Showing papers in "Journal of Non-newtonian Fluid Mechanics in 1976"
TL;DR: In this article, a non-linear constitutive equation for polymer melts and concentrated solutions is presented, which is based on known results of network theories, and the model contains a distinctive feature: that of letting the relaxation times depend upon the existing structure.
Abstract: A non-linear constitutive equation for polymer melts and concentrated solutions is presented. Based on known results of network theories, the model contains a distinctive feature: that of letting the relaxation times depend upon the existing structure. The model extends the constitutive equation of linear viscoelasticity to the non-linear region in a well-defined way, with the uncertainty of just a single adjustable parameter. Predictions of the model for common cases of non-linear response are derived and discussed.
150 citations
TL;DR: In this article, a theoretical analysis for an inelastic liquid having an arbitrary viscosity/shear rate relationship is developed for a commercial rheometer that can be adapted to perform in a squeeze-film mode.
Abstract: This paper contains theoretical and experimental work on the squeeze-film situation for Newtonian and non-Newtonian liquids. To facilitate the interpretation of experimental results, a theoretical analysis is developed for an inelastic liquid having an arbitrary viscosity/shear rate relationship. In the process, it is found necessary to relax the common assumption that material planes which are initially horizontal remain so during the subsequent deformation. It is also shown how the inertia of the moving plate can be accommodated in the analysis. The experimental work contains a description of how a commercial rheometer can be adapted with ease to perform in a squeeze-film mode. Experimental data on polymer solutions indicate that under light-loading conditions, the behaviour of the liquids is predictable from a knowledge of the shear dependent viscosity only. However, under conditions of heavy loading ( i.e. high Deborah number), viscoelastic effects are in evidence. Under these conditions, the liquids behave as better lubricants than one would predict from viscosity considerations.
84 citations
TL;DR: In this paper, a model of nonlinear viscoelasticity with relaxation times which depend on the structure is compared with experimental results reported in the literature for a L.D. polyethylene.
Abstract: A model of non-linear viscoelasticity with relaxation times which depend on the structure is compared with experimental results reported in the literature for a L.D. polyethylene. The single parameter of the model is determined by comparison with steadystate shear results. The model is then used to interpret various transient data. These are: tangential and normal stress growth in shear, stress growth in elongation, normal stresses in shear creep. The comparison shows a good general agreement, thus supporting the suggestion of relating the change in time of the relaxation spectrum to structural variables.
59 citations
TL;DR: In this paper, the second normal stress difference for elastic liquids was determined from a study of the direction and strength of the secondary flows in pipes of non-circular cross-section, and it was concluded unequivocally that the direction of the streamlines is no indication of the sign of v2.
Abstract: In this paper, we are concerned with the possibility of determining the second normal stress difference v2 for elastic liquids from a study of the direction and strength of the secondary flows in pipes of non-circular cross-section.
Experimental results show that the direction of the secondary flows in pipes of square cross-section is the same for six different elastico-viscous systems, although there is some evidence to suggest that the sign of v2 may not be the same for all the test liquids. The situation is clarified by a theoretical analysis which involves the numerical solution of coupled partial differential equations. From the analysis, it is concluded unequivocally that the direction of the streamlines is no indication of the sign of v2.
57 citations
TL;DR: In this article, the intrinsic stress contribution from closely-spaced, rod-like rigid particles suspended in a non-Newtonian liquid subject to a simple extensional flow was derived.
Abstract: A formula is derived for the intrinsic stress contribution from closely-spaced, rod-like rigid particles suspended in a non-Newtonian liquid subject to a simple extensional flow. The present results indicate that, compared to the Newtonian-fluid result given previously by Batchelor [1], a much smaller particle-stress effect may occur in non-Newtonian fluid, owing to shear thinning and possible tensile stiffening in the fluid itself. While unsubstantiated by any critical experimental test, this prediction appears to agree qualitatively with some recent experimental observations of Charrier and Rieger [29] on glass fibers in polymer melts. Further work is suggested for the relevant slender-body theory, particularly the far-field body influence in non-linear fluids.
54 citations
TL;DR: In this article, the growth and decay of the corner eddies as a function of the Reynolds and Weissenberg numbers have been examined, and the numerical method makes use of non-symmetric first-and third-order derivatives for maximizing the diagonal terms of the iterative systems.
Abstract: The flow of a fluid of second grade through three types of plane channels with a contraction has been studied. The numerical method makes use of non-symmetric first- and third-order derivatives for maximizing the diagonal terms of the iterative systems. In particular, the growth and decay of the corner eddies as a function of the Reynolds and Weissenberg numbers have been examined.
51 citations
TL;DR: In this paper, the shear thickening observed in laminar Poiseuille flow for dilute solutions of polystyrene of high molecular weight in decalin is studied as a function of concentration, molecular weight and shear rate.
Abstract: The shear thickening observed in laminar Poiseuille flow for dilute solutions of polystyrene of high molecular weight in decalin is studied as a function of concentration, molecular weight and shear rate. The experiments show the existence of four invariants, i.e. two critical Deborah numbers, a critical viscosity ratio, and a critical hydrodynamic dissipated energy, in terms of which a mechanism of this shear thickening can be proposed. It involves a permanent deformation of the macro molecules above a critical shear.
47 citations
TL;DR: In this article, the application of modern non-linear continuum thermodynamics to the analysis of energy dissipation in both isothermal and non-isothermal flow of polymers is discussed.
Abstract: The application of modern non-linear continuum thermodynamics to the analysis of energy dissipation in both isothermal and non-isothermal flow of polymers is discussed. A fundamental simplification of the field equation for the balance of energy arises if the assumption is made that the flowing polymer is a material with entropic elasticity. This assumption is partly justified by structural arguments, and its validity can be checked experimentally. The results of the experiments performed support the validity of the assumption.
34 citations
TL;DR: In this paper, both polymeric and fibrous drag-reducing additive can be used to suppress the torque of agitators operating in a turbulent regime, but their combined action seems to be additive.
Abstract: Both polymeric and fibrous drag-reducing additives can be used to suppress the torque of agitators operating in a turbulent regime. Unlike the tubular flow, the influence of the fibrous additive is stronger than that of the polymer but their combined action seems to be additive. The action of the drag-reducing additives is further enhanced by the presence of a second phase (gaseous or liquid) in the stirred tank. In aerated tanks, the torque on the shaft of the agitator is not only reduced by more than 70% but also the flooding point is shifted to higher gas rates. An application of this phenomenon to fermentation tanks and similar technologies has been suggested.
29 citations
TL;DR: Two-dimensional creeping sink flow of a Maxwell fluid is an accurate approximation to a bounded converging flow of contraction ratio at least 5 : 1 and covergence half-angles of up to 45°.
Abstract: Two-dimensional creeping sink flow of a Maxwell fluid is an accurate approximation to a bounded converging flow of contraction ratio at least 5 : 1 and covergence half-angles of up to 45°. In this range, a perturbation expansion in Weissenberg number can be used over most of the flow field in the range where stable processing can be carried out.
27 citations
TL;DR: In this paper, it is shown that the Part 1 concentration on the half-time t 1 2 sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied.
Abstract: The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t 1 2 sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied. The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading. The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.
TL;DR: Godfrey et al. as mentioned in this paper proposed a procedure for calculating the variation of power consumption with time when a thixotropic liquid is agitated from rest using an impeller which rotates at constant speed.
Abstract: A procedure has recently been proposed by Godfrey et al . [1] for the calculation of the variation of power consumption with time when a thixotropic liquid is agitated from rest using an impeller which rotates at constant speed. This procedure requires a knowledge of the power requirement for Newtonian and time-independent non-Newtonian liquids together with viscometric data for the thixotropic liquid obtained under constant shear rate conditions. Experimental work to test the procedure has been carried out in a 0.126-m-diameter cylindrical vessel with anchor, helical ribbon and helical screw impellers. Power consumption data were obtained for a range of Newtonian and time-independent non-Newtonian liquids and this was then used to make predictions of the measured power input to the thixotropic liquids: salad cream, tomato ketchup, yoghurt, paint and Laponite solutions. The agreement between experiment and theory was usually better than 10%. However, for the case of the helical screws rotating in Laponite it was observed that regions existed close to the vessel wall where there was no fluid circulation. In such cases, the predicted power input was greatly in excess of the measured value. This is not a serious limitation of the predictive procedure since efficient industrial mixers would keep the entire fluid in circulation.
TL;DR: In this paper, a laser anemometer has been used to study the developing flow both upstream and downstream from the entry plane in a re-entrant tube geometry, and the measured fully developed velocity profiles were found to be in excellent agreement with those calculated.
Abstract: A laser anemometer has been used to study the developing flow both upstream and downstream from the entry plane in a re-entrant tube geometry. A 0.75% polyacrylamide/water solution was used and Reynolds numbers (based on wall conditions in the fully developed downstream flow) in the range 100–500 were obtained in 1.82-cm and 2.40-cm-diameter tubes. The shear stress-shear rate relationship for the fluid was measured using a cone and plate geometry in conjunction with a Weissenberg rheogoniometer. Theoretical fully developed velocity profiles were calculated numerically from these measurements. The measured fully developed velocity profiles were found to be in excellent agreement with those calculated. Velocity profiles measured at the tube entry plane showed the pronounced wall region distortion typically predicted by recent numerical solutions of the flow of purely-viscous fluids through an abrupt tube contraction. It was found that the major velocity rearrangements were achieved within only a few diameters (both upstream and downstream) of the entry plane. In particular, the velocity distribution near the tube wall varied negligibly over the relatively longer distance (many diameters) that it took for the centreline velocity to achieve its fully developed value. Entry lengths were found to be only about half those for purely-viscous fluids. Calculation of the time of flight along the central streamline confirmed that the major rearrangements of velocity suffered by the fluid occurred over a relatively short time period. This indicates that hereditary integral constitutive equations may have to be used in theoretical analyses of this type of flow situation.
TL;DR: In this article, the authors studied the die-swell at the exit of a cylindrical tube and determined axial velocity profiles from the nearly parabolic profile in the tube (compared to the Poiseuille curve computed with the Le Roy-Pierrard model).
Abstract: In the scope of a study on non-viscometric flows of viscoelastic liquids, the authors have studied die-swell at the exit of a cylindrical tube. The work presents the determination of axial velocity profiles, by laser anemometry, from the nearly parabolic profile in the tube (compared to the Poiseuille curve computed with the Le Roy-Pierrard model) to the flat profile in the jet. The main conclusions are that the axial velocities are modified at more than one diameter upstream in the die and rigid body flow is attained at about one radius downstream.
TL;DR: In this paper, a triple jet system was used to measure axial stress, strain and strain rate in a kerosene-based aircraft safety fuel and aqueous solutions of poly (ethylene oxide) and polyacrylamide.
Abstract: A kerosene-based aircraft safety fuel and aqueous solutions of poly (ethylene oxide) and polyacrylamide are examined using the “triple jet” system. This device allows the solution to be stretched as it flows from a capillary tube and the axial stress, strain and strain rate in the liquid are measured. The shear history of the solution is altered by placing cylindrical inserts in the capillary tube. This is shown to have a large effect on the extensional behaviour of aircraft safety fuel, a moderate effect on the extensional behaviour of poly (ethylene oxide) solution and little effect on the behaviour of polyacrylamide solution. The extensional viscosity of the aircraft fuel is raised by an order of magnitude when a long period of high shear is used; the effects last for periods of up to one second, though traditional methods suggest a relaxation time of the order of 10−3 seconds. A liquid of shear viscosity 4 centipoise may have an extensional viscosity of over 100 poise. Plots of the extensional modulus of the jet as a function of distance along the jet emphasize the importance of shear history for the first two types of solution and suggest that the latter stages of the stretching process are elastic in character. Typical extensional moduli for the solutions tested are in the range 1.3–5.0 × 104 dyn.cm−2. The relevance of the interplay between shearing and stretching flow to the phenomena of lubrication and turbulence suppression is mentioned.
TL;DR: The Torsional-balance rheometer as discussed by the authors was developed for measuring the viscometric functions of elastic liquids and was shown to have significant advantages over other rheometers in the very high shear-rate range.
Abstract: The observation made in Part 2 that squeezing flow with a superimposed rotation results in an equilibrium situation with the applied load just balancing the normal stresses generated in the test fluid is used to develop a new technique (the Torsional-Balance Rheometer) for measuring the viscometric functions of elastic liquids. The Rheometer utilizes conventional torsional flow and its novel feature is that the applied load is fixed and the associated shear rate at the rim determined, in contrast to the usual situation where the shear rate is fixed and the total normal force measured. It is argued that the Torsional Balance has significant advantages over other rheometers in the very high shear-rate range, since the normal stresses being measured themselves supply a mechanism for keeping the top plate (which is free to float on the test fluid) at a constant separation from the rotating bottom plate, hence allowing very small gaps to be considered. Consistent data are shown to be possible for shear rates in excess of 10 5 s −1 .
TL;DR: In this article, the linear viscoelastic properties of polybutadiene and three butadiene-styrene copolymers were determined using stress relaxation measurements using Tobolsky-Murakami Procedure X to characterize a discrete relaxation spectrum.
Abstract: The linear viscoelastic properties of a polybutadiene and three butadiene-styrene copolymers were determined using stress relaxation measurements. The Tobolsky-Murakami Procedure X was applied to the data to characterize a discrete relaxation spectrum. The nonlinear behavior was studied using transient shear stress buildup experiments. A single integral nonlinear constitutive equation was able to represent both the linear and nonlinear data if a Bogue-type relaxation time was used which involves a time-averaged second invariant of the rate of deformation matrix. The linear viscoelastic behavior is in essence represented by a single function G(t) (or the parameters Gm, τm, Gm−1, τm−1, Gm−2, τm−2, ⋯). Similar studies were carried out on a butadiene-styrene copolymer based tread and sidewall compounds but the shear stress transient data were incompatible with the stress relaxation results. The reasons for this are discussed.
TL;DR: In this paper, a laser anemometer was used to study the region of accelerating shear flow near the exit of a vertical tube, where the transition between steady laminar shear flows in the upstream tube and elongational flow in the downstream liquid jet takes place.
Abstract: A laser anemometer has been used to study the region of accelerating shear flow near the exit of a vertical tube. It is in this region that the transition between steady laminar shear flow in the upstream tube and elongational flow in the downstream liquid jet takes place. Downstream velocity profiles were measured for solutions of 0.9% polyacrylamide in 85% glycerol/water and 0.9% polyacrylamide in water. Reynolds numbers (based on wall conditions in the fully developed upstream flow) ranged from 45 to 310 and Froude numbers from 0.294 to 4.11. Tubes, having sharpedged and rounded exit corners, with diameters of 1.25 cm and 1.90 cm were used Upstream velocity profiles were measured for a solution of 0.9% polyacrylamide in water. Reynolds numbers ranged from 16 to 670. Only tubes having sharp-edged exit corners were used. It was found that the transition region did not extend upstream into the tube but was confined to the downstream jet. The transition took place over a distance of about 3–5 tube diameters depending upon the value of the Froude number. The axial distance downstream from the tube exit plane at which the velocity profile first became flat increased with increasing Froude number. The magnitude of the jet velocity at this point decreased with increasing Froude number. The condition of the tube exit corner was found to influence the flow in the transition region. Downstream velocity profiles obtained using tubes having rounded exit corners initially develop more slowly than, but soon catch up with and eventually overtake, the corresponding profiles obtained using tubes with sharp-edged exit corners. Downstream velocity profiles obtained for the 0.9% polyacrylamide in 85% glycerol/water solution were found to develop smoothly. The transition from steady shear flow in the tube to elongational flow in the jet took place through the combined processes of acceleration of the outer layers of the jet due to radial transfer of momentum with adjacent inner layers, the process spreading steadily inwards with increasing axial distance from the tube exit plane, and acceleration of the whole due to gravity. However, the velocity profiles obtained for the 0.9% polyacrylamide in water solution did not always develop so smoothly. At a Reynolds number of 310 and Froude number of 2.06 the radial momentum transfer process was restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tube diameters was reached. Thereafter, the transition to a flat profile took place smoothly.
TL;DR: In this article, the steady flow of an infinite mass of elastico-viscous liquid and the generation of this steady state are considered, in the case when an infinite disc is impulsively given a constant angular velocity.
Abstract: Both the steady flow of an infinite mass of elastico-viscous liquid and the generation of this steady state are considered, in the case when an infinite disc is impulsively given a constant angular velocity. It is found that this flow situation, unlike other rotational flows, does not exhibit marked departures from the steady Newtonian velocity profile, a conclusion which contradicts previously published results. During the time the steady state is being generated, however, certain transient elastic effects are evident.
TL;DR: In this article, the authors proposed a method for approximating the shear stress/rate-of-strain curve (called here the rheogram) by a series of piecewise continuous linear segments.
Abstract: The treatment of non-linear partial differential equations of unsteady flows of non-Newtonian fluids generally leads to the use of numerical methods. The present method consists in approximating the practical shear stress/rate-of-strain curve (called here the rheogram) by a series of piecewise continuous linear segments. This method involves the solution of the linear differential equation system using a computer. The study is on unsteady laminar flows of pseudoplastic, dilatant and Bingham fluids. The results obtained by this method are compared with those determined by Laser Doppler anemometry using the Bragg cell. The results are concordant.
TL;DR: In this paper, an analytical description for the divergent radial flow of a viscoelastic liquid between parallel disks is given, where a five-constant Oldroyd model characterizes fluid rheology and yields several other models as limiting cases.
Abstract: An analytical description is obtained for the divergent radial flow of a viscoelastic liquid between parallel disks. A five-constant Oldroyd model characterizes fluid rheology and yields several other models as limiting cases. Predictions are made for the pressure profile P(r,z) and the velocities vr(r,z) and vz(r,z) in terms of truncated series which include inertial effects through the first power of the Reynolds number and elastic effects through the second power of time constants. It is found that quite different predictions arise for models with and without τ (stress) derivatives δτ δt or stress/strain rate (·γ) couplings such as τ · ·γ. Pressures may increase or decrease, and retrograde motions may or may not appear, depending on whether such terms are included in the model and whether time constants are large. The role of non-Newtonian viscosity and possible elastic fracture effects are also discussed.
TL;DR: In this article, the rotational flow in a region bounded by two axisymmetric closed surfaces has been studied by variational methods, and the approach is described in detail for the global power-law representation of a viscosity function.
Abstract: The rotational flow in a region bounded by two axisymmetric closed surfaces has been studied by variational methods. The approach is described in detail for the global power-law representation of a viscosity function. Numerical, as well as simple, analytical results applicable to immersion rotational viscometry are given for the spheroidal geometry of bounding surfaces.
TL;DR: In this paper, a derivation of viscoelastic fluid theory is given which includes the effect of the entropy dependence upon deformation history, and the resulting theory predicts the observed effects of shear thinning and stress overshoot.
Abstract: A derivation of viscoelastic fluid theory is given which includes the effect of the entropy dependence upon deformation history. The theory is then specialized to model second-order effects. The resulting theory predicts the observed effects of shear thinning and stress overshoot. Comparison is made with experimental data.
TL;DR: In this paper, it was shown that if an incompressible Newtonian fluid of viscosity 1/2 flows in a straight pipe of uniform non-circular cross-section under a constant pressure gradient P, unless the velocity is so large that the flow becomes turbulent, each fluid particle moves down the tube in a rectilinear path with velocity v 3 which depends on its position on the transverse crosssection and is given by the solution of the equation.
Abstract: If an incompressible Newtonian fluid of viscosity 1/2 flows in a straight pipe of uniform non-circular cross-section under a constant pressure gradient P, then, unless the velocity is so large that the flow becomes turbulent, each fluid particle moves down the tube in a rectilinear path with velocity v 3 which depends on its position on the transverse cross-section and is given by the solution of the equation:
TL;DR: In this paper, the authors measured pressure profiles for a highly non-Newtonian polymer solution undergoing divergent radial flow between circular disks and found that the liquid is designed to be highly elastic, unlike others examined in radial flow.
Abstract: Pressure profiles P(r) are measured for a highly non-Newtonian polymer solution undergoing divergent radial flow between circular disks. The liquid is designed to be highly elastic, unlike others examined in radial flow, with (τ11 − τ22)/τ21 > 1 in viscometric shear. Also unlike earlier results, pressure data are found to lie above the predictions of the power-law viscosity model. This may be attributed to elastic effects, as predicted with an Oldroyd model in Part I, providing the fluid time spectrum obeys certain restrictions. The Oldroyd model is incapable of quantitative accuracy, but good curve-fits of P(r) data are possible with empirical modification of the rigorous analysis.