Normal Stress Difference Coefficient Research Articles

Non‐uniformity of gas dispersion in turbine‐generated viscoelastic circulation flow

AbstractGas dispersion has been analysed experimentally by measuring local gas hold‐up in two lab‐scale conventional turbine‐agitated vessels loaded with viscoelastic fluids. The gas hold‐up radial profiles obtained in various axial positions close and far from the impeller reveal high non‐uniformity of gas dispersion. Non‐uniformity is quantified and related to elasticity, shear dependent viscosity, shear rate, first normal stress difference coefficient and type of fluid. The low values of gas hold‐up obtained for the vessel central area are interpreted as hindered gas dispersion caused by the flow pattern specific for the viscoelastic properties of the circulating fluid. The paper is complementary to the effort for collecting up data for benchmarking numerical CFD simulations (Kumar et al., 1997).

Experimental studies on the Taylor instability of dilute polymer solutions

In this research the stability of Taylor-Couette flow of Newtonian and drag reducing fluids was experimentally studied. The primary concern was to investigate the effect of added polymer on the critical Taylor number. The polymer chosen in this study was polyacrylamide, xanthan gum and polyacrylic acid and the solvent was 1:1 mixture of glycerine and distilled water. The concentration was in the range of 0–0.5 of the coil overlap concentration. The result showed that the qualitative characteristics of transition from the stable Couette flow to the turbulent flow was found to be similar to that of Newtonian fluid. In the cases of polyacrylamide and xanthan gum solutions, the critical Taylor number decreased as the polymer concentration increased. Some polyacrylic acid solutions showed increase in the critical Taylor number. The Couette flow was stabilized by the added polymer when the solvent viscosity was used in determining the critical Taylor number regardless of polymer type. This implies that the wall layer of turbulent pipe flow should be stabilized and hence manifest drag reduction. By using a dilute solution theory, the second normal stress difference coefficient was estimated. It was found that the ratio of second to first normal stress difference coefficients was negative and in the range of 0.0–0.2, and was closely related to the flexibility of polymer chain. It was also found that even a very weak shear rate dependency of viscosity could play an important role in determining the stability of Couette flow.

A constant-contour-length reptation model without independent alignment or consistent averaging approximations for chain retraction

A reptation model for the primitive chain that does not assume independent alignment or consistent-averaging for the retraction process, or equilibrium relaxation for the reptation process is proposed and compared to the analytical expressions of Doi and Edwards in single-step, double-step strains and steady-state shear flow. The Doi and Edwards model with independent alignment approximation underpredicts the magnitude of the relaxation modulus by 25%, and consistently overpredicts the magnitude of the damping function; for steady shear flow, it predicts the correct shape for the steady-state viscosity and the first normal stress difference coefficient, although the magnitude is incorrect. The analytical expressions of Doi and Edwards without independent alignment approximation are excellent approximations to the damping function. In double-step strains, the expressions of Doi assuming consistent averaging, but no independent alignment, predict well the stress decay following the second strain. Linear response theory is found to be invalid for describing the stress relaxation following single-step strain for the models considered. Similar to the Doi and Edwards model, no overshoot for the first normal stress difference is observed for the simulation model. Unlike the Doi equation derived without the independent alignment approximation but restricted to double-step strains, the simulation model proposed here can be easily generalized to complex flow fields. No contour length fluctuation or constraint release is considered in this model, and chain retraction is assumed to be instantaneous.

Non-newtonian effects on the drag of creeping flow through packed beds

Slow flows of non-Newtonian fluids through packed beds of solid particles are studied numerically and analytically using the free-surface cell model to account for the interactions between particles. The flow problem of a Carreau fluid is solved by the finite difference method and that of a second-order fluid by the perturbation method. It is shown that the flow drag decreases with a decrease in the flow behavior index and with an increase in the characteristic time. The degree of this reduction is found to be more significant at low voidages. The numerical results are found to be closer to the lower bounds obtained using variational principles by earlier investigators. The perturbation solutions predict that the second normal stress difference coefficient has a significant influence on the flow resistance. The flow resistance can either increase or decrease with an increase in the Deborah number, according to the values of the second normal stress difference coefficient. The results are found to be in agreement with the experimental findings that the viscoelastic flow through packed beds can exhibit a rapid increase in the flow resistance, over and above that expected for a comparable viscous fluid, in the second normal stress difference coefficient range for most real viscoelastic fluids.

Structure and dynamics of a polymer solution subject to flow-induced phase separation

Experiments combining mechanical rheometry with polarimetry (birefringence and scattering dichroism) have been conducted on a 6% solution of polystyrene (1.86x106 molecular weight) in dioctyl phthalate. Birefringence is used to measure the extent of segmental orientation, whereas the dichroism is sensitive to orientation and deformation of concentration fluctuations associated with the process of flow-induced phase separation. The results indicate that these fluctuations grow predominately along the neutral (or vorticity axis) of a simple shear flow. At higher rates of shear, orientation in the flow direction is favored. The transition in orientation direction is accompanied by time-dependent behavior in the optical properties of the solution during shear and the onset of shear thickening of the viscosity and the first normal stress difference coefficient.

Dynamics of colloidal particles in sheared, non-Newtonian fluids

The dynamics of colloidal, axisymmetric particles suspended in non-Newtonian, polymeric liquids are examined by using optical techniques that are capable of following their average orientation. Polarimetry measurements of birefringence and dichroism are used on dispersions of colloidal iron oxide spheroids and small-angle light scattering is used on hardened, prolate human red blood cells. A variety of suspending liquids are used including Boger fluids designed to have some of the attributes of a second order fluid under steady state flow conditions, and concentrated polystyrene solutions with rheological properties similar to more commonly encountered polymer liquids. As a result of the non-Newtonian properties of the polymeric suspending fluids, and in particular the presence of normal stresses, the particles were observed to drift out of the orbits associated with their motion in Newtonian fluids. As predicted by Cohen and coworkers, in flows where the elastic forces on the particles dominate Brownian forces, the particles tend to drift away from orientations favoring the flow direction and towards the vorticity axis. When Brownian forces are large compared with the elastic forces, alignments in the flow direction are more probable. This trend, however, was only observed when the Boger fluids were used as suspending fluids. Although the polystyrene solutions possessed shear viscosities and first normal stress difference coefficients that were comparable in magnitude to those of the Boger fluids, only orientations along the flow direction were found.

Generalized hydrodynamics and Reynolds-number dependence of steady-flow properties in the cylindrical Couette flow of Lennard-Jones fluids.

The Reynolds-number dependence is calculated for the shear stress, normal stress difference, velocity, effective viscosity, and effective normal stress coefficient by using the generalized hydrodynamic equations in the case of cylindrical Couette flow of a Lennard-Jones gas. The drag coefficient is shown to decrease more rapidly than the inverse Reynolds number and rapidly vanish on passing a critical Reynolds number. The shear stress and normal stress difference also vanish beyond the critical Reynolds number, and there is a region in space where the effective shear viscosity and normal stress coefficients become so small that the fluid behaves like an almost inviscid fluid in the region. These numerical results indicate that the generalized hydrodynamic equations tend to the Euler equations in the limit of sufficiently large Reynolds number at a rate faster than that for the Navier-Stokes equation and that the fluid is almost inviscid in some parts where the velocity changes steeply, and viscous in the rest.

Dependency of the Steady‐State and Transient Viscosity and First and Second Normal Stress Difference Functions on Molecular Weight for Linear Mono and Polydisperse Polystyrene Solutions

Experimental data are reported for solutions of polystyrene in normal butyl‐benzene for concentrations, C, of 450 kg/m3 (with two exceptions). Polystyrene molecular weights, Mw, varied from 37,000 to 2×106, and molecular weight distributions, Mw/Mn, varied from 1.06 to 22. The solutions were sheared in cone‐plate geometry in steady‐state, oscillatory, and stress growth and decay shearing modes. For CMw>35,000, the viscosities depend on a molecular weight, M3,4. The first and second normal stress difference and related function dependences are similar. Both the first and second normal stress difference functions of the shear rate are similar and approach a constant “zero” shear rate asymptote. The ratios of the second to first normal stress difference coefficients for steady‐state shearing and the same ratios for the displacement and amplitude coefficients for oscillatory shearing have average values of −0.25, −0.225, and −0.22, respectively. The steady‐state shear compliances, Je0, computed from steady‐state “zero” shear rate first normal stress difference coefficients and the corresponding second normal stress difference properties are proportional to Mw(Mz/Mw)3,5 for CMw<critical. The normalized first and second normal stress differences grow and decay at the same relative rates upon the abrupt inception and discontinuance of steady‐state shearing.

Dependence of the primary normal-stress difference coefficient on molecular weight for concentrated solutions of rodlike molecules

The primary normal-stress difference has been measured for concentrated solutions of rodlike and flexible chain polyamides. The solutions studied here consist of poly-p-phenyleneterephthalamide (PPT) which is believed to be rodlike in 100% H2SO4 and the flexible chain polyamide, polyhexamethylene adipamide (nylon 6.6), which is also dissolved in 100% sulfuric acid. For the PPT solutions the primary normal-stress difference coefficient in the limit of zero shear rate (ψ 1(0)) is observed to be proportional toM 12.9 whereM w is the weight average molecular weight. For concentrated solutions of nylon 6.6,ψ 1(0) is found to be proportional toM 7.6 which is in reasonable agreement with values reported for other flexible chain polymers. These data point out that there are some interesting differences in the primary normal stress differences for rodlike and flexible chain polymers and at the same time provide support of the recent theoretical predictions ofDoi andEdwards for concentrated solutions of entangled rodlike molecules.

Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation

Flow visualization experiments have been carried out on these melts flowing from a reservoir into a capillary die. The existence and magnitude of vortices at the die entrance have been determined over a range of extrusion conditions. The vortex size is interpreted in terms of the theory of viscoelastic fluid mechanics. It is found that the second-order fluid-perturbation solution cannot represent the observed experimental results. The data are correlated with (i) a Weissenberg number τ ch V L ≡ \\ ̄ gt( γ ̇ w) γ ̇ w ≡ Ψ 1 γ ̇ w/2η  ( N 1) w/ 2(σ 12) w measured at the die wall and (ii) with the deformation-rate dependence of relaxation time. Interpretation of vortex formation and size in terms of elongational viscosity is offered. Several polystyrene and polyethylene melts have been rheologically characterized as part of this study with measurements of viscosity η and principal normal stress difference N 1. The zero shear viscosity η 0 of the polystyrenes varies with the 3.5 power of the weight-average molecular weight M w while the principal normal stress difference coefficient Ψ 1 varies with the sixth power of M w when evaluated at a shear rate of 1 sec −1.

The influence of titanium dioxide on the rheological and extrusion properties of polymer melts

An experimental study of the influence of titanium dioxide (TiO2) on the rheological and extrusion properties of five polymer melts (two low-density polyethylenes, two high-density polyethylenes, and a polystyrene) has been carried out. Increasing TiO2 loading increases the shear viscosity η, with the extent of increase being greater at lower shear rates. At moderate and high TiO2 loadings, the filled melts may possess yield values. Empirical equations relating viscosity to filler loading have been developed. The first normal stress difference was measured for the melts and found to increase with increasing TiO2 loading. However, the extent of increase was less than found for the viscosity function and interpretation in terms of the theory of viscoelasticity suggests that the characteristic relaxation time of the melts decreases with increasing TiO2 level. Empirical equations relating the first normal stress difference coefficient to volume fraction of the filler have been developed. Addition of TiO2 is found to decrease extrudate swell and retard the occurrence of extrudate distortion.

Melt viscosity of poly(vinyl chloride) formulations

AbstractCommercial, suspension‐type PVC resin, poly (vinyl chloride), molecular weight Mw × 10−4 = 8.6 ± 0.9, polydispersity Mw/Mn = 2.26, was mixed with plasticizer di(2‐ethyl hexyl)phthalate (DOP) and organo‐tin stabilizer in four different proportions. The mixtures were milled and pressed into sheets for testing. The polymer content in these samples was 97, 80, 60, and 40 wt percent. The viscoelastic properties of the materials were investigated using a Weissenberg rheogoniometer in a cone‐and‐plate, steady‐state shearing mode. The viscosities and primary normal stress difference coefficients were measured at shear rates of 10−2 ≤ \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^. $\end{document} (sec−1) ≤ 102 and at temperatures from 151 to 246°C. The zero shear viscosities, plotted as log η0 vs 1/T (T is the absolute temperature) did not follow either a straight line dependence, reported for PVC melts at low shear rates at 170–190°C, nor was any discontinuity found near 195°C as by others; the data follows a continuous concave curve. The apparent activation energy of flow increases steeply with decreasing temperature. The data can be represented by a WLF type of equation, but the magnitudes of the parameters of this relation differ from expected values. A crossplot of log η0 (T = const.) vs log w (where w is the polymer content) also demonstrates a faster increase of η0 with w than expected from the straight line dependence. The primary normal stress difference coefficient was found to increase with w and decrease with T, paralleling the observed dependencies of η0.

An integral rheological model and its application in steady and unsteady viscoelastic data of a polyisobutylene-dekahydronaphthalin solutions

A simple integral model is proposed which predicts the viscosity and the first normal stress difference coefficient in conditions of steady simple shearing flow. The model also predicts the time dependent viscosity during the buildup of the shear stress after suddenlyimposed constant shear rate. The experiments performed with aWeissenberg Rheogoniometer on a PIB-Dekahydronaphthalin solution support the integral model in a considerable range of the shear rate.