Showing papers in "Journal of Non-newtonian Fluid Mechanics in 1980"
TL;DR: In this article, a detailed formulation for simulating the injection-molding filling of thin cavities of arbitrary planar geometry is presented, in terms of generalized Hele-Shaw flow for an inelastic, non-Newtonian fluid under non-isothermal conditions.
Abstract: A detailed formulation is presented for simulating the injection-molding filling of thin cavities of arbitrary planar geometry. The modelling is in terms of generalized Hele-Shaw flow for an inelastic, non-Newtonian fluid under non-isothermal conditions. A hybrid numerical scheme is employed in which the planar coordinates are described in terms of finite elements and the gapwise and time derivatives are expressed in terms of finite differences. The simulation is applied to the filling of a two-gated plate mold having an intentionally unbalanced runner system. Good agreement is obtained with experimental results in terms of short-shot sequences, weldline formation and pressure traces at prescribed points in the cavity.
474 citations
TL;DR: In this article, a complete rheological equation of state for dilute polymer solutions is obtained by modelling the polymer molecules as bead-spring chains, in which the springs are finitely extensible.
Abstract: A complete rheological equation of state for dilute polymer solutions is obtained by modelling the polymer molecules as bead—spring chains, in which the springs are finitely extensible. Hydrodynamic interactions among the beads are accounted for approximately by using the equilibrium-averaged Oseen tensor as in the Zimm theory. The rheological equation of state thus obtained may be regarded as an extension of the Lodge—Wu equation. Consequently, it incorporates all the good features of the Zimm theory, but it can describe nonlinear viscoelastic phenomena as well, such as monotone decreasing shear-rate-dependent viscosity. Comparisons with experimental data on shear flows are encouraging.
393 citations
TL;DR: In this article, the authors presented numerical results on the calculation of die swell at the exit of slit, circular and annular dies, where the material is an upper-convected Maxwell fluid.
Abstract: Numerical results are presented on the calculation of die swell at the exit of slit, circular and annular dies. The material is an upper-convected Maxwell fluid (rubberlike liquid); the numerical method is of the mixed finite element type.
135 citations
TL;DR: In this article, the memory kernel is chosen to be a single exponential in the time lapse (Maxwell model) and the formulation is such that it can easily be generalized to more realistic models such as the BKZ theory.
Abstract: A finite element simulation has been carried out for a viscoelastic fluid of the single integral, memory type. In the current work, the memory kernel is chosen to be a single exponential in the time lapse (Maxwell model). However, the formulation is such that it can easily be generalized to more realistic models such as the BKZ theory. From the point of view of numerical analysis, differential models are appealing because they avoid the complexities of memory integrals. However, in these models the viscoelastic effect always enters through terms having the highest-order derivatives. The disadvantage of this situation for numerical analysis appears to be borne out in the experiences reported recently by several workers. In a memory integral formulation, the demand on differentiability of the velocity field is no greater than for the Newtonian fluid. The basic idea in the formulation is the approximation of the memory integral by a Laguerre numerical quadrature formula. The kinematical problem is the computation of the displacement vector from every node to the Laguerre points upstream along particle paths. Since this operation requires the velocity field to be known, the method is restricted to the calculation of non-linear effects as body forces. Thus, the equations being solved in any iteration are those of linear viscous flow with an arbitrary body force. In spite of this limitation, the method converges at dimensionless relaxation times greater than the largest values attained with formulations based on differential models. In the present paper, the method is illustrated with the die entry flow in which the fluid is forced through a four-to-one axisymmetric contraction.
127 citations
TL;DR: In this article, the effect of fluid elasticity on the drag coefficient in the absence of any significant shear-thinning effects is clearly demonstrated, and a continuous reduction in drag below the Stokes value is observed until an asymptotic reduction of 26% is reached for We >/ 0.7.
Abstract: Drag coefficients are measured for the creeping motion of a sphere in nonshear-thinning elastic fluids. The data obtained cover a Weissenberg number range of 1.66 × 10−4 to 2.02. For 0 ⩽ We ⩽ 0.1 no significant deviation from the Stokes drag is observed as a result of fluid elasticity. For We > 0.1 a continuous reduction in drag below the Stokes value is observed until an asymptotic reduction of 26% is reached for We >/ 0.7. Existing theoretical analyses are inadequate for predicting the reduction in drag observed and its asymptotic value. The effect of fluid elasticity on the drag coefficient in the absence of any significant shear-thinning effects is clearly demonstrated.
106 citations
TL;DR: In this article, a photograph is taken of a straight edge and its reflection in the fluid surface and the shape of the surface is then calculated from the shape using geometrical optics.
Abstract: The second normal stress difference N2 can be determined by measuring the free surface shape of fluids flowing down a semicircular channel. A photograph is taken of a graduated straight edge and its reflection in the fluid surface. The shape of the surface is then calculated from the shape of the straight edge reflection using geometrical optics. We show that the surface shape can be measured to an accuracy of ±0.01 mm by this photographic method, leading to a value of N2 which is accurate to about ±0.1 N m−2.
A study was made of a number of moderately concentrated polymer solutions. The shear stress τ and the first nomral stress difference N1 were measured on a cone—plate rheometer while the second normal stress difference N2 was determined from the pipe flow experiment. The normal stress ratio N2/N1 was found to be negative and independent of shear rate in the range 1 < γ. < 100 s−1. The ratio N2/N1 is weakly dependent on concentration and our values are consistent with those obtained by other investigators using different methods to measure N1 and N2.
82 citations
TL;DR: In this article, the motion of a rigid particle in a viscoelastic fluid under the assumption of negligible inertial effects (based on particle dimensions) was studied, focusing solely upon situations where no change of orientation or position of the particle is possible in a Newtonian solvent.
Abstract: In this paper, we study the motion of a rigid particle in a viscoelastic fluid under the assumption of negligible inertial effects (based on particle dimensions). Concentrating solely upon situations where no change of orientation or position of the particle is possible in a Newtonian solvent, it suffices to consider the low Weissenberg number limit. By employing the concept of a second-order-fluid, the theoretical predictions for a single particle in an essentially unbounded domain correlate quite well with experimental results. As soon as interaction effects (particle—particle and particle—wall, respectively) are included in the theory, all predictions are at odds with the observations.
66 citations
TL;DR: In this paper, the authors present an analysis and critical appraisal of capillary and slit methods of normal stress measurement, and conclude that a detailed understanding of velocity profile rearrangement of viscoelastic liquids is essential before exit flow measurements can be used for normal stress measurements.
Abstract: The procedures currently in use for measurement of normal stresses at high shear rates utilize the flow at the exit of a capillary or slit. These methods, which follow from a common analytical framework, require the measurement of the excess exit pressure, in the case of a polymer melt or concentrated solution, and the jet swell or reactive thrust, in the case of a dilute solution. In both cases, the normal stresses calculated from experimental data are excessively high, and the methods provide only a weak upper bound to the actual viscometric functions. We present here an analysis and critical appraisal of capillary and slit methods of normal stress measurement. The source of error is shown to be associated with the velocity profile rearrangement at the exit. Available data for both solutions and melts show that velocity rearrangement calculations for inelastic and second-order fluids are inadequate for real viscoelastic liquids, and that available experimental results for velocity rearrangement in viscoelastic liquids are not sufficiently precise for the calculations required. We conclude that a detailed understanding of velocity profile rearrangement of viscoelastic liquids is essential before exit flow measurements can be used for normal stress measurements.
65 citations
TL;DR: In this paper, an experimental study of the quiescent and flow birefringent characteristics of two liquid crystalline polymer solution systems, poly-p-phenylene terephthalamide (PPD-T) in sulfuric acid and hydroxypropyl cellulose (HPC) in water over a range of concentrations.
Abstract: An experimental study is reported of the quiescent and flow birefringent characteristics of two liquid crystalline polymer solution systems, poly-p-phenylene terephthalamide (PPD-T) in sulfuric acid and hydroxypropyl cellulose (HPC) in water over a range of concentrations. It is shown that for the quiescent state the dilute solutions are optically isotropic while the concentrated solutions consists of negatively birefringent domains. During flow at low deformation rates, moving domains are still seen. At higher deformation rates, a homogeneous highly birefringet fluid is obtained. The birefringence increases with concentration at constant deformation rate and exhibits a major increase as the liquid crystalline state is formed. The source of the birefringence is due to (1) anisotropy of polarizability of oriented macromolecules, and (2) difference in refractive index of solvent and oriented macromolecules (form birefringence). The results are interpreted in terms of the level of polymer orientation which may be developed in flow for liquid crystalline polymer solutions as compared to solutions of flexible macromolecules.
64 citations
TL;DR: In this paper, the authors presented numerical solutions for the two-dimensional flow past a circular cylinder in an infinite domain, where the flow is assumed to be uniform at infinity and the cylinder is allowed to rotate with a constant angular velocity Ω.
Abstract: Numerical solutions are presented for the two-dimensional flow past a circular cylinder in an infinite domain. The flow is assumed to be uniform at infinity and the cylinder is allowed to rotate with a constant angular velocity Ω. Ω is chosen to be in the range (0 to 5 W / a ) where a is the radius of the cylinder and W is the mainstream velocity at infinity. To incorporate viscoelastic properties into the flow, an implicit four-constant Oldroyd model is used, and the resulting nonlinear constitutive equations are solved in parallel with the equations of motion as a coupled set of partial differential equations. The method of solution used is a finite difference technique with block over-relaxation. The results are compared with those of other numerical computations as well as with available experimental data. In particular, consideration is given lift experienced by the cylinder and on the streamline patterns and vorticity distribution.
64 citations
TL;DR: In this paper, an inelastic mechanism for extrudate swelling in a variable-temperature Newtonian fluid was proposed, which depends on the increased resistance to deformation (viscosity) of strongly elongated filaments on the outside of an extruder.
Abstract: The recent discovery of extrudate swelling in a variable-temperature Newtonian fluid shows that a new theory of extrudate swelling is needed. Here we display an inelastic mechanism that dependes on the increased resistance to deformation (“viscosity”) of strongly elongated filaments on the outside of an extrudate. The theory agrees with available numerical computations and with experimental results. Several suggestions for improvements in the theory are made, and it is considered that three separate mechanisms (called, respectively, Newtonian, elastic and inelastic) all contribute to swelling.
TL;DR: In this article, a theoretical approach to the shear viscosity of concentrated suspensions of small particles in a non-Newtonian fluid has been developed using a cell theory model involving particle-particle interaction.
Abstract: A theoretical approach to the shear viscosity of concentrated suspensions of small particles in a non-Newtonian fluid has been developed using a cell theory model involving particle-particle interaction. The cell theory of Frankel and Acrivos was first generalized to power-law fluid matrices without particle interaction. Particle-particle interaction was then taken into consideration. The theory suggests that the flow behavior of such systems at low shear rates is chiefly dependent upon non-hydrodynamics interparticle interaction such as van der Waals—London and electrostatic forces which induce flocculation and yield stresses. The flow properties at high shear rates are determined by hydrodynamics interaction essentially dependent upon particle concentration and shape.
TL;DR: In this paper, a power-law fluid with a suddenly applied arbitrary time-dependent pressure gradient in a horizontal circular tube is presented, and the momentum equations are solved numerically using a implicit finite-difference technique for the timedependent velocity profiles for both start up and oscillating pressure gradients.
Abstract: Flow of a power-law fluid with a suddenly applied arbitrary time-dependent pressure gradient in a horizontal circular tube is presented. The momentum equations are solved numerically using a implicit finite-difference technique for the time-dependent velocity profiles for both start up and oscillating pressure gradients. Dimensionless curves are presented showing velocity profile development and phase angle variation for a variety of power-law index values.
TL;DR: In this paper, the radial profile of the shear rate is described by means of Dirac's delta function and the slip at the wall is held to be a consequence of and not the cause of phenomenon.
Abstract: During the oscillatory flow of linear polyethylene (HDPE) melt through a capillary, the shape of the extrudate varies periodically with time: sharkskin, plug and rough. This paper deals with the transition between increasing and decreasing pressure. At that transition, the flow rate through the tube is suddenly and intensively increased. We present a theoretical analysis which is in good agreement with experiment. The “slip” at the wall is held to be a consequence of and not the cause of phenomenon. The radial profile of the shear rate is described by means of Dirac's delta function.
TL;DR: In this article, a flow of elastico-viscous liquids through an axisymmetric pipe whose diameter is slowly varying in the axial direction is considered, and the idealized equations of state chosen to characterize these liquids are of the implicit Oldroyd type and include a constant viscosity model.
Abstract: Consideration is given to the flow of elastico-viscous liquids through an axisymmetric pipe whose diameter is slowly varying in the axial direction. The idealized equations of state chosen to characterize these liquids are of the implicit Oldroyd type and include a constant viscosity model, for which analytic solutions to second order are presented. A variable viscosity model is also considered and solutions in closed form are still possible. These solutions are evaluated numerically.
TL;DR: In this paper, an anomalous stress magnitude, a relaxation time and the exponent of the power law were derived for dilute polymer solutions of HPAM at various concentrations using the above three flow geometries, and it was shown that pressure anomalies occurred when critical values of the ratio between a velocity and length scale, representative of the strain rate, are exceeded.
Abstract: Previous work has shown that anomalous pressure behaviour occurs when dilute polymer solutions are subjected to elongational flows such as those existing upstream of a capillary tube entrance, of an orifice and of the stagnation point of a Pilot tube probe. Tests have been conducted with aqueous solutions of HPAM at various concentrations using the above three flow geometries. It is shown that pressure anomalies occured when critical values of the ratio between a velocity and length scale, representative of the strain rate, are exceeded. They are proportional to the power of the strain rate with an exponent larger than one. Based on previous and present results, it appears that the polymer solutions' behaviour may be characterized by three parameters: an anomalous stress magnitude, a relaxation time (inverse of the critical strain rate) and the exponent of the power law. The first two parameters depend on the polymer type, concentration and molecular weight, while the third depends only on the polymer type. The anomalous pressure is also affected by the conformation of the molecules as shown by results obtained with HPAM solutions containing varied amounts of NaCl.
TL;DR: In this paper, the authors provide a common basis for both Newtonian and non-Newtonian systems in situations where anomalous wall effects are absent, and use the wall viscosity in defining the Reynolds number as a test for anomalous behaviour.
Abstract: For Newtonian fluids, the engineering predictions for pressure drop in turbulent pipe flow are well established. However, in the case of non-Newtonian liquids, only a few design techniques have been proposed and these do not share a common basis with the approach for Newtonian systems. This present work attempts to provide a common basis for both Newtonian and non-Newtonian systems in situations where anomalous wall effects are absent. Previously published experimental data suggest that if the Reynolds number is calculated on the basis of the apparent viscosity at the wall then the standard Newtonian correlations can be used for the prediction of pressure drop. The use of the wall viscosity in defining the Reynolds number also serves as a test for anomalous behaviour. Any departure of the experimental data from the Newtonian turbulent friction factor correlation indicates anomalous behaviour.
TL;DR: A finite element routine for analysing non-Newtonian flows that is capable of handling a wide range of implicit constitutive relations and is illustrated by its application to some extrusion problems.
Abstract: We describe a finite element routine for analysing non-Newtonian flows. The routine uses a mixed method and is capable of handling a wide range of implicit constitutive relations. A new sort of pressure interpolant is used and leads to very accurate results. The routine is illustrated by its application to some extrusion problems.
TL;DR: In this paper, a review of the expressions that have been proposed for the Newtonian case and demonstrate how these can be extended to non-Newtonian and drag-reducing flows is presented.
Abstract: An understanding of turbulent pipe flow is of extreme importance in engineering. The eddy viscosity concept has been used by many workers to describe fully developed, Newtonian, turbulent pipe flow. A knowledge of the eddy viscosity can then be used to predict mean velocity and temperature profiles (and hence heat transfer coefficients). In addition to the Newtonian case, there are many instances when the turbulent flow of non-Newtonian and drag-reducing fluids are encountered. Although several workers have proposed eddy viscosity expressions for Newtonian pipe flow, few comparative studies are available. Little work has been done to develop eddy viscosity expressions for non-Newtonian and drag-reducing pipe flow. The objective of the present study is to review the expressions that have been proposed for the Newtonian case and demonstrate how these can be extended to non-Newtonian and drag-reducing flows.
TL;DR: In this paper, the authors consider the case where the stress σ(X, t) at a particle X at time t is given by σ (X,t) = F [F[X, τ] where F[F(T, τ)] denotes a functional of the history of the deformation gradient matrix [F, τ], from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant
Abstract: We consider constitutive expressions which the stress σ(X, t) at a particle X at time t is given by σ (X, t) = F [F[X, τ)] where F [F(X, τ)] denotes a functional of the history of the deformation gradient matrix [F(X, τ)] from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant under the group of unimodular transformations, i.e. F [F(X, τ)] = F [F(X, τ) H] must hold for all matrices H such that det H - 1. We employ results from the classical theory of invariants in order to determine the general form of the expression F [F(X, τ)] which is consistent with these restrictions. Special cases are considered where the functional is replaced by a function of the strain, rate of strain, ⋯ matrices. The case of shear flow is briefly discussed.
TL;DR: In this paper, the authors derived the load bearing capacity of a continuous flow version of squeeze film with no moving parts and showed that a 0.1 per cent aqueous polyacrylamide solution gives well defined load enhancement and (quite independently) the jet thrust method gives the relationship between normal stress and shear rate.
Abstract: A form of squeeze film apparatus was recently described in which the movement of one plate towards the other was simulated by the continuous volume generation of liquid over the plate area. The liquid exuded from a large number of holes in the lower plate surface and formed a “continous flow” version of squeeze film apparatus with no moving parts [1]. A later paper gave derivations of equations from which squeeze film load bearing capacity could be evaluated, taking into account viscous, inertial and normal stress effects in the liquid film [2]. In order to find the total load in a squeeze film system, it was necessary to obtain the relationship between the first normal stress difference and shear rate for the liquid in use, using an experimental method. At high shear rates, the jet thrust method provided these data [3,4] and from them the load bearing capacity of squeeze films of hot, polymer-thickened oil were predicted [2]. A more complete test of the method is possible with a highly elastic liquid because considerable load enhancement due to extra stress is present at moderate deformation rates in squeeze film systems [1,5,6,7]. Thus a 0.1 per cent aqueous polyacrylamide solution gives well-defined load enhancement and (quite independently) the jet thrust method gives the relationship between normal stress and shear rate from which predictions of load enhancement may be made. Furthermore, convergent nozzles may be used in the jet thrust apparatus [3] to measure the stress development in an elastic liquid which is being simulateneously sheared and stretched, a situation which more closely resembles the squeeze film case than that of steady shear.
TL;DR: In this article, the motion of a slender rod near a vertical wall in a viscoelastic liquid is investigated experimentally, and it is shown that the theory and experiment are in qualitative agreement provided the particle is at least one length from the wall.
Abstract: The motion of a slender rod near a vertical wall in a viscoelastic liquid is investigated experimentally. When the particle is not too near the wall, its motion is compared to existing theoretical solutions for the motion of a slender rod in an unbounded second-order fluid. In this case, it is found that the theory and experiment are in qualitative agreement provided the particle is at least one length from the wall. When the particle is nearer to the wall, its motion is shown to correspond qualitatively to a simple superposition of the motion in a second-order fluid of infinite extent and the motion in a Newtonian fluid near a vertical wall.
TL;DR: In this paper, the authors proposed an accurate method for determination of the transient viscosity, η−(t; γ 0), after stopping a steady shearing flow of shear rate γ. 0.
Abstract: The purpose of this study is the proposal of an accurate method for determination of the transient viscosity, η−(t; γ . 0), after stopping a steady shearing flow of shear rate γ . 0. It is known that η−(t; γ . 0) cannot be obtained accurately from tangential stress measurements during a relaxation test in a Weissenberg Rheorgoniometer, for the following reasons: 1. (a) lag time tr and braking time ts are not zero, 2. (b) a coupling between the mechanical properties of the sample and those of the measuring system arises. With respect to (a), our modifications of the Theogoniometer allowed the accurate determination of tr and the reduction of ts (ts With repsect to (b) we present: 1. (i) a numerical method to obtain η−(t; γ . 0) from the coupled experimental signal, provided tr is accurately determined and tr is accurately determined and ts negligible. 2. (ii) measurements performed with a piezoelectric sensor. The coupling phenomenon is then strongly reduced. For t > 2ts the signal was found to be independent of ts and radily gave η−(t; γ . 0). Good agreement was obtained between (i) and (ii). These functions η−(t; γ . 0) which were determined for a polystyrene solution proposed by the NBS were then used to test Yamamoto's equation It was found to be valid in the power-law region.
TL;DR: In this paper, a strain-dependent constitutive equation for polymeric liquids is considered and a memory function factorized into time and strain dependent components is developed in simple shear flow which allows one to directly test the factorability assumption with experimental data.
Abstract: A recently proposed form for a constitutive equation for polymeric liquids is considered. The equation is of a strain-dependent type with a memory function factorized into time- and strain-dependent components. Relations are developed in simple shear flow which allow one to directly test the factorability assumption with experimental data. An exponential form for the strain-dependent component is then compared with experimental data on a polystyrene solution.
TL;DR: A theoretical and experimental study of heat transfer to polymer melts flowing through ducts is presented in this paper, where a mathematical model provides for shear heating and expansion cooling effects for various wall boundary conditions.
Abstract: A theoretical and experimental study of heat transfer to polymer melts flowing through ducts is presented. The mathematical model provides for shear heating and expansion cooling effects for various wall boundary conditions. Experimental results, obtained in tube flow, show reproducible temperature and velocity profiles and the data confirm the predictions of the magnitude of the shear heating and expansion cooling effects. It is concluded that the method can be justified for predicting heat transfer in more complex geometries and some preliminary results are presented.