A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid

A boundary element method is used to simulate the unsteady motion of a sphere falling under gravity along the centreline of a cylindrical tube containing a viscoelastic fluid. The fluid is modelled by the upper-convected Maxwell constitutive equation. Results show that the viscoelasticity of the liquid leads to a damped oscillation in sphere velocity about its terminal value. The maximum sphere velocity, which occurs in the first overshoot, is approximately proportional to the square root of the Weissenberg number when the ratio of the sphere radius to the tube radius is sufficiently small. Particular attention is also paid to the wall effects. It is shown that a closer wall reduces the oscillatory amplitude of the sphere velocity but increases its frequency. The results suggest that the falling-ball technique, which is now widely used for viscosity measurement, might also be used for the determination of a relaxation time for a viscoelastic fluid.     

On the derivation of the constitutive equation of the simple fluid from that of a simple material

It is shown, by the use of the symmetry group of a fluid, that the stress in an elastic fluid depends on the current value of the deformation gradient through the absolute value of the determinant of the deformation gradient, whereas in a simple fluid it depends on this value of the determinant and the history of the relative deformation gradient and nothing else. This derivation is then used to show some of the errors in the criticisms made about the work of Noll. Additionally, a lacuna in Noll's derivation is removed to overcome the only valid criticism made about this theory of simple fluids.     

Comments on the paper “On the derivation of the constitutive equation of the simple fluid from that of a simple material” by R.R. Huilgol

Huilgol's paper [1] is discussed critically. It is shown that there are significant errors in each section of the paper. His argument depends critically on a refusal to recognize that when a statement is made that a functional of a tensor function is invariant under a group of transformations, these transformations must be constants and that otherwise the statement is meaningless. Furthermore, he consistently ignores the contradictions to which this refusal leads.     

The motion of rodlike particles in the pressure-driven flow between two flat plates

The motion of freely suspended rodlike particles has been observed in the pressure-driven flow between the two flat plates of a Hele Shaw flow cell at low Reynolds numbers. Data are reported for rodlike particles with aspect ratios of 12.0 suspended in a Newtonian fluid for gap thickness to particle length ratios of 3, 6, and 20; and for rodlike particles with aspect ratios between 5 and 8 in a non-Newtonian fluid (79.25 wt.% water, 20.2 wt.% glycerine, and 0.55 wt.% polyacrylamide). For the Newtonian fluid, the time-dependent orientation of the particles near and far from walls was shown to be in quantitative agreement with Jeffery's theory for ellipsoids suspended in a simple shear flow if an effective aspect ratio is calculated from the experimental period of rotation. Particles aligned with the flow direction and less than a particle half-length from a wall interacted irreversibly with the wall. For the non-Newtonian fluid, the timedependent orientation far from a wall was shown to be in qualitative agreement with Leal's theory for a second-order fluid; however, particles that were aligned with the flow direction and were near walls did not rotate.     

Stress optical measurement of the third normal stress difference in polymer melts under oscillatory shear

Results are reported for the dynamic moduli,G andG, measured mechanically, and the dynamic third normal stress difference, measured optically, of a series bidisperse linear polymer melts under oscillatory shear. Nearly monodisperse hydrogenated polyisoprenes of molecular weights 53000 and 370000 were used to prepare blends with a volume fraction of long polymer, _L, of 0.10, 0.20, 0.30, 0.50, and 0.75. The results demonstrate the applicability of birefringence measurements to solve the longstanding problem of measuring the third normal stress difference in oscillatory flow. The relationship between the third normal stress difference and the shear stress observed for these entangled polymer melts is in agreement with a widely predicted constitutive relationship: the relationship between the first normal stress difference and the shear stress is that of a simple fluid, and the second normal stress difference is proportional to the first. These results demonstrate the potential use of 1,3-birefringence to measure the third normal stress difference in oscillatory flow. Further, the general constitutive equation supported by the present results may be used to determine the dynamic moduli from the measured third normal stress difference in small amplitude oscillatory shear. Directions for future research, including the use of birefringence measurements to determineN ₂/N ₁ in oscillatory shear, are described.     

Estimates of apparent elongational viscosity using the fibre spinning and “pure” methods

A comparison is made between elongational properties predicted by the FENE-P dumbbell rheological equation of state for fibre spinning and true values obtained from pure elongation. True values are found by using a constant elongation rate in the model and calculating the steady state values of the stresses. Two new averaging procedures are suggested for extracting elongational properties from fibre spinning experiments. It is suggested that a variety of flow conditions be used and care be taken when determining these properties. The rheological model is found to predict adequately shear and elongational flow properties of the polyacrylamide-type of Boger or ideal elastic fluid.     

Orientation and rheology of rodlike particles with weak Brownian diffusion in a second-order fluid under simple shear flow

An analysis of particle orientation in a dilute suspension of rodlike particles in a second-order fluid was performed to examine the effects of the elasticity of the fluid and of weak Brownian diffusion of the particle on its orientation. Distributions of particle orientation under a simple shear flow with rate of shearg have been obtained as a function of a single nondimensional parameter, ^* =/r _e ² (D/g), which combines the effects of the particle aspect ratior _e , the weak fluid elasticity, and the weak Brownian rotation diffusion coefficientD of the particle. In the limit of larger _e , when the fluid elasticity is strong enough to overcome the rotational diffusion effect on the particle motion, most of the particles will orient close to the vorticity axis. A new shear-thinning mechanism of the shear viscosity of such systems is predicted by the theory.     

Transient stress responses predicted by the internal viscosity model in elongational flow

Predictions are made for the elongational-flow transient rheological properties of the dilute-solution internal viscosity (IV) model developed earlier by Bazua and Williams. Specifically, the elongational viscosity growth function _e ⁺ (t) is presented for abrupt changes in the elongational strain rate . For calculating _e ⁺, a novel treatment of the initial rotation of chain submolecules is required; such rotation occurs in response to the macroscopic step change of at t = 0. Representative are results presented for N = 100 (N = number of submolecules) and = 1000 f and 10000 f (where is the IV coefficient and f is the bead friction coefficient), using h ^* = 0 (as in the original Rouse model) for the hydrodynamic interaction. The major role of IV is to cause the following changes relative to the Rouse model: 1) abrupt stress jump at t = 0 for _e ⁺; 2) general time-retardance of response. There is no qualitative change from the Rouse-model prediction of unbounded il growth when exceeds a critical value ( ), and calculations of submolecule strains at various show that the unbounded- _e ^– behavior arises from unlimited submolecule strains when . However, the time-retardance could delay such growth beyond the timescale of most experiments and spinning processes, so that the instability might not be detected. Finally, _e ⁺ (t) and _e ( ) in the limit are presented for N = 1 and compared with exact predictions for the analogous rigid-rod molecule; close agreement lends support for the new physical approximation introduced for solving the transient dynamics for any N.     

Analysis of incompressible three-dimensional flows using the concept of stream tubes in relation with a transformation of the physical domain

This paper presents an analysis of three-dimensional flows based on the concept of stream tubes and streamlines. In the case of incompressible liquids, the physical domain can be transformed, under certain assumptions, into a cylinder where the streamlines are parallel straight lines. In contrast to classical methods, the unknown to be determined is the transformation between the two domains. This analysis generalizes the formulation already proposed for plane and axisymmetric flows.     

New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows

The paper describes pertinent laboratory tests to characterize the rheological properties of paper coatings with regard to blade coating over a very wide range of shear rates in both transient and steady-state shear flows. Shear rates as high as 10⁶ s^–1 can be reached by means of a gas-driven capillary rheometer. Examples for the evaluation of end effects, wall effects, and coating thixotropy are given. A stiff and fast Couette rheometer is used to determine flow curves and the shear stress overshoot in step shear rate tests. The primary normal stress difference can be measured up to 10⁴ s^–1 by means of a high shear cone-plate rheometer with piezo transducer. A correct evaluation of the measurements has to take into account inertia contributions to the normal force. First results using a sinusoidal modulation of the shear rate are presented.Paper presented at: International Symposium on Pigment Coating Structure and Rheology, Helsinki, Febr. 8–9, 1989     

An anomalous phenomenon occurring in the flow of viscoelastic fluids out of ducts

When viscoelastic fluids flow out from horizontal ducts many cracks and protruding ridges are formed on the jet surface. As far as the authors know, this phenomenon has not yet been reported. The occurrence of the anomalous phenomenon is not affected by inlet flow condition or duct shape.The phenomenon may be divided into three regimes, namely, a stable state, a breakage state, and a multiplication state. In the breakage state one ridge divides into two parts after its growth, and in the multiplication state a new ridge is suddenly formed at a crack point. With increasing shear rate, the flow pattern of the jet changes from the stable state to the breakage state, and then to the multiplication state.Furthermore, a recoil effect is observed. In this effect many air bubbles rush into the duct from the exit when the flow is quickly stopped.     

Accurate evaluation of the transverse shear modulus in unidirectional fibrous composites

A new relation for the prediction of the transverse shear modulus in unidirectional fiber composites has been derived. The theoretical results of this relationship are in better agreement with the experiments than those of other relations, existing in the literature. The discrepancies, which are observed among the theoretical predictions and the experimental values, are explained by the consideration of the boundary layers existing between the matrix and the fibers of the composite. A new model, which includes the intermediate phase between the matrix and the fiber, called the mesophase, is considered in order to take into account the above-mentioned layers.     

Distribution of stresses in a pre-cracked bone specimen

Of concern in the paper is an analysis of the stress-distribution in a specimen of bone having an exterior crack. In conformity to experimental findings, the osseous medium is treated as a composite transversely isotropic material. By considering a numerical example, values of the longitudinal stress at different locations of the bone specimen are obtained by computing the analytically derived expressions, using experimental data for the various physical constants of osseous tissues.     

On the role of stress-induced migration on time-dependent terminal velocities of falling spheres

There is experimental evidence to suggest that even under steady-state conditions the velocity of solid spheres or bubbles moving through viscoelastic fluids can become time dependent. One of the possible explanations offered for interpreting this phenomenon has been the generation of a polymer depleted layer in the line of passage of the particles, which disappears due to the counterbalancing effect of molecular diffusion in the long range. We have done some careful experiments and measured these concentrations to show that no such depletion layers are formed. Alternative explanations of the phenomenon have been examined and the importance of the possible effects of microstructures generated through temporary associations has been emphasised.     

A photographic study of shapes of bubbles and coalescence in non-Newtonian polymer solutions

This communication reports (photographically) on the shapes of bubbles of different gases in several Newtonian and polymer solutions, encompassing a wide range of rheological behavior. Effects of a surface active agent are mentioned and bubble coalescence in viscoelastic solutions is visualized.     

On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.     

A note on the coefficient of viscosity of pure molten 2,4,6-trinitrotoluene (TNT)

The viscosity of pure molten TNT has been investigated over the temperature range 82.0°–95.4°C. The temperature dependence of viscosity was found to be best represented by a relation of the type = A e ^B/T whereA = 0.000541,B = 3570, is the viscosity in mPa s andT is the temperature in Kelvin. Earlier work, which suggests an inverse temperature dependence of the flow activation energy, is shown to include an error in the published equation for the temperature dependence of the viscosity of molten TNT.