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This study compares the predictions of several “suspension temperature” models of particle migration to laser Doppler velocimetry measurements in a concentrated suspension of noncolloidal spheres. We compare the shear rate, concentration, and suspension temperature profiles in narrow-gap Couette flow. The models predict the observed macroscopic shear rate and concentration profiles well at moderate bulk particle concentration but diverge from one another and from the data at high concentrations. In addition, the predictions of the models compare poorly with suspension temperature measurements. Most of the models greatly underpredict the magnitude of the scalar temperature, capturing instead only the magnitude of the smaller two diagonal components of the temperature tensor. Also, the models do not predict the observed variation of the suspension temperature with particle concentration. Our investigation shows that both neglect of suspension temperature anisotropy and qualitative choices of model coefficie...

High-resolution laser Doppler velocimetry and flow-induced birefringence measurements were used to determine velocity and stress fields and to calculate a transient planar elongation viscosity profile for viscoelastic flow on the shear-free centerplane of a planar contraction geometry. A novel, two-component Boger fluid was developed that enabled Weissenberg numbers, We, as high as 2.9 to be attained and allowed the stress-optical coefficients of the fluid components to be measured. The experimental apparatus was designed to subject the fluid traveling on the centerplane to Hencky strains, e, as great as 3.5. Despite the high We and e, the flow induced only a weakly nonlinear viscoelastic stress response in the fluid. A spectral decomposition of predictions of linear and nonlinear constitutive equations was used to demonstrate that the weakly nonlinear response was a result of the nonhomogeneous form of the centerplane strain-rate profile in conjunction with the finite response time of the individual relaxation modes. The degree of nonlinear stress response in this experiment cannot be increased by increasing the contraction ratio to increase the total Hencky strain experienced by a fluid element traveling along the centerplane, increasing the flow rate to increase the strain rate at a given point along the centerplane, or introducing a taper to the contraction geometry to change the strain rate profile. Thus, the experimental configuration of flow through a contraction appears to be of limited value in evaluating the accuracy of nonlinear constitutive models in predicting the stress response of a viscoelastic polymer solution to a generalized elongational flow.

Recent statistical constitutive models of suspensions of neutrally buoyant, noncolloidal, solid spheres in Newtonian fluids suggest that the particles migrate in response to gradients in “suspension temperature,” defined as the average kinetic energy contained in the particle velocity fluctuations. These models have not yet been compared systematically with experimental data. In addition the “temperature'’ models assume isotropic particle velocity fluctuations, since the suspension temperature is given as a scalar. However, highly anisotropic particle velocity fluctuations have been observed experimentally, which suggests that a suspension temperature tensor is more realistic. We use laser Doppler velocimetry to measure particle velocity fluctuations arising from interparticle collisions in a concentrated suspension under nearly homogeneous shear flow in a narrow-gap concentric cylinder Couette device. We compare the relative sizes of the fluctuating velocity components and determine the variation of each...

A wide variety of bead-spring kinetic theory models have been proposed to explain the stress growth and hysteretic behavior of dilute polymer solutions in uniaxial extension. We analyze the Kramers chain, a fine-scale model for polymer dynamics, in order to assess the validity of the coarser-grained bead-spring models in these deformations. Whereas the spring force is a simple function of the dumbbell length for the FENE spring, we find that the relationship between the ensemble-averaged end-to-end force and the extension for a Kramers chain depends on the kinematic history to which it has been subjected. In a quiescent fluid, the Kramers chain force–extension relationship is identical to the FENE force law. However, during start up of elongational flow, the ensemble-averaged end-to-end force for a given (end-to-end) length of the molecule increases with strain until steady state is reached. If the extensional flow is suddenly stopped, the Kramers chain force–extension relationship relaxes back to the FEN...

Laser Doppler velocimetry and flow‐induced birefringence are used to measure the rate of deformation and the principal components of the refractive index tensor in a 5% polyisobutylene (PIB) solution in tetradecane (C14) flowing along the centerplane of an abrupt 3.97:1 planar contraction. The stress optical law is used to interpret the birefringence data in terms of the normal stress difference, which is used to calculate a transient elongational viscosity defined along the centerplane. These measurements are compared directly to predictions of six multimode, differential constitutive models (Oldroyd‐B, White–Metzner, Acierno et al., Giesekus, Bird–DeAguiar, and Phan‐Thien–Tanner) that are fit to steady and small amplitude oscillatory shear flow data for the PIB/C14 solution. The fluid exhibits slight elongational thickening followed by apparent extensional thinning at higher elongation rates. We believe that this ‘‘thinning’’ behavior is due to the decreased residence time of the polymer molecules in the high‐strain‐rate region as the flow rate (and maximum elongation rate) is increased. The nonlinear constitutive equations, except for the White–Metzner model, are virtually indistinguishable in their description of the dynamical response of the fluid in this experiment; however, the Phan‐Thien–Tanner model gives the best quantitative fit to the data. These results point to the need for experiments in which the fluid flowing along the centerline is subjected to a greater total elongational strain.

Finite‐element predictions of steady, two‐dimensional viscoelastic flow based on the elastic‐viscous split stress formulation of Rajagopalan et al. [Rajagopalan, D., R. C. Armstrong, and R. A. Brown, ‘‘Finite Element Methods for Calculation of Steady, Viscoelastic Flow Using Constitutive Equations with a Newtonian Viscosity,’’ J. Non‐Newt. Fluid Mech. 36, 159–192 (1990)] are compared to birefringence measurements of stress for flow between eccentric rotating cylinders of a solution of polystyrene in tricresyl phosphate. Steady and small‐amplitude oscillatory shearing measurements are used to characterize the shear flow rheology of the fluid, and a four‐mode Giesekus model with a Newtonian solvent viscosity is fit to the rheological data. Comparison of the stress fields measured by flow‐induced birefringence in the one‐dimensional flow between concentric cylinders to the mechanically measured viscometric data show that the optical data compare reasonably well with the mechanical data. Comparison of optical...

Results are presented of a detailed rheological study of two different polyisobutylene systems in steady and transient shear flows. Shear‐thinning solutions are prepared by dissolving high molecular polyisobutylene in tetradecane, a low viscosity solvent. Nearly constant viscosity solutions are obtained by adding a third component of highly viscous polybutene. The linear viscoelastic behavior of these fluids is characterized in small‐amplitude oscillatory shearing flow and the nonlinear behavior is studied by means of steady and transient shear flows. The spectrum of relaxation times for both solutions has been determined. Temperature and concentration effects on the material functions are reported. Three differential constitutive equations with four relaxation modes were tested for fitting the experimental data in steady and transient shear flows: they are the quasilinear Oldroyd‐B model and the nonlinear Giesekus and Bird–DeAguiar models. The presence of nonlinear stress terms is found to be important f...

Structural models for foams having two‐dimensional cells with straight line segments and rounded corners are used to study the effect of gas volume fraction (φ) on small shearing deformations. A stress tensor expression is presented which is valid for all gas volume fractions between unity (dry foam) and φ=0.9069 (hexagonally close‐packed cells). Two models are studied. In the first, the linear foam films are considered infinitesimally thin with all liquid in the foam confined to the Plateau borders. In the second model, the films have a finite thickness; the distribution of liquid between the Plateau borders and films varies with strain and is determined from an equal pressure drop requirement across all gas‐liquid interfaces. Identical results are obtained for the two models studied. The stress‐strain relation is a strong function of the gas volume fraction and, in general, the yield or minimum (negative) stress decreases with decreasing gas fraction. Unlike the dry foam model, initial cell orientation ...

A rheological equation of state is developed for a semiconcentrated suspension of stiff fibers in a Newtonian solvent with volume fraction φ in the range (D/L)2

Measurements are reported for the shear stress of a semiconcentrated suspension of rigid fibers in a Newtonian solvent in the startup of steady shear flow. There are n fibers each of length L per unit volume. A variety of different volume fractions of fibers was used; the fiber aspect ratio was varied as well. It is found that the viscous resistance of a randomly oriented fiber suspension is roughly (nL3) times the resistance observed when the particles lie in the planes of shear. The transient shear stress measurements show the rheological properties depend on the total strain and not on shear rate or time separately. Experiments were done in torsional flow between parallel plates separated by distance H to determine the effect of H/L on the measured properties. It is found that there is no effect for H/L>1. Two different models are presented to account for wall effects for H/L

A dilute solution of flexible macromolecules is modeled as a dilute suspension of dumbbells with slightly nonlinear elastic connectors (nearly‐Hookean dumbbells) and subject to Browian motion. The molecular variables are eliminated directly from the kinetic theory by means of a perturbation scheme to give an expression for the stress tensor which is good for flows that do not deform the dumbbells drastically. The stress tensor is expressed in terms of several structure tensors, each of which is given by an auxiliary differential equation. This Hand form of the rheological equation of state can also be expressed in an Oldroyd form or as a memory integral expansion. Calculations of rheological properties for the model show substantial improvements over Hookean dumbbell predictions. In an appendix the distribution function for the nearly‐Hookean dumbbell is derived, and it can be used to obtain the same rheological equation of state by traditional methods.

The experimental study of foam is complicated by its inherently unstable nature and by the presence of a liquid film slip layer at the wall. A method is described for generating reproducible and stable foam capable of retaining its structure for prolonged times. An experimental technique has also been devised which eliminates problems associated with wall slippage and allows measurement of material functions without use of any empiricisms for the wall region. In a steady‐shear flow, foam behaves like a Bingham plastic with a viscosity inversely proportional to shear rate indicating the presence of a yield stress τ0. The value of the viscosity, which is significantly higher than the parent liquid viscosity, is an increasing function of gas volume fraction φ. Yield stress values obtained by extrapolating viscosity versus shear stress data agree with direct τ0 measurements obtained by using a stress relaxation technique. The yield stress is also found to increase with φ. Small amplitude oscillatory shearing ...