Predictions of the excess pressure drop of Boger fluids through a 2:1:2 contraction–expansion geometry using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamics are used to generate the flow of polymer solutions, specifically of Boger fluids, through a planar 2:1:2 contraction–expansion geometry. The solvent molecules are represented by Lennard–Jones particles, while linear molecules are described by spring-monomers with a finite extensible non-linear elastic spring potential. The equations for Poiseuille flow are solved using a multiple time-scale algorithm extended to non-equilibrium situations. Simulations are performed at constant temperature using Nose–Hoover dynamics. At simulation conditions, changes in concentration show no significant effect on molecular conformation, velocity profiles, and stress fields, while variations in the Deborah number have a strong influence on fluid response. Increasing the magnitude of the Deborah number (De), larger deformation rates are developed in the flow region. For a Deborah number of one, the non-dimensional pressure drop presents values lower than the correspondent Newtonian case. However, for large Deborah numbers, the pressure drop increases above the Newtonian reference. An effective excess pressure drop above the Newtonian value is predicted for Boger fluids along this geometry.     

Spectral element method for viscoelastic flows in a planar contraction channel

A new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic fluid flows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four‐to‐one contraction channel for a fluid governed by an Oldroyd‐B constitutive equation. The behaviour of the Oldroyd‐B fluids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Constitutive equations from Gaussian slip-link network theories in polymer melt rheology

Allowing for flow-dependent slip in the junctions of a temporary junction network, we derive the constitutive equations of temporary slip-link networks. The stress tensor is determined by three material functions, namely, the time-dependent linear-viscoelastic memory function, and two strain-dependent functions describing slip and disentanglement of network strands. Slip and disentanglement are related via a mass balance for network strands.By specifying slip and disentanglement, the constitutive equations of Lodge, Wagner, Doi-Edwards, and Marrucci are shown to be special temporary slip-link constitutive equations. To demonstrate the predictive power of temporary slip-link network theories, we compare predictions and extensional flow data with step change in flow direction.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

On the fully developed tube flow of a class of non-linear viscoelastic fluids

The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson-Segalman and some versions of the Phan-Thien-Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage.     

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.     

Slip and induced magnetic field effects on peristaltic transport of Johnson-Segalman fluid

The peristaltic flow of a Johnson-Segalman fluid in a planar channel is investigated in an induced magnetic field with the slip condition.The symmetric nature of the flow in a channel is utilized.The velocity slip condition in terms of shear stresses is considered.The mathematical formulation is presented,and the equations are solved under long wavelength and low Reynolds number approximations.The perturbation solutions are established for the pressure,the axial velocity,the micro-rotation component,the stream function,the magnetic-force function,the axial induced magnetic field,and the current distribution across the channel.The solution expressions for small Weissenberg numbers are derived.The flow quantities of interest are sketched and analyzed.     

Perturbation theory for viscoelastic fluids between eccentric rotating cylinders

The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.     

Planar elongation of soft polymeric networks

A new test fixture for the filament stretch rheometer (FSR) has been developed to measure planar elongation of soft polymeric networks with application towards pressure-sensitive adhesives (PSAs). The concept of this new geometry is to elongate a tube-like sample by keeping the perimeter constant. To validate this new technique, soft polymeric networks of poly(propylene oxide) (PPO) were investigated during deformation. Particle tracking and video recording were used to detect to what extent the imposed strain rate and the sample perimeter remained constant. It was observed that, by using an appropriate choice of initial sample height, perimeter, and thickness, the planar stretch ratio will follow l(t) = h(t)/h₀ = exp([(e)\dot] t)\lambda(t) = h(t)/h_0= \exp({\dot{\varepsilon}} t), with h(t) being the height at time t and [(e)\dot]{\dot{\varepsilon}} the imposed constant strain rate. The perimeter would decrease by a few percent only, which is found to be negligible. The ideal planar extension in this new fixture was confirmed by finite element simulations. Analysis of the stress difference, σ _zz  − σ _xx , showed a network response similar to that of the classical neo-Hookean model. As the Deborah number was increased, the stress difference deviated more from the classical prediction due to the dynamic structures in the material. A modified Lodge model using characteristic parameters from linear viscoelastic measurements gave very good stress predictions at all Deborah numbers used in the quasi-linear regime.     

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

The goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan-Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov-like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non-linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re-entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔP_ent) varied from model to model. The UCM and WM models predicted negative values for ΔP_ent.     

PIV measurements of slow flow of a viscoelastic fluid within a porous medium

Particle image velocimetry (PIV) and pressure loss measurements were used to investigate slow flow through a square array of cylinders having a solid fraction of 10%. The test fluids were a Newtonian fluid and a Boger fluid, both of high viscosity such that the Reynolds number did not exceed 0.1. The pressure loss data reveal that the onset of elastic effects occurred at a Deborah number around 0.5 and that flow resistance was up to several times Newtonian values at Deborah numbers up to 3. PIV showed that the transverse velocity profiles for the Newtonian and non-Newtonian fluid were the same at Deborah numbers below onset. Above onset, the profiles became skewed, increasingly so as the Deborah number increased. In the wake regions between cylinders in a column, periodic flow structures formed in the spanwise direction. The structures were staggered from column to column, consistent with the skewing and were offset. These flow patterns are the result of an apparent elastic instability.     

Steady spinning of the oldroyd fluid B: I: Theory

Equations are derived for the isothermal, low-speed fiber spinning of the Oldroyd fluid B. These equations involve three dimensionless groups whose numerical values are restricted to narrow ranges by the boundary conditions that have to be imposed. Asymptotic results are obtained for limiting values of these groups by means of a perturbation analysis. Also, numerical solutions are presented both in terms of velocity profiles and the admissible values of the three dimensionless groups for different fiber draw ratios. These results are discussed in the context of their application to Boger fluids and are also contrasted with the predictions of the Maxwell fluid model. It is found that under appropriate conditions the predictions of the Oldroyd model B can be quite different from those of the upper convected Maxwell model. In particular, the fluid behaves in a Newtonian manner for both very low and very high Deborah number situations.     

Numerical simulation of viscoelastic flows with Oldroyd-B constitutive equations and novel algebraic stress models

The purpose of the present study is to compare numerical simulations of viscoelastic flows using the differential Oldroyd-B constitutive equations and two newly devised simplified algebraic explicit stress models (AES-models). The flows of a viscoelastic fluid in a 180° bent planar channel and in a 4:1 planar contraction are considered to illustrate and support the underlying theory. The flow in the bent channel is used to illustrate the frame-invariant property of the new models in a pure shear flow exhibiting strong streamline curvature. The flow in the 4:1 contraction serves as a benchmark test in a situation where strong elongation occurs. For both geometries, it is found that the predictions of the new AES-models are in good agreement with Oldroyd-B up to Deborah numbers of order 0.5, with a significant reduction in computational effort.     

The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to the Doi-Edwards model

The nonlinear strain measure of a polyisobutylene (PIB) melt as determined by analysis of uniaxial, planar, ellipsoidal, and equibiaxial extensions is compared to the predictions of the molecular model of Doi and Edwards. It is found that the universal strain function of the Doi-Edwards model is unable to predict the nonlinear behavior of this polymer melt in general extensional flow. The qualitative agreement between predictions and experimental data for the strain dependence of shear stress and first normal stress difference in shear flow that was considered as powerful evidence for the correctness of the Doi-Edwards model seems to be accidental. The exaggerated strain dependence of the model suggests a need to reconsider the assumptions concerning the chain retraction process.Presented at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.Dedicated to Professor F.R. Schwarzl on the occasion of his 65th birthday     

Effect of viscoelasticity on the rotation of a sphere in shear flow

When particles are dispersed in viscoelastic rather than Newtonian media, the hydrodynamics will be changed entailing differences in suspension rheology. The disturbance velocity profiles and stress distributions around the particle will depend on the viscoelastic material functions. Even in inertialess flows, changes in particle rotation and migration will occur. The problem of the rotation of a single spherical particle in simple shear flow in viscoelastic fluids was recently studied to understand the effects of changes in the rheological properties with both numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346] and experiments [Snijkers et al., J. Rheol. 53 (2009) 459–480]. In the simulations, different constitutive models were used to demonstrate the effects of different rheological behavior. In the experiments, fluids with different constitutive properties were chosen. In both studies a slowing down of the rotation speed of the particles was found, when compared to the Newtonian case, as elasticity increases. Surprisingly, the extent of the slowing down of the rotation rate did not depend strongly on the details of the fluid rheology, but primarily on the Weissenberg number defined as the ratio between the first normal stress difference and the shear stress.In the present work, a quantitative comparison between the experimental measurements and novel simulation results is made by considering more realistic constitutive equations as compared to the model fluids used in previous numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346]. A multimode Giesekus model with Newtonian solvent as constitutive equation is fitted to the experimentally obtained linear and nonlinear fluid properties and used to simulate the rotation of a torque-free sphere in a range of Weissenberg numbers similar to those in the experiments. A good agreement between the experimental and numerical results is obtained. The local torque and pressure distributions on the particle surface calculated by simulations are shown.     

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.     

The characteristic‐based split (CBS) scheme for viscoelastic flow past a circular cylinder

A fully explicit, characteristic‐based split (CBS) method for viscoelastic flow past a circular cylinder, placed in a rectangular channel, is presented. The pressure equation in its explicit form is employed via an artificial compressibility parameter. The constitutive equations used here are based on the Oldroyd‐B model. No loss of convergence to steady state was observed in any of the results presented in this paper. Comparison of the present results with other available numerical data shows that the CBS algorithm is in excellent agreement with them at lower Deborah numbers. However, at higher Deborah numbers, the present results differ from other numerical solutions. This is due to the fact that the positive definitiveness of the conformation matrix is lost between a Deborah number of 0.6 and 0.7. However, the positive definitiveness is retained when an artificial diffusion is added to the discrete constitutive equations at higher Deborah numbers. It appears that the fractional solution stages used in the CBS scheme and the higher‐order time step‐based convection stabilization clearly reduce the instability at higher Deborah numbers. The Deborah number limit reached in the present work is three without artificial dissipation and two with artificial dissipation. Copyright © 2007 John Wiley & Sons, Ltd.     

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a?1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.     

VISCOELASTIC MODELLING OF ENTRANCE FLOW USING MULTIMODE LEONOV MODEL

A simulation of planar 2D flow of a viscoelastic fluid employing the Leonov constitutive equation has been presented. Triangular finite elements with lower-order interpolations have been employed for velocity and pressure as well as the extra stress tensor arising from the constitutive equation. A generalized Lesaint–Raviart method has been used for an upwind discretization of the material derivative of the extra stress tensor in the constitutive equation. The upwind scheme has been further strengthened in our code by also introducing a non-consistent streamline upwind Petrov–Galerkin method to modify the weighting function of the material derivative term in the variational form of the constitutive equation. A variational equation for configurational incompressibility of the Leonov model has also been satisfied explicitly. The corresponding software has been used to simulate planar 2D entrance flow for a 4:1 abrupt contraction up to a Deborah number of 670 (Weissenberg number of 6·71) for a rubber compound using a three-mode Leonov model. The predicted entrance loss is found to be in good agreement with experimental results from the literature. Corresponding comparisons for a commercial-grade polystyrene, however, indicate that the predicted entrance loss is low by a factor of about four, indicating a need for further investigation. © 1997 by John Wiley & Sons, Ltd.