FENE Dumbbell Research Articles

Relaxation of dilute polymer solutions following extensional flow

The relaxation of dilute polymer solutions following stretch in uniaxial extensional flow is investigated via Brownian dynamic simulations of a flexible freely-draining bead-rod chain. The bead-rod chain simulations are compared to Brownian dynamic simulations of a FENE dumbbell and numerical calculations of a FENE-PM chain. A universal relaxation curve for the stress decay from steady-state is found by shifting the results to lie on the curve described by the relaxation of an initially straight chain. For all the models investigated, the initial rapid decay of the polymer stress decreases at a rate which scales for large Weissenberg number, Wi as Wi2. Our universal curve is in good qualitative and in some cases quantitative agreement with the available experimental data: it is particularly good in predicting decay after stretch at the largest strains. We find hysteresis in comparing the stress versus birefringence during the startup of flow and subsequent relaxation for the bead-rod chain and FENE dumbbell, but not for the FENE-PM chain. The hysteresis in the latter model is lost in the preaveraging of the nonlinear terms. The bead-rod model also displays a configuration hysteresis. The hysteresis observed in these models is in qualitative agreement with recent experiments involving polystyrene-based Boger fluids.

Numerical simulations of the flow of dilute polymer solutions in a four-roll mill

We study the startup flow of dilute polymer solutions in a four-roll mill by using FENE dumbbell models. Because this flow is inhomogeneous and contains an extension-dominated region at the stagnation point, strong coupling between the flow field and the polymer configuration can be expected. Our effort at simulating such flows and making meaningful comparisons with experiments is a major step toward assessing and improving molecularly-based constitutive models for dilute solutions. The first objective of the paper is to examine the behavior of the Chilcott-Rallison version of the FENE model (FENE-CR) with varying parameters c, De and L. At moderately high values of De and L, the polymer stretching mainly happens within a ‘birefringent strand” along the exiting flow axis emanating from the stagnation point. The coupling between flow and polymer stretching is exhibited by concerted flow suppression and reduction of polymer extension. In particular, the model predicts double-humped velocity and strain rate profiles across the outflow axis, in qualitative agreement with experiments. The second objective of the paper is to examine the effects of two additional features that can be added to the basic FENE-CR model: shear-thinning and an extra viscous stress. For shear-thinning we use the FENE-P model and for the viscous stress we adopt the expression recently proposed by Rallison [J. Non-Newtonian Fluid Mech., 68 (1977) 61–83]. For the Deborah numbers studied here, shear-thinning causes a small increase in the steady-state polymer stretching at the stagnation point. The strain rate there is also somewhat higher than the corresponding FENE-CR value. The extra viscous stress tends to reduce polymer stretching and the strain rate at the stagnation point. In previous studies, the FENE-CR model has been shown to over-predict polymer stretching. Thus, the extra viscous stress will bring predictions of the model closer to experiments.

Macromolecular dynamics in extensional flows: 2. The evolution of molecular strain

The opposed jets apparatus has been used to investigate the dynamics of the coil-stretch transition of polymer solutions in an idealized stagnation point extensional flow field. Flow simulations generated fluid strain profiles for different geometries. Assuming a molecular uncoiling model, true birefringence profiles have been converted to molecular strains for closely monodisperse, high molecular weight aPS under θ-conditions. This has enabled macromolecular deformation to be followed as a function of position and residence time. Non-linear hydrodynamic friction FENE dumb-bell simulations give qualitative agreement. Initially, molecular uncoiling is non-affine, consistent with changing hydrodynamic screening on extension. Deformation in a good solvent is more affine. Results are compared with PEO/water to investigate the effect of molecular parameters. The effective extensional viscosity has been ascertained by correction for the area of high molecular extension. The increase in extensional viscosity due to molecular stretching is substantial, of the order of the number of equivalent flexible units in the chain. © 1997 Elsevier Science Ltd.

2‐D time‐dependent viscoelastic flow calculations using connffessit

AbstractTwo‐dimensional time‐dependent calculations for a molecular model of finite extensibility in the journal‐bearing geometry are presented. The flow is considered to be incompressible and isothermal. The momentum conservation equation is integrated using a time‐marching procedure in which local ensembles of dumbbells act as stress calculators. The calculations are based on the Calculation of non‐Newtonian flows: finite elements and stochastic simulation technique (CONNFFESSIT) and combine deterministic (finite elements) and stochastic techniques to advance the velocity and stress fields in time. The ability of CONNFFESSIT to treat models for which no closed‐form constitutive equation can be derived is illustrated by performing calculations using FENE dumbbells. Significant differences in the stress field between the true FENE and the linearized FENE‐P are found, especially during the inception period. Steady‐state kinematics are, however, identical within error bars for both FENE and FENE‐P and for the Newtonian fluid. The essential algorithm of 2‐D CONNFFESSIT is detailed, as well as experience gathered from its parallel and vector versions.

Dynamic simulation of freely draining flexible polymers in steady linear flows

We present a study of the rheological and optical behaviour of Kramers bead–rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coefficient. We have developed a computer algorithm to differentiate the Brownian from the viscous stress contributions which also avoids the order (δt)−1/2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the effective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we find an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead–rod chain scales as N3. Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi, Wi ≈ 0.06N2, where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi−1/2 and the first normal stress as Wi−4/3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead–rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi.

On the Peterlin approximation for finitely extensible dumbbells

For the simplest non-linear kinetic theory of dilute polymeric solutions (FENE dumbbells), the pre-averaging Peterlin approximation used to derive a macroscopic constitutive equation (FENE-P) is shown to have a significant impact on the statistical and rheological properties of the model. This is illustrated in simulations of transient elongational flows by means of standard and stochastic numerical techniques.

Computational studies of the FENE dumbbell model with conformation-dependent friction in a co-rotating two-roll mill

The flow of a viscoelastic fluid in a co-rotating two-roll mill is studied using a modified Chilcott-Rallison constitutive model, which includes the Hinch-DeGennes formulation of a conformation-dependent friction coefficient. One well known feature of this model, for homogeneous flows, is that there is a hysteresis loop in the dependence of the steady-state configuration on the strain-rate. However, for inhomogeneous flows, at finite polymer concentrations, there is a strong coupling between the velocity and configuration fields, which is shown by present calculations to mask the hysteresis loop, and the associated non-uniqueness of the steady-state configuration. This is a consequence of the reduction in extensional strength of the velocity field with increasing polymeric contribution to the stress field. For small polymer concentrations the transition from the coiled-state to the stretched-state is abrupt in time and the transitional Deborah number range is relatively narrow. However, with increasing polymer concentration, both the time taken to reach the steady-state and the transitional Deborah number range increase significantly. For relatively large values of c, L and De a birefringent pipe-like structure emerges along the outflow axis of the two-roll mill. This and the overall spatial distribution of the configuration qualitatively agree with birefringence photographs for real polymeric liquids.

Cross‐stream migration of bead–spring polymers in nonrectilinear pore flows

AbstractDeterministic cross‐stream migration of FENE dumbbells, cyclic trimers and bicyclic tetramers in nonhomogeneous, nonrectilinear flows representative of tortuous pores is analyzed. Identifying the crucial feature of misalignment between a tumbling dumbbell and the surrounding streamlines, Brunn (1983) showed that the dumbbell model requires three reflections in the bead–bead hydrodynamic interaction (HI) for lateral migration to occur: lower‐order approximations of the HI are insufficient because they lead only to alignment with the flow rather than tumbling. In any orientation the trimer (tetramer) has at least two (three) “bonds” out of alignment with the flow. Radial migration in rotary Couette flow between concentric cylinders occurs in the freely draining limit, and the simplest, first‐order HI is sufficient to cause lateral migration in rectilinear tube flow. Flow through a sinusoidally corrugated pore brings a new convective timescale on which the bead–spring entity moves between converging and diverging flow environments. Since this process outpaces the dumbbell's alignment, even a freely draining dumbbell spends most of its time slightly misaligned with the surrounding streamlines, and migrates toward the walls (higher shear). Tumbling occurs on a much longer timescale, with the dumbbell traveling through many wavelengths of the wall corrugations (and fluctuating in orientation) between successive (rapid) end‐for‐end flips in the shear field. The flipping time seems to scale inversely with the length of the dumbbell. The trimer and tetramer rotate largely as in rectilinear shear, and exhibit somewhat stronger migration for the same bond length. As a simple model of pore entrance effects, net drift in an oscillatory Sampson flow through a thin orifice is also considered.

An extended FENE dumbbell theory for concentration dependent shear-induced anisotropy in dilute polymer solutions

The original FENE dumbell kinetic theory is extended to describe concentration dependent shear-induced anisotropy in dilute polymer solutions. A mean-field term is introduced into the model equations to take into account intermolecular forces. For the case of stationary shear flow the corresponding coupled non-linear relaxation equations for the components of the tensor of gyration are solved numerically. We present results for the shear and concentration dependence of different quantities related to the tensor, i.e. the orientation angle, radius of gyration, the eigenvalues, and different pseudospherical components. They are in good qualitative agreement with data from light-scattering experiments. Corresponding results for the rheological quantities are briefly discussed.

Nonisothermal polymeric fluids

The phase-space kinetic theory for polymeric liquid mixtures has been further developed for molecular models without internal constraints. The theory provides expressions for the mass, momentum, and energy fluxes, each of which may in general be influenced by concentration, velocity, and temperature gradients. To illustrate the use of these results, the thermal conductivity for a dilute polymer solution is derived; the FENE dumbbell model is used to describe the mechanical behavior of the polymer chains. The Hookean dumbbell results can be obtained by letting the “finite extensibility parameter” b tend to infinity.

Finite element simulation of flow around a [formula omitted] corner using the FENE dumbbell model

In this paper we numerically simulate flow of the FENE dumbbell model around a 3π 2 corner. By refining the radial mesh at the corner in both the radial and tangential directions, we show that a bounded converged solution can be obtained for large values of the Deborah number, and for arbitrarily large values of the finite-extensibility parameter. For non-zero polymer concentrations, our numerical results show that for r ≠ 0 in the vicinity of the corner both velocity and configuration fields can be approximated by generalized power laws with exponents depending on the angular position θ. Away from the two walls there is an angular region within which the exponents of the governing power laws are approximately constant. We also find that the polymeric stress singularity is stronger than the Newtonian stress singularity. These results are in agreement with the analytical results obtained by Hinch (J. Non-Newtonian Fluid Mech., 34 (1993) 319–349).

Spinning viscosity

The use of fibre spinning and related experiments is examined and the presentation of results in terms of a “spinning viscosity” is proposed. A re-examination of results for convected Maxwell and Jeffreys models and FENE dumbbell models shows the importance of the flow history prior to the spinning. Some undesirable ways of treating data are discussed. Out of this comes a discussion of the appropriateness of using viscosity or modulus as a measure of a material response and of using rate of strain or strain as a variable. The work is theoretical, but leads to a proposal for testing the ways in which experimental data are treated. A definition of a “spinning viscosity” is offered, and the idea that one should only obtain one value from one spinning experiment is emphasized.

Direct modeling of flow of FENE fluids

Direct simulations of macromolecular fluids are carried out for flows between parallel plates and in expanding and contracting channels. The macromolecules are modeled as FENE dumbbells with soft disks or Lennard-Jones dumbbell-dumbbell interactions. The results are presented in terms of profiles and contour plots of velocity, pressure, temperature, density, and flow fields. In addition the data for potential energy, shear stress, and the normal components of the stress tensor are collected. In general, an excellent agreement is found between the simulated profiles and the well-known flow structures, such as flow separation and formation of viscous eddies, indicating that micro-hydrodynamics is a viable tool in linking macroscopic phenomena with the underlying physical mechanisms. The simulations are performed in the Newtonian regime, for medium-size systems comprising up to 3888 dumbbells. This number is sufficiently large to control boundary and particle number effects. The flow is induced by gravity. The traditional stochastic (thermal) and periodic boundary conditions are employed. Also, diffusive boundary conditions, which could include a stagnant fluid layer and repulsive potential walls, are developed. The scaling problems, which are related to the application of a large external force in a microscopic system (of the size of the order 100 A), result in extreme pressure and temperature gradients. In addition, the viscosity and thermal conductivity coefficients obtained from velocity and temperature profiles of the channel flow are presented. These results are confirmed independently from modeling of Couette flow by the SLLOD equations of motion and from the Evans algorithm for thermal conductivity.

Large and small force limits for spinning with FENE dumbbell models

Theoretical results for spinning in the limiting cases of very small and very large applied force are obtained. The small force limit shows how the Deborah number dependence of the ‘spinning viscosity’, predicted for the FENE dumbbell model at low rates of strain, is due to the decay of initial stresses rather than to the extensional flow. The large force limit is consistent with expected Newtonian behaviour for fully extended dumbbells. The merit of using physically relevant quantities in the mathematical manipulation is discussed.

Quantitative prediction of the viscoelastic instability in cone-and-plate flow of a Boger fluid using a multi-mode Giesekus model

Analysis and experiments have shown that the flow of a viscoelastic fluid between a rotating cone and plate is unstable to a three-dimensional time-dependent instability which results in a secondary motion that, close to onset, has the form of a Bernoulli spiral. Here we report results of a linear stability analysis for a multi-mode formulation of the Giesekus constitutive equation with parameters determined by regression to the linear relaxation spectrum and steady-state shear flow properties of the constant viscosity solution of polyisobutylene, polybutene and tetradecane first characterized by Quinzani et al. (J. RheoL, 34 (1990) 705-748). The numerically calculated stability boundaries for the onset of the elastic instability are compared to flow visualization experiments and provide quantitative agreement for both the critical Deborah number and the non-axisymmetric azimuthal structure of the spiral instability. The results are qualitatively similar to the predictions obtained from non-linear models with a single relaxation time, such as a single-mode Giesekus model or the FENE dumbbell model proposed by Chilcott and Rallison; however, the more precise fit of the first normal stress coefficient supplied by the multi-mode model appears necessary for quantitative prediction of the experiments. Quasi-linear models lacking shear-thinning behavior in the first normal stress coefficient Ψ 1(γ), such as single- or multi-mode formulations of the Oldroyd-B model, are incapable of even qualitative prediction of the dependence of the stability boundaries on Debroah number observed in experiments with viscoelastic polymer solutions.

Uncoiling a polymer molecule in a strong extensional flow

It has been understood for many years that a polymer molecule will uncoil in a coil-to-stretch transition when it is placed in an extensional flow with a strain rate exceeding the slowest molecular relaxation rate. This paper looks at the effort which must be exerted to achieve this uncoiling, in other words the transient stress in a dilute solution of such uncoiling polymer molecules. Numerical simulations have been performed of an isolated linear chain of inextensible links which are freely hinged, the chain being placed in an axisymmetric extension flow. Hydrodynamic interactions between the many beads are not included. During the uncoiling, the stress is found to be mainly dissipative rather than elastic, i.e. the stress is proportional to the instantaneous strain rate rather than being independent of it. A rapid build up of this viscous stress with the total strain is shown to come from the growth of segments of fully stretched chain. The evolution of these segments, the growth in their size along with the reduction of their number, is examined with a simplified ‘kinks dynamics’ model. The above rheological behaviour in transient strong extensional flows is not described by the standard constitutive relations which have been used in the past for dilute polymer solutions, e.g. the Oldroyd-B fluid and FENE dumbbell models. A suitable modification is suggested, which gives large strain-dependent viscous stresses.

Computational studies of the FENE dumbbell model in a co‐rotating two‐roll mill

The motion of a viscoelastic liquid, modeled via the Chilcott–Rallison version of the FENE dumbbell model, is studied numerically for a co‐rotating two‐roll mill. Solutions are obtained for values of the Deborah number up to 13.6, and for the dimensionless contour length 300≤L2≤2000 at three different values of the concentration parameter c≤1. The emphasis of the study is on start‐up flow from a state of rest. Of particular interest is the onset of a complicated overshoot and undershoot of the dumbbell configuration at very large values of De and L. This phenomenon is due primarily to the fact that the flow is significantly modified as the polymer contribution to the stress increases, but there is also a contribution due to the closed nature of the flow system which means that all streamlines of the flow are closed, and the deformation history experienced by a polymer molecule is therefore periodic in time.

Elongational Stress of the Flow of Polymer Solutions.

This review describes methods and techniques developed to date for measuring elongational stresses produced in elongational flows. Experimental results are presented and it is shown that in many cases the strain exerted on a fluid element is important for the stress in elongational flow fields of dilute and nondilute polymer solutions. In several cases, however, it is demonstrated that the elongational stress is correlated with the elongational rate and agrees with the prediction of the FENE dumbbells model of the second-order type.

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c−1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.

Polymer stretch in dilute fixed beds of fibres or spheres

A theory is developed to describe the conformation change of polymers in flow through dilute, random fixed beds of spheres or fibres. The method of averaged equations is used to analyse the effect of the stochastic velocity fluctuations on polymer conformation via an approach similar to that used in our previous analysis of particle orientation in flow through these beds (Shaqfeh & Koch 1988 a, b ). The polymers are treated as passive tracers, i.e. the polymeric stress in the fluid is neglected in calculating the stochastic flow field. Simple dumbbell models (either linear or FENE) are used to model the polymer conformation change. In all cases we find that the long-range interactions provide the largest contribution (in the limit of vanishingly small bed volume fraction) to an evolution equation for the probability density of conformation. These interactions create a conformation-dependent diffusivity in such an equation. Solutions for the second moment of the distribution demonstrate that there is a critical pore-size Deborah number beyond which the radius of gyration of a linear dumbbell will grow indefinitely and that of the FENE dumbbell will grow to a large fraction of its maximum extensibility. This behaviour is shown to be related to the development of ‘algebraic tails’ in the distribution function. The physical reasons for this critical condition are examined and its dependence on bed structure is analysed. These results are shown to be equivalent to those which we derive by the consideration of a polymer in a class of anisotropic Gaussian flow fields. Thus, our results are explicitly related to recent work regarding polymer stretch in model turbulent flows. Finally, the effect of close interactions and their modification of our previous results is discussed.