The numerical simulation of boger fluids: A viscometric approximation approach

A general-purpose finite element program has been used to simulate the flow of nonshear-thinning, highly elastic polymer solutions (Boger fluids). In particular, the creeping flow through an abrupt 4:1 circular and planar contraction is studied, as well as the flow at the exit of a capillary die for the determination of extrudate swell. Experimentally measured normal stress and viscosity data are included in a simple rheological model, based on the viscometric simplification of the CEF constitutive equation. Vortex size and intensity in the die entry and extrudate swell at the die exit increase rapidly, with elasticity level, in general agreement with experimental findings. It is shown that despite the limitations of the model, the viscometric approximation can be used to study the effect of normal stresses in cases where a main flow direction can unambiguously be defined. In die exit Flows, it can also provide an upper limit for the determination of extrudate swell, while Tanner's theory of elastic recovery provides the lower limit.     

Estimation of rheological parameters using velocity measurements

In the present investigation, we develop a method for estimating rheological parameters of viscoelastic fluids using velocity measurement in a square straight channel. It is believed that a somewhat complicated patterns of secondary flows due to the non-zero second normal stress difference are more useful than the simple viscometric flows traditionally adopted in the determination of rheological parameters. The inverse problem of determining the rheological parameters from a set of velocity measurements is solved using a conjugate gradient method. When applied to a general constitutive equation encompassing the UCM model, the Oldroyd-B model and the PTT model, the present method is found to yield a reasonably accurate estimation of five rheological parameters simultaneously even with noisy velocity measurements.     

Huggins viscometric constant of semirigid and rigid-chain aromatic polyamides

The absolute values of the Huggins viscometric constant for semirigid- and rigid-chain aromatic polyamides are high, but not very high in comparison to the values for flexible-chain polymers. The Huggins viscometric constant virtually does not change with some, at least two-fold, change in the molecular weight of semirigid- and rigid-chain polyamides, which is similar to the results known for flexible-chain polymers. Either the thermodynamic rigidity of the macromolecules or the thermodynamic affinity between polymer and solvent as a function of the nature of the solvent have the predominant effect on the values of the Huggins viscometric constant in solutions of aromatic polyamides. Translated from Khimicheskie Volokna, No. 1, pp. 3–7, January–February, 1998.     

A STUDY OF SPINLINE DYNAMICS USING ORTHOGONAL COLLOCATION

Previous work on slow flow of non-Newtonian fluids past particles assemblages has been reviewed. Using a combination of Happel's free surface model and variational principles, bounds on the drag have been obtained for the creeping flow of a Carreau Model fluid past an assemblage of rigid spheres. The bounds are related to friction factor for flow through fixed beds of spherical particles. Numerical results covering a wide range of model parameters and bed voidages are presented.

Theoretical predictions are validated by comparing with experimental results reported in the literature that involve viscoelastic fluids. Arithmetic averages of the two bounds compare well for 182 data points with an average error of 12%. It is demonstrated that the present analysis, though based on a purely viscous model, can predict creeping flow behaviour in rigid particles assemblage for both inelastic and viscoelastic fluids.     

VISCOMETRIC MEASUREMENT OF CHROMIUM(III)-POLYACRYLAMIDE GELS†

Gelled polymers are being used increasingly to modify the movement of injected fluids in secondary and enhaced oil recovery processes. A common gelation process involves the reduction of Cr(VI) to Cr(III) in the presence of polyacrylamide. The Cr(III) reacts or interacts with the polymer to form a gel network. Although correlations of gelation time with principal process variables have been obtained, viscometric data have not been reported during or after gelation. These data are needed for fluid flow calculations in surface equipment and estimation of flow behaviour in reservoir rocks.

A Weissenberg Rheogoniometer, with cone and plate geometry, was used to obtain viscometric data for the gelation of polyacrylamide and chromium (III). Solutions consisting of polyacrylamide polymer, sodium dichromate-dihydrate and sodium bisulfite were gelled under a steady shear field at constant temperature. The shear stress versus time profile for the galation process was interpreted to define a gelation time and to determine the apparent viscosity of the gelled fluid. The gelation time decreased as the applied shear rate increased up to about 14.25 sec^?1 and was affected by shear rate history. Viscometric properties of the gelled solutions were determined. Apparent viscosity of the gelled solutions decreased as the shear rate under which they were formed increased.

Post gelation studies indicated that gels exhibited a residual stress at zero shear rate and behaved as Bingham plastics under steady shear. Gels formed at low shear rates were more viscous than gels formed at high shear rates. However, the structure of these gels was susceptible to shear degradation.     

Extensional flow of non-newtonian fluids—A review

Extensional flows have been the object of study in several laboratories in recent years. Polymeric systems have been studied in most cases because of their interesting behaviour and also because of the importance of their rheological properties to the plastics engineer. Controlled, steady elongation is more difficult to achieve in the laboratory than the more traditional viscometric flows. Moreover, it is not in general possible to predict the response of a viscoelastic material to steady extension based on knowledge of its viscometric functions. This review begins by presenting some useful expressions describing the kinematics of extensional flows. Then some results of interest from rational mechanics are presented and the behaviour predicted by a number of constitutive equations for viscoelastic fluids are discussed. After presenting the pertinent relations of linear viscoelasticity for extensional flows, experimental methods and results for steady simple extension are reviewed and some possible implications for the processing of molten polymers are discussed.     

Viscometric behavior of ternary concentrated solutions of hydroxypropyl cellulose and ethyl cellulose in m-cresol

The shear viscosity of blend solutions of hydroxypropyl cellulose (HPC) and ethyl cellulose (EC) in m-cresol (both HPC/m-cresol and EC/m-cresol systems form lyotropic liquid crystals) was determined by cone-plate-type and capillary-type viscometers. The textures for the same systems at rest and undergoing shear were also observed with a polarized microscope. At shear rate of 1 s^?1, viscosity exhibited a maximum and a minimum with respect to temperature, and this suggested that the phase of the matrix dominated the viscometric behavior of the ternary systems; the blend composition dependence of the viscosity was not additive, and this suggested that HPC and EC were immiscible. At relatively high shear stress, the blend composition dependence of the viscosity greatly depended on the total polymer concentration of the solutions and was quite different from that at low shear rate; the texture of the anisotropic solutions was also different from that at low shear rate. Our findings suggested that the dependence of viscosity on shear and concentration for pure HPC solution was different from that for pure EC solution.     

Compatibility studies on solution of polymer blends by viscometric and phase‐separation technique

Solution blending of polystyrene (PS) and natural rubber (NR) was carried out in toluene and chloroform to determine the compatibility. Experimental evidence for the compatibility of these blends was derived from viscometric and phase‐separation studies. The viscometric method was based on the intrinsic viscosities of transfer of polymer in pure and “mixed” solvents. The compatibility of these blends based on the heat of mixing was also examined theoretically. All the experimental and theoretical evidence show that the blends are incompatible at the compositions studied. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 880–889, 2000     

The effects of compatibility on the mechanical properties and fatigue resistance of butyl/EPDM rubber blends

In this study, compatibility of butyl/ethylene‐propylene‐diene rubbers in different ratios was investigated byviscometric technique. For evaluating polymer compatibility, the parameters Δb suggested by Krigbaum&Wall, ΔB and μ that were suggested by Chee were calculated. Data obtained from viscometric study were used for preparing new compound recipe. Mechanical and rheological properties of the new compounds were measured and the changes in the properties before and after thermal aging were determined. Replacement of ethylene‐propylene‐diene rubber with butyl rubber in appropriate ratio improved the fatigue resistance of the vulcanizates. Improvement of dynamic mechanical properties of polymer blends is expected only in the case of blending compatible polymers in appropriate ratios. The compatibility of blends in different ratios has been studied by viscometric technique and mechanical measurements and obtained very similar results. It has been concluded that viscometric determination of butyl/ethylene‐propylene‐diene rubbers compatibility could be used as a simple technique for predicting the mechanical properties of the same rubber blend that was related with solid phase compatibility. POLYM. COMPOS., 31:1869–1873, 2010. © 2010 Society of Plastics Engineers.     

Analysis of Shear Stress Growth Experiments for Linear Constitutive Equations

Shear stress growth curves for viscoelastic fluids at low shear rates are analyzed using two linear rheological constitutive equations, an integral constitutive equation and a mixed type constitutive equation. It is shown that some published solutions do not satisfy all of the pertinent boundary conditions. For the low shear rate region, available experimental shear stress curves show a monotonic increase with decreasing slope in the shear stress. Shear stress curves calculated using a mixed type constitutive equation are found to exhibit this type of behavior while curves calculated using an integral constitutive equation do not. For the mixed type constitutive equation, the calculated developing velocity distribution is used to examine its effect on the developing shear stress distribution. For low values of E (the elasticity number), there is a moderate effect, but, for sufficiently large values of E, the developing velocity distribution has a negligible effect. It is also shown that results consistent with experimental data obtained at low shear rates can be attained using a single relaxation time. Additionally, incompressible Newtonian fluids are considered, and it is found that there can be single maxima in some shear stress curves with no maxima occurring in the velocity curves. Multiple maxima were not obtained in the Newtonian shear stress results unlike some published results.     

Melt transformation coextrusion. II: Flow analysis

Melt Transformation Coextrusion for power law fluids was analyzed with particular attention being paid to the influence of the die's geometrical shape on flow patterns, and to skin layer effect on the velocity profile and shear stress at the interface between layers in a three-layered system. Volumetric flow rate expressions were also generated for the aforementioned system, and stream and potential functions were derived for a free boundary converging film geometry. Newtonian as well as non-Newtonian fluids were considered for this analysis in which the streamline shape changes their convergence downstream into the land section. Also, a finite element solution technique was utilized to reveal shear stress profiles that arise from the flow of high density polyethylene at 200°C in a converging fixed boundary geometry.     

SLOW NON-NEWTONIAN FLOW PAST AN ASSEMBLAGE OF RIGID SPHERES

Previous work on slow flow of non-Newtonian fluids past particles assemblages has been reviewed. Using a combination of Happel's free surface model and variational principles, bounds on the drag have been obtained for the creeping flow of a Carreau Model fluid past an assemblage of rigid spheres. The bounds are related to friction factor for flow through fixed beds of spherical particles. Numerical results covering a wide range of model parameters and bed voidages are presented.

Theoretical predictions are validated by comparing with experimental results reported in the literature that involve viscoelastic fluids. Arithmetic averages of the two bounds compare well for 182 data points with an average error of 12%. It is demonstrated that the present analysis, though based on a purely viscous model, can predict creeping flow behaviour in rigid particles assemblage for both inelastic and viscoelastic fluids.     

TRANSVERSE LAMINAR FLOW OF NON-NEWTONIAN FLUIDS OVER A BANK OF LONG CYLINDERS

The creeping transverse flow of incompressible Carreau model fluids past an array of infinitely long cylinders has been analysed theoretically. The hydrodynamic interactions between cylinders have been simulated using a simple cylinder-in-cylinder free surface cell model. In this formalism, the overall mean porosity characterises an array of cylinders without any assumption regarding the actual geometrical arrangement of the individual cylinders. The resulting non-linear field equations have been solved approximately by employing the well known velocity and stress variational principles. The resulting upper and lower bounds are non-coincident and diverge increasingly with the rising extent of non-Newtonian behaviour of the liquid medium. However, the mean value of the drag coefficient deviates from the individual bounds by no more than 22% and therefore, the use of the arithmetic mean of the upper and lower bounds is suggested. The theoretical predictions reported herein encompass wide ranges of physical and kinematic conditions as follows: 1 ≥ n ≥ 0.2; 0.9 ≥ ε ≥ 0.3 and ∧ ≤ 500. The paper is concluded by presenting comparisons between the present predictions and the scant experimental results for two limiting cases, namely, for the flow of Newtonian fluids (n = 1) through assemblages of solid rods and random fibrous beds, and for the flow of power law liquids (i.e., for large values of ∧) through banks of rods. The correspondence is found to be satisfactory.     

THERMODYNAMIC EQUILIBRIUM OF NONHOMOGENEOUS AND NONISOTROPIC CHEMICALLY REACTIVE SYSTEMS

Thermodynamic equilibrium conditions for nonhomogeneous and nonisotropic chemically reactive systems are developed. In a preliminary discussion it is shown that the fundamental equilibrium criteria are the constancy of the temperature and of the generalized chemical potentials, and that these are the only independent equilibrium criteria in the absence of a chemical reaction. The main result is the mechanical equilibrium criterion for this general case, in which a weighted average of the stress tensor components replaces the hydrostatic pressure of simple fluids. This criterion is compared with the mechanical equilibrium condition which is derived from Newton's laws and the implications of this comparison for simple fluids and for elastic systems are discussed. The equilibrium constant for a chemical reaction is independent of the nonisotropy and nonhomogeneity of the system.     

A Viscometric Approach to the Study of the Kinetics of Polyurethane Reactions

Kinetics and rheological studies were carried out on polyurethane reactions involving a new polyol and TDL Conversion of isocyanate by uncatalyzed polymerization under isothermal conditions was monitored by IR spectrometry and titration. Viscosity changes of the reaction mixture were followed by a digital cone-and-plate viscometer under identical conditions. Conversion and viscosity were found to follow a power relationship. A relationship involving the order of the reaction n that connects the viscosity with time was derived. This provides a new viscometric method to find the order of the reaction. The order of the reaction determined by this method was very close to the values obtained by a fractional-life method already reported.     

DIMENSIONAL ANALYSIS OF HEAT AND MASS TRANSFER CORRELATIONS USING MICRO- AND MACRO-SCALES

A new kind of dimensional analysis, which uses two kinds of scales, provides a powerful tool for analyzing heat and mass transfer correlations as has been demonstrated by Ruckenstein (1986) in the case of forced convection of Newtonian fluids. In this work, the approach based on micro- and macro-scales is extended to free convection of Newtonian fluids, as well as to forced and free convection of power-law fluids.

As the micro-participation and macro-participation dominate in laminar and turbulent flow respectively, bounds on the exponents of the Reynolds number Re and the Grashof number Gr can be established in the case of forced and free convection respectively. An upper bound is obtained for laminar flow and a lower bound for turbulent flow. Thus, for free convection in Newtonian fluids, it is demonstrated that the bounds on the exponent of Gr for laminar and turbulent flow are 0.25 and 0.375 respectively.     

The compressive deformation of polymeric foams

An alternative approach has been developed to evaluate compressive stress on polymeric foams. Compressive stress depends primarily on three factors: foam density, deformation strains, and deformation rates. The density dependency derived from this approach agrees closely with the empirical models reported in the literature. Correlations between the dynamic compression and shear deformation are also derived. Experimental data are presented which show that G′ and G″ increase with increasing strain rates, while the damping factor reaches a maximum at low strain rates. Note that the proposed model for the prediction of cell-wall rupture would not apply to foams with high open-cell contents.     

Axial pressure profiles in non-newtonian flow: Part I: Theoretical and experimental considerations

The flow behavior of polymeric fluids in tubes can be determined from measurements of the radial stress profile in the flow direction. This subject, the focus of our research, will be examined in Part II of this communication. Part I describes a research designed to test the accuracy of stress measurement with pressure transducers in a variety of recess mount geometries. No mount geometry errors were detected for the fluids used in this work over a shear rate range of 1 to 2000 sec^?1. Part I also contains a survey of the literature and theoretical considerations pertinent to the entire work.     

VISCOMETRIC MEASUREMENT OF CHROMIUM(III)-POLYACRYLAMIDE GELS†

Gelled polymers are being used increasingly to modify the movement of injected fluids in secondary and enhaced oil recovery processes. A common gelation process involves the reduction of Cr(VI) to Cr(III) in the presence of polyacrylamide. The Cr(III) reacts or interacts with the polymer to form a gel network. Although correlations of gelation time with principal process variables have been obtained, viscometric data have not been reported during or after gelation. These data are needed for fluid flow calculations in surface equipment and estimation of flow behaviour in reservoir rocks.

A Weissenberg Rheogoniometer, with cone and plate geometry, was used to obtain viscometric data for the gelation of polyacrylamide and chromium (III). Solutions consisting of polyacrylamide polymer, sodium dichromate-dihydrate and sodium bisulfite were gelled under a steady shear field at constant temperature. The shear stress versus time profile for the galation process was interpreted to define a gelation time and to determine the apparent viscosity of the gelled fluid. The gelation time decreased as the applied shear rate increased up to about 14.25 sec^-1 and was affected by shear rate history. Viscometric properties of the gelled solutions were determined. Apparent viscosity of the gelled solutions decreased as the shear rate under which they were formed increased.

Post gelation studies indicated that gels exhibited a residual stress at zero shear rate and behaved as Bingham plastics under steady shear. Gels formed at low shear rates were more viscous than gels formed at high shear rates. However, the structure of these gels was susceptible to shear degradation.