Rubber-like elasticity by boundary element method: Finite deformation of a circular elastic slice

We briefly review the phenomenological theory of rubber-like elasticity and report a microstructural model that leads us to eventually adopt a particular constitutive equation, which includes the Neo-Hookean and the Mooney materials. A numerical implementation of the Boundary Element method for solving a general two-dimensional or axisymmetric finite deformation problem is described and tested with some simple deformations. The resulting program is used to analyse the finite deformation of a circular elastic slice perfectly bonded to two parallel rigid end plates; the bottom plate is stationary and the top plate is given a constant displacement. The problem has a free surface which must be found as part of the solution. The results indicate that the Boundary Element method can be an efficient tool for stress-strain analyses with rubber-like materials.     

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.     

The squeezing flow of a model suspension fluid

The paper is concerned with the squeezing flow of a model suspension fluid. The numerical solution obtained by a time-dependent Boundary Element Method is compared to an asymptotic solution at large radius. It is found that the kinematics are Newtonian in character, and the fibres quickly align themselves radially. Consequently, the squeezing force is only weakly dependent on the initial orientations of the fibres and the device can be used for measuring the effective viscosity of the suspension. The effective viscosity found from the squeezing flow agrees surprisingly well with experimental data and numerical data derived from the falling sphere geometry at low volume fractions ( < 0.1).     

Translation and rotation of spheres settling in square and circular conduits: Experiments and numerical predictions

The terminal settling velocities and rotation rates of spherical particles settling in circular and square conduits were investigated experimentally and numerically with the aim to benchmark the numerical predictions using the boundary element method. Spheres were allowed to settle in viscous Newtonian fluid under conditions such that only hydrodynamic forces exerted an appreciable effect. The terminal settling velocities and the rotation rates were measured as a function of the size and density of the falling sphere, the drop position of the sphere in the conduit and the dimensions and geometry of the containing vessel or conduit. The experimental measurements were subjected to an exacting error analysis and compared with fully three-dimensional boundary element calculations. We found that the results of the experiments and numerical simulations showed remarkable agreement within the bounds of experimental error.     

The flow past a sphere in a cylindrical tube: effects of intertia,shear-thinning and elasticity

Numerical solutions are presented for the flow past a sphere placed at the centreline of a cylindrical tube for Reynolds numbers ranging from 0 to 150, using a boundary element method. Fluids are modelled by a variety of constitutive equations including the Newtonian, the Carreau and the Phan-Thien-Tanner models. The influence of inertia, shear-thinning and fluid elasticity on the flow field, drag and the pressure drop force-drag ratio is examined. Some results are compared with available experimental data.     

Three-dimensional study of extrusion processes by Boundary Element Method.

This paper describes an implementation of a Boundary Element method to solve a general three-dimensional viscoelastic flow problem. The Boundary Element method is formulated in terms of unknown boundary velocity and traction fields. The fluid is incompressible and is modelled by a differential constitutive equation. The steady-state stress field is obtained by a time marching process of integration. For the first time, some results for steady state isothermal creeping flow extrusion of a viscoelastic fluid from triangular and square dies are described. The concept of an axisymmetric-equivalent swell ratio is introduced to compare the present results with the results of axisymmetric extrusion studies reported in the literature. It is shown that reasonable agreement is achieved.     

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.     

Three-dimensional study of extrusion processes by Boundary Element Method.

In this paper we report an implementation of a Boundary Element Method (BEM) dealing with three-dimensional extrusion problems of viscous fluid. Die shapes with and without sharp corners and their influence on the final shape of extrudate have been studied: both shrinkage and swelling of the extrudates relative to the original die shapes have been discovered. The first half of this paper deals with the problem of implementing isoparametric boundary elements. It is followed by results of test problems, and finally the results of extrusion study are presented.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

On the relationship between swelling of myofibrils and their suspension rheology

The swelling of myofibrils extracted from white bovine muscle was followed by measuring their suspension rheology. Swelling of the myofibril with increasing pH and ionic strength was accompanied by an increase in both the steady shear viscosity of the suspension and the dynamic viscoelastic properties. Swelling was continuously monitored by measuringG while the ionic strength of the suspension was being changed by dialysis. The relationship between the degree of swelling and the rheological parameters is complicated because myofibrils are rodshaped and swell radially and therefore swelling results in a change in shape. To allow for this an attempt was made to generalize the data by plotting viscosity andG againstcS/ø _m , wherec is the protein concentration in the suspension,S is the swollen volume of the myofibrils per weight of protein, and ø _m is the maximum packing fraction.The best fit to the data was represented by the equations _sp = 1.05 (cS/ _m – 0.84)^1.23 Pa · sG = 8.78 (cS/ _m – 0.67)^2.22 N m^–2. The scatter was greatest forG, possibly because at low degrees of shear the myofibrils were associated and this was confirmed by optical microscopy. Pronounced non-Newtonian behavior was observed and it was suggested that this was due to the disruption of aggregate structures, although at low concentration, orientation of the rods in the shear field may also be important.     

Shear viscosity of polybutyleneterephthalate/liquid crystal polymer blends

Viscosity measurements have been carried out on blends of polybuty-leneterephthalate (PBT) and a liquid crystalline copolyesteramide (LCP). The flow curves of the blends with LCP content larger than 20%, show a behavior similar to that of the pure LCP, with a rapid rise of the viscosity at low shear rates. The viscosity-composition curves exhibit a deep minimum at low LCP content which may be mainly attributed to the lack of interactions between the two phases.     

Lubricated squeezing flow of a Newtonian liquid between elastic and rigid plates

Theoretical force-time relationships were derived for squeezing flow of a Newtonian liquid between lubricated rigid and elastic plates. It is shown how the elastic number, representing the ratio between the elastic plate's rigidity and the specimen's viscosity, affects the force-time curve and under what circumstances the fluid specimen's thickness becomes a significant factor. Potential implications of the analysis in oral sensory evaluations of viscous foods are also considered.     

Measurement of the shear viscosity using a newly developed GRLDA-viscometer

A two-point laser-Doppler anemometer is used to determine velocity gradients. Measuring additionally the pressure drop in channel flow allows one to use this instrument as a viscometer. This is demonstrated by measuring two polymer solutions as well as water. Besides the velocity gradient, the system also furnishes the velocity as well as velocity fluctuations. For surfactant solutions the sudden increase in these fluctuations go hand-in-hand with the sudden shear thickening reported. This behavior thus seems to be caused by a change in type of flow field (structural turbulence) rather than a change in the rheology of the surfactant solutions.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

A study of swelling and collapse in polyelectrolyte gels

Swelling and collapse of a polyelectrolyte are shown to be the result of a delicate balance of three thermodynamic forces, viz. osmotic pressure, polymer elasticity, and van der Waals attraction. The behaviour of the polyelectrolyte gel is studied under isotropic pressure and under uniaxial and biaxial loading.     

Stabilität der ebenen Couette-Strömung bezüglich Störungen in der viskosimetrischen Ebene

Zusammenfassung Die Stabilität der ebenen Scherströmung eines einfachen Fluids wird im Rahmen der Kurzwellenapproximation für Störungen in der viskosimetrischen Ebene untersucht. Für kurze Wellen ist eine Stabilitätsanalyse unabhängig von der speziellen Form der Stoffgleichung möglich. Die vorliegende Analyse stellt einen ersten Schritt in diese Richtung dar und führt zu einem hinreichenden Stabilitätskriterium. Für kurze Wellen sind die Maxwell-FlüssigkeitenA undB bezüglich ebener Störungen stabil.

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The stability of a plane shear flow of simple fluids is investigated for perturbations in the viscometric plane within the framework of the short wave approximation. For short waves it is possible to carry out a stability analysis, which is independent of the type of constitutive equation. The analysis presented is the first step in this direction and leads to a sufficient stability criterion. For short waves Maxwell fluidsA andB are stable for perturbations in the viscometric plane.

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