Showing papers in "Journal of Non-newtonian Fluid Mechanics in 1982"

A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility

Abstract: Some time ago a theory of elastic fluids such as concentrated polymer solutions and melts was presented based on the concept of a multitude of penetrating statistical continua. The configurations of these depend on the deformation history, and their relative motion is determined by the coordinated part-stress in connection with a tensorial mobility which is a function of all configuration tensors. The simples special case of thise theory is given by a model with only one configuration tensor which both the stress and mobility tensors depend on. In this paper a further specialization is introduced by assuming a linear dependency of stress as well as of mobility tensors on the configuration tensor. Some relationship to a modified retation model, recently presented by Curtiss and Bird, is hereby established. This model which is of the rate-type and is non-linear in the configuration and stress tensors, respectively, nevertheless permits analytic solutions of some of the most important types of flow. These are explicitly given for both steady and transient extensional and shear flows. Besides shear thinning and non-vanishing first and second normal-stress differences, an extensional viscosity with finite asymptotic values as well as non-exponential stress relaxation and start-up curves are predicted, with stress-overshoot at least in the case of shear flow.

Finite-element Analysis of Die Swell of a Highly Elastic Fluid

Abstract: A mixed finite element method is applied to the die swell calculation of ana Oldroyd fluid B. The use of large entry lengths together with the presence of the retardation time in the constitutive equations allow us to reach values of the recoverable shear as high as four for the flow emerging from slit and circular dies, with swelling ratios of the order of 2. The numerical results are in good agreement with some available experimental data.

Constitutive equations for polymer melts predicted by the Doi—Edwards and Curtiss—Bird kinetic theory models

Abstract: The constitutive equations derived by Doi and Edwards and by Curtis and Bird from kinetic theory models are shown to possess a potential function U , which completely determines the strain-dependent memory fading. An exact expression for U is found, together with a good approximation expressed as a simple combination of the strain invariants. This result is compared with empirical results for low-density polyethylene, leading to speculation on the cause of the observed differences.

Approximation error in finite element calculation of viscoelastic fluid flows

Abstract: Numerical calculations of complex, two-dimensional flows of viscoelastic fluids fail when the elastic contribution to the stresses, measured by a Deborah number, becomes comparable to the viscous contribution. Reasons for the limit on Deborah number are explored by a sequence of finite element calculations with contravariant convected Maxwell and second-order fluid models. The possibilities of bifurcation or loss of a steady, two-dimensional flow field are ruled out by employing continuation methods for flow through a planar contraction and by existence and uniqueness proofs for a second-order fluid in a driven cavity. Error in the finite element approximation to steep gradients of stress causes the breakdown of all calculations. This is most clearly seen in the calculations for a second-order fluid, because the exact flow field for any Deborah number is known.

Strong flow criteria based on microstructure deformation

Abstract: The dynamics of fluid systems which consist of a suspended material in a Newtonian continuous phase is investigated theoretically. Criteria are derived to predict conditions under which the strength of a flow, i.e. a measure of the form and magnitude of the velocity gradient tensor, is sufficient to induce significant deformation and/or orientation of the fluid microstructure, that is, the elements which collectively comprise the suspended phase. The development relies upon the choice of a model to describe the microstructure, and the form of the criteria reflects this choice. Once the choice is made, however, the detailed material properties of a particular fluid system enter only as parameters in the resulting equations, and thus, the results encompass a large class of systems, including particulate suspensions and macromolecular solutions. Two microstructure models are investigated here. When the microstructure is characterized by a vector, the flow strengths of all linear flows are displayed in a single figure from which the strength of a particular flow can be evaluated directly. A comparison is then made for selected flows between these results and those for the case where an irreducible second order tensor is employed to describe the microstructure. A significant difference between the two models derives from the fact that the “volume” of the microstructure must be conserved in the second-order tensor case. The criteria are finally used to predict the degree of macromolecular stretching in a model turbulent flow and the breakup of immiscible liquid drops in simple shear flow. A comparison between the flow strength predictions and experimental data yields good qualitative agreement in the latter case.

Multiaxial elongational flows of polymer melts—classification and experimental realization

Abstract: For the class of multiaxial elongational flows with constant strain rates, a new classification system is proposed. Ordering the main strain rates such that ∈ . 0  ∈ . 11 ⩾ ∈ . 22 ⩾ ∈ . 33, the constant ratio m  ∈ . 22/ ∈ . 11 characterizes the special type of flow, e.g. in simple elongation m = − 1/2, in equibiaxial elongation m = +1, and in planar elongation (“pure shear”) m=0. For the representation of the test results, three elongational viscosities μi = Aiσi/ ∈ . 0 are defined from the three normal stress differences σi such that for the linear viscoelastic limit the time-dependent linear viscoelastic shear viscosity follows: μi(t)→ t. For the realization of such multiaxial elongational flows, rotary clamps are used. They are modified such that the force components along and normal to the direction of motion at the clamp can be measured separately. The investigation was performed with polyisobutylene in simple, equiviaxial, and planar elongations. The latter mode of testing is of special interest because two different elongational viscosities are determined simultaneously. The results show that for small deformations the material behaviour is linear viscoelastic. For larger deformations there are different deviations the character of which depends on m, i.e. the type of the elongational flow. From the general single-integral constitutive equation it follows that the predicted stress differences for the different m are very differently connected with the different parts of the equation, as in the case of the two normal stress differences in simple shear flow.

On Numerical Die Swell Calculation

Abstract: A few recent theoretical papers have been devoted to the effect of the elastic properties of a fluid upon die swell. A rapid survey might suggest that the range of Weissenberg numbers covered by the calculations and the values of swelling ratio and exit pressure loss depend upon the numerical algorithm. In the present note, we show the importance of mesh refinement on the analysis; we also show that mixed and displacement techniques lead to slightly different results.

Compressive flow between parallel disks:: I. Newtonian fluid with a transverse viscosity gradient

Abstract: Compressive flow between parallel disks when a transverse viscosity gradient exists is studied numerically and analytically. There are two distinct flow regimes, depending on the group = ηmaxH2/ηminR2. Whe is small, parallel squeezing occurs and the maximum velocity is in the high viscosity “core” fluid. For large values of , parallel squeezing does not occur, the velocity maximum occurs in the low viscosity fluid near the disks, and the low viscosity material is preferentially expelled. Both flow regimes have been reported in compression molding experiments on sheet molding compounds.

Flow of dilute polymer solutions through converging channels

Abstract: Extensional motion of non-Newtonian fluids was studied by pushing dilute solutions of polyethylene oxide through miniature converging channels. The channels were conical or smoothly converging in shape, and each had an exit diameter around 0.13 mm to make Reynolds number effects comparable. Channels were also made by cementing small beads together, which created passages similar to those in packed beds. Pressure drop and flow rate were measured in each channel and the flow field was observed. Flow patterns and data for the solutions were the same as for water when the maximum rate of extension was below about 103 s−2. Above this, the pressure drop for a solution exceeded that of water, and when solution values reached twice the water values, the flow pattern changed abruptly from a simple converging motion to one dominated by a large unsteady vortex ring surrounding the channel opening on the upstream side. This secondary motion persisted at high flow rates, had a variety of forms, and at times produced tiny filaments of polymer, most likely by strain-induced crystallization. Non-Newtonian effects were found to increase with concentration and to decrease with temperature and ago; in each case, much of the effect could be traced to that parameter's influence on the onset of non-Newtonian behaviour. In bead channels with successive convergent sections, onset strain rates were much lower, around 102 s−1, indicating that strain history has a strong effect on onset. Disparate onset strain rates for the various single channels, from 300 s−1 to 6000 s–1, were accounting for by considering total extension of the polymer molecules in each flow. In fact, analyses suggest that molecules were stretched about six times their coil size in the converging flows. The large non-Newtonian effect when secondary motion is present is thought to result from high molecular extension or possibly from a contraction of the central flow.

Pulsating flow revisited

Abstract: We report some new data on the pulsating flow of polymeric fluids. The data show that the flow enhancement decreases with the frequency of the pulsating pressure gradient noise. Theoretical predictions using a non-affine network model agree both qualitatively and quantitatively with the experimental data. Based on the power-law model, it is found that the extra power required in pulsating the flow is always positive and thus there is no economic advantage in pulsating the flow.

A numerical study of the motion of a viscoelastic fluid past rigid spheres and spherical bubbles

Abstract: Numerical solutions are obtained for an Oldroyd-type fluid in streaming flow past a rigid sphere and a spherical bubble. Both co-rotational and co-deformational time derivatives are considered for Weissenberg numbers in the range O

Matched eigenfunction expansions for slow flow over a slot

Abstract: We solve the problem of plane flow of a second-order fluid over a rectangular slot when inertia is neglected by matching biorthogonal eigenfunction expansions in different regions of flow. The method appears to be cheaper and more accurate than direct numerical methods. The effect of normal stresses on pressure measurements at the bottom of the slot is discussed.

Newtonian and non-Newtonian flow near a re-entrant corner

Abstract: The Laser Doppler technique is used to study flow characteristics in the vicinity of a re-entrant corner for both Newtonian and elastic liquids. An L-shaped channel is chosen as the test geometry and a constant-viscosity Boger fluid as the elastic liquid. It is found that near the re-entrant corner the flow characteristics are virtually independent of both fluid inertia and fluid elasticity. The observed influence of fluid inertia is in agreement with existing theoretical predictions, whilst that of fluid elasticity is consistent with the elastic liquid behaving asymptotically like a Newtonian viscous liquid near the corner, suggesting the use of Oldroyd- (rather than Maxwell-) type models in future simulation studies. Numerical predictions for the Newtonian cases using finite-difference techniques are in satisfactory agreement with experiment, whilst those for an Oldroyd model are in qualitative agreement with experiment results for the Boger fluid.

Non-isothermal flow of a molten polymer in a narrow rectangular cavity

Abstract: An accurate and stable numerical scheme is presented for the calculation of the non-isothermal flow of a non-Newtonian liquid in a narrow rectangular cavity. The effects of solidification of matter on the walls of the cavity during filling and viscous heating are taken into account. The viscosity of the liquid model used is shear rate and temperature dependent.

The stringiness of dilute polymer solutions

Abstract: Macromolecular solutions flowing downwards out of a vertical pipe from drops at the exit end. As a drop falls it pulls out a “string” of fluid, which ultimately breaks. The broken piece still attached to the pipe retracts. Another drop then forms and falls and so on. This has been observed using a Hyspeed camera operating at 1500 frames per second and a sequence of events lasting for only about 0.1 second can be examined in detail. Hence, a value of the apparent elongational viscosity can be determined.

Constant force elongational flow of a low-density polyethylene melt—experiment and theory☆

Abstract: Analysing the melt spinning process, uniform deformation experiments at constant force are of particular interest. For a LDPE melt, which has been well characterized by experiments at constant elongation rate, start-up experiments at constant force and uniform deformation have been performed. Experimental data are compared in this paper with theoretical predictions using a Maxwell model, a power-law approach, and the rubber-like-liquid constitutive equations of Lodge and Wagner.

Calculation of hole pressure:: I. Newtonian fluids

Abstract: Numerical calculations using the finite element method are reported for the pressure hole problem in which flow between two flat plates is interrupted by a slit. Predicted are the effects of flow velocity and geometry: height between the flat plates and width and depth of the slit. Results compare favorably to the experimental results of Tong for a circular hole.

Steady planer extension with lubricated dies

Abstract: We demonstrate that planar stagnation flow can be achieved using molten polystyrene in a die with walls lubricated by silicone oil. The boundary conditions and flow in the lubrication layer are discussed. Wall pressures drop by nearly an order of magnitude under lubricated conditions. Pressure values were below the accuracy of our transducers. However, both normal stress differences were measured by birefringence and at low extension rate were found in good agreement with the values expected for a Newtonian fluid. Even unlubricated flow gave similar stress differences at the same flow rates, including that the shear is confined to a thin layer near the die wall. Tracer pictures confirm this. Extrudate well was also measured and found to be in a reasonable agreement with total recoverable strain predicted from shear measurements.

Modelling of strain histories for memory integral fluids in steady axisymmetric flows

Abstract: A path line tracking procedure is derived which determines the strain history of infinitesimal material elements in steady axisymmetric flow. Path line tracking requires a pointwise knowledge of the kinematics as it is gained in numerical calculations (or may be available from measurements). Tabulated values of density as a function of position are also required. Path line tracking then interpolates spatial positions along path lines at residence times ( t — t ′), magnitude of velocity, velocity gradient normal to the path line, and slope of path line at time t . This knowledge is then condensed into the time dependent Finger strain components C −1 ( t ′, t ) for a given material element in steady axisymmetric flow. The tracking is applicable to compressible flows of arbitrarily large strains, since the strain history C −1 is determined for a material point (infinitesimal material element). Path line tracking avoinds differentials in space, except when determining the shear strain and when interpolating between the given point values of velocity. The method is tested in a specific example (planar stagnation flow) and it is applied to modelling of processing flows.

A study of parison development in extrusion blow molding

Abstract: The dynamics of parison development in the extrusion blow molding process have been studied by employing a pinch-off mold technique in conjunction with experimental photography The experiments yielded quantitative information of the parison length, diameter, and weight distribution as a function of extrusion time and axial distance At a given extrusion time, parison shape is controlled by the two competing influences of swell and drawdown The initial phase of parison extrusion is dominated by swell effects, which are then gradually overshadowed by the growing influence of drawdown arising from the weight of the parison In addition, the results indicate the strongly time-dependent nature of both the diameter and weight swell functions The relationship between these swell functions suggests that the deformation is slightly anisotropic for the polyethylene resin used in this study Moreover, a modelling approach, based on a combination of the competing effects due to swell and drawdown, has been proposed in order to develop an approximate quantitative analysis capable of predicting parison dimensions

Birefringence and pressure drop for the orifice flow of a polymer solution

Abstract: By means of laser illumination, we have measured the birefringence and tractracer-particle trajectories in the converging flow of a concentrated poly-acrylamide solution just upstream of a small circular orifice. For different orifice diameters D and mean discharge velocities V we obtain a close correlation between the birefringence, based on D, and the strain-rate parameter V/D for values of V/D up to 50s−1. However, the local values of axial extension rate, as deduced approximately from tracer streakline photographs, appear much smaller in magnitude than V/D. Our birefringence measurements also reveal what appear to be flow instabilities of the type observed in previous experiments on converging flow of polymer solutions. As an especially surprising and interesting result, we find that measured pressure drops across various orifices do not conform to predictions based on general dimensional arguments for inertialess viscoelastic flow from infinite reservoirs. This leads to the tentative conclusion that a filamentary- core structure associated with viscoelastic flow may generally give rise to pronounced hydrodynamic interactions with the upstream flow-chamber walls, even for nominally large contraction (chamber/orifice diameter) ratios.

Calculation of hole pressure:: II. Viscoelastic Fluids

Abstract: Numerical calculations using the finite element method are reported for the pressure hole problem caused by flow between two flat plates interrupted by a slit. Predicted are the effects of flow rate and geometry for viscoelastic fluids modeling solutions of polystyrene in dioctylphthalate. Model results are compared to experimental data from the literature for corotational and codeformational differential models having one time constant, when the models are adjusted empirically to give exact shear stress and first normal stress differences in viscometric flows.

Normal and reverse squeezing flows

Abstract: In this paper experimental results are presented for the flow of elasticoviscous fluids in thin films held between parallel horizontal circular flat plates. Specifically, we consider the case when the plates are forced to separate by an applied load. The results are discussed in the context of lubrication properties as well as from a fundamental fluid mechanics point of view. On the theoretical side, existing analyses are discussed in the context of the present experimental study. Recent theories are considered, with particular emphasis placed on the effect of including the plate inertia in the corresponding analyses. It is found that the theories predict opposing results and that most are not capable of even qualitative agreement with the present experimental results and those published earlier by other authors. One theory, however, does provide qualitative agreement with experimental findings in both the standard squeeze-film flow and the flow considered in the present report, when the Deborah number is not too large. The attempt to include the plate inertia in some of the existing analyses does not meet with success since physically unrealistic results are obtained.

A note on the simple fluid

Abstract: In 1957 Green and Rivlin [1] discussed the restrictions which can be imposed on the constitutive equation for a material with memory, in which the stress is assumed to be a functional of the deformation gradient history for all times up to and including the instant at which the stress is measured. Certain of these restrictions result from considerations of the manner in which the stress is changed by the superposition on the assumed deformation of an arbitrary time-dependent rigid rotation.

Multiaxial elongational flow of sheets and annular cylinders: growth disturbances

Abstract: The growth of disturbances in sheets and annular cylinders undergoing multiaxial elongational flows is described. Detailed analyses are given for a Newtonian fluid undergoing constant stretch rates. It is shown that lateral stretching of sheets decreases the size of disturbances but at a less rapid rate than the homogeneous decrease in thickness. Elongations to break (in the machine direction) are actually decreased. Preliminary considerations are given for viscoelastic fluids which suggest a more rapid relative growth of disturbances.

Influence of the contraction ratio in elongational flow of polymer solutions

Abstract: Using upstream chambers and capillary tubes of various diameters D0 and D respectively, it is shown that the excess pressure drop ΔPe, associated with the elastic properties of polymer solutions, varies with the contraction ratio D0/D. This variation is such that ΔPe first increases and then decreases as D0/D increases. The analysis of previous and present results indicates that the polymer solution's flow configuration in the upstream chamber is responsible for the observed dependency.

Stretching the Phan-Thien—Tanner model: stress growth and creep

Abstract: A detailed investigation of the predictions of the constitutive equation due to Phan-Thien and Tanner in simple (uniaxial) extension has been undertaken. Computed results and a qualitative analysis of the differential equations are presented for the case of a single relaxation time in stress growth (constant rate of strain) and creep (constant rates). The main conclusions are that in stress growth a steady stress in always attained, and that stress overshoot is predicted for some reasonable values of the parameters. Comparison with a four-parameter Oldroyd model is briefly discussed.

The interaction of yield stress and viscoelasticity on the Weissenberg effect

Abstract: The interaction of fluid yield stresses and viscoelasticity upon the Weisenberg effect has been investigated in several ways. Firstly, a simple theoretical model for the Weissenberg effect has been developed (assuming the Weissenberg hypothesis) and used to predict surface profiles for solutions without yield stresses and for solutions of similar rheologies but with yield stresses superimposed. Secondly, the surface profiles for pairs of polymer solutions were observed and compared. Each comparison consisted of two solutions of similar experimentally determined rheologies except that one solution exhibited a yield stress whereas the other did not. Thirdly, the model for the Weissenberg effect was used to predict the surface profiles for these pairs of solutions using their experimentally determined rheologies. The results of each of these analyses were in agreement, i.e., the Weissenberg effect is suppressed when yield stresses and stagnant fluid regions are present. In many processes, for example fluid mixing, it is important to know whether a fluid is viscoelastic or not and the Weissenberg effect is an indication of this viscoelasticity. The effects of stirred tank scale-up criteria upon the presence of a noticeable Weissenberg effect around a rotating impeller shaft are discussed.