainable with these calculations, We≤0.75.