Lewis E. Wedgewood

On coil–stretch transitions in dilute polymer solutions

In this paper we examine molecular stretching in the inception of uniaxial elongational flow of dilute polymer solutions. The polymer molecules are modeled as bead–spring chains with finitely extensible nonlinear elastic springs, and we use the Peterlin approximation. This work is distinguished from earlier work because we model the macromolecules with chains instead of dumbbells, and we examine the time dependence of three average quantities describing the chain conformation in unsteady flows: root‐mean‐square end‐to‐end distance, root‐mean‐square extensions of the individual links, and mean moment of inertia about the axis of elongation. We observe a gradual transition from the coiled equilibrium state of the chain to the stretched state after the inception of strong uniaxial elongational flow, and we describe the nature of this transition which takes place in roughly four stages: I equilibrium coil; II deformed coil; III spring stretched (‘‘locally unraveled’’); and IV unfolded chain. Inclusion of hydr...

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid elements net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, ω = ωR + ωD. The deformational vorticity, ωD, arises from shear deformation and is frame invariant while the rigid vorticity, ωR, is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.

Internal viscosity in polymer kinetic theory : shear flows

The Gaussian closure method and Brownian dynamics simulations have been used to calculate the shear material properties of a dilute solution of Hookean dumbbells with internal viscosity. Results for the zero-shear-rate material properties and small amplitude oscillatory shear material properties have been found analytically, and numerical results for the steady state shear material properties are also presented. Two interpretations of the stress tensor are investigated and results are compared. Brownian dynamics simulations are used to obtain material properties of the Hookean dumbbell with internal viscosity without approximations. These simulation results are compared with the perturbation solution of Booij and van Wiechen as well as with a new Gaussian closure solution. Also presented are the contracted distribution functions as derived from the Gaussian closure method and from Brownian dynamics simulations.

A finitely extensible network strand model with nonlinear backbone forces and entanglement kinetics

In an earlier paper, a nonaffine network model of polymer melts was presented in which the rotation caused by shearing as well as the extension of the test strand are hindered by interactions with the network itself. In that work, it was shown that such a strand motion leads to qualitatively correct steady shear and elongational material properties, even though the strand disentanglement rate was constant and the strand force law was linear. These simplifications were accepted in order to emphasize the effects of the strand motion on material properties. In this paper, however, we show that these idealizations cause the model to fail in the start-up of shearing flow because no overshoot is seen in the shear stress growth function. To address this failure, the finitely extensible nonlinear elastic (FENE) network model is introduced in which the FENE force law replaces the Hookean force law used in the earlier finitely extensible network strand (FENS) model. Also, a nonlinear expression for the kinetics of ...

A domain perturbation study of steady flow in a cone-and-plate rheometer of non-ideal geometry

The method of domain perturbation developed by Joseph is used to calculate velocity and stress profiles in a slightly misaligned cone-and-plate rheometer where the cone is spinning and the plate is stationary. Results for a Newtonian fluid, a Criminale-Ericksen-Filbey fluid, an upper-convected Maxwell fluid, and a White-Metzner fluid are presented and compared with earlier results in which the cone is stationary and the plate is spinning (Dudgeon and Wedgewood, 1993). Streamlines calculated for the Newtonian fluid show a very small recirculation region near the stationary plate. Velocity and stress contours are symmetric around the plane of largest gap width. For the elastic fluids studied, streamlines are asymmetric. The fluid response lags where the fluid is dominated by memory effects. Much larger recirculation regions are calculated for fluids dominated by shear thinning. These recirculation regions contain a large fraction of the fluid in the apparatus and have the effect of changing the shape of the flow domain for the remaining fluid that rotates around the cones axis. Elasticity also has a pronounced effect on the stress profile, indicating that the accuracy of the cone and plate may be compromised even for small mis-alignments.

Flow in the misaligned cone-and-plate rheometer

Abstract The effects of small misalignment in the cone-and-plate rheometer on the flow of a Newtonian fluid, an upper-convected Maxwell fluid, and a White-Metzner fluid are examined in detail. The method of domain perturbation and the lubrication approximation are employed to calculate velocity and stress profiles to first order in the misalignment parameter ϵ. Results for the Newtonian fluid show symmetric velocity and stress profiles around the plane at the widest gap. The two elastic fluids studied exhibit asymmetry in their velocity and stress profiles caused by fluid memory effects. Misalignment effects on torque and thrust measurements in the cone and plate are also discussed, and results are compared with a similar calculation made for flow in the journal bearing.

A Brownian dynamics study on ferrofluid colloidal dispersions using an iterative constraint method to satisfy Maxwell’s equations

Ferrofluids are often favored for their ability to be remotely positioned via external magnetic fields. The behavior of particles in ferromagnetic clusters under uniformly applied magnetic fields has been computationally simulated using the Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. However, few methods have been established that effectively handle the basic principles of magnetic materials, namely, Maxwell’s equations. An iterative constraint method was developed to satisfy Maxwell’s equations when a uniform magnetic field is imposed on ferrofluids in a heterogeneous Brownian dynamics simulation that examines the impact of ferromagnetic clusters in a mesoscale particle collection. This was accomplished by allowing a particulate system in a simple shear flow to advance by a time step under a uniformly applied magnetic field, then adjusting the ferroparticles via an iterative constraint method applied over sub-volume length scales until Maxwell’s equations were satisfied. The resultant...

Laser Doppler measurements of flow in a cone-and-plate rheometer and the effect of cone misalignment

Laser Doppler velocimetry (LDV) is used to measure the flow profile of a Newtonian fluid in a cone-and-plate rheometer. The primary and secondary flow patterns are measured in the ideal geometry. The results confirm prior predictions of flow patterns. Flow profiles are also measured in the misaligned geometry in which the cone axis of rotation is tilted slightly off the perpendicular with the plate surface. Numerical predictions of these flow patterns (Dudgeon and Wedgewood, 1994) are also confirmed.

Application of nonuniform magnetic fields in a Brownian dynamics model of ferrofluids with an iterative constraint scheme to fulfill Maxwell’s equations

Ferrofluids are steadily rising in applications across many fields, preferred for their ability to be remotely positioned and controlled via external magnetic fields. In magnetic separation operations, nonuniform magnetic fields elicit a phenomenon known as magnetophoresis so that the ferroparticles will undergo migration toward areas of higher magnetism. To comprehend this behavior, the authors developed a Brownian dynamics simulation of particles in ferromagnetic clusters under the influences of a simple shear flow and an applied magnetic field gradient. An iterative constraint mechanism was implemented to satisfy Maxwell’s equations throughout the dense colloidal suspension, ensuring that essential laws of magnetostatics are rigorously fulfilled at all times over small, finite sub-volumes of the system. Because of the presence of nonuniform magnetic fields, magnetophoresis and magnetic separation behavior were analyzed to assess the effectiveness of the model. Results showed that, when compared to “unc...

Effects of viscosity variations in steady and oscillatory couette flow

A Newtonian fluid with small variations in the viscosity in the primary flow direction of steady and oscillatory Couette flow is studied. These variations in viscosity create a coupling of the components of the momentum equations between the flow-direction component and the gradient-direction component. The coupling leads to secondary flows even in planar Couette flow where a rectilinear flow may be expected for a purely viscous fluid under creeping flow conditions. A perturbation solution has been applied for small-amplitude oscillations in the viscosity in both steady and oscillatory Couette flow. Because many rheological measurements are made assuming rectilinear flow, these results may have important consequences and may allow error caused by heterogeneity to be estimated. Finally, the relation between the momentum and the assumption of a symmetric stress tensor is discussed by introducing an alternative constitutive equation that is linear in the velocity gradient tensor and objective, but gives an asymmetric stress tensor. By adjusting the degree of asymmetry for the stress tensor, the secondary flows can be altered or eliminated.