On the burst of branched polymer melts during inflation

Two molten low-density polyethylene melts, shaped as plates, have been inflated into a circular cylinder during isothermal conditions. Lowering the inflation rates allow the plates to be inflated into a larger volume of the cylinder before bursting. Numerical simulations of the inflations have been performed, using a time-strain separable constitutive K-BKZ equation based on the potential function from the Doi–Edwards theory. The material parameters in the constitutive model are based on liner viscoelastic and time dependent uniaxial elongational viscosities. The numerical calculations show quantitative agreement with the experiments, including the appearance of the burst, for a wide range of experimental conditions. This strongly suggests that the initiation of the burst in the polymer melts is a hydrodynamic phenomenon.     

Mapping of the relaxation patterns of polymer melts with linear flexible molecules of uniform length

The linear viscoelastic material functions of linear flexible polymers of uniform length are calculated from the BSW spectrum (Baumgaertel et al., 1990, 1992), and explicit analytic expressions are presented for several of the most common material functions for transient and dynamic experiments. However, numerical calculations are presented whenever needed. The BSW spectrum was determined from experimental G, G data of two sets of molten polymers of narrow molecular weight distribution, polystyrene and polybutadiene. The purpose of the mapping is to show a wide range of viscoelastic behavior which otherwise is not available in such comprehensive form. Experimental check of these predictions is still needed in most cases. Also, some insight into the predictions for the non-linear (including the non-equilibrium) viscoelastic behavior is achieved by studying two particular experiments: the start-up of uniaxial extension at constant rate and the start-up of shear flow at constant rate.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

Resonant torsional vibrations: an application to dynamic viscometry

Summary A family of sensors which measure the complex coefficient of dynamic viscosity ( ^*) is discussed. Using a first order approximation to a general viscoelastic fluid, it is shown how these instruments determine ^* for a general linear viscoelastic fluid. The measurement technique employed relates the linear viscous damping and resonant frequency of these instruments in the presence and absence of fluid to ^*. This analysis also provides the inherent limitations of the sensors.

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Resonante Torsionsschwingungen: Eine Anwendung zur Dynamischen Viscometrie

Übersicht Eine Familie von Sensoren wird diskutiert, welche den komplexen Koeffizienten ( ^*) der Viskosität misst. Mit Hilfe einer Approximation erster Ordnung eines allgemeines viskoelastisches Fluid wird gezeigt, wie diese Instrumente verwendet werden können, um ^* für ein allgemeines linear viskoelastisches Fluid zu bestimmen. Die Messtechnik verwendet die Tatsache, daß Resonanzfrequenz und linearviskose Dämpfung der Sensoren abhängig sind vom Fluid, welches sich in der direkten Umgebung des Sensors befindet. Die Analyse, die hier verwendet wird, zeigt Möglichkeiten und Grenzen der Sensoren auf.

     

Linear viscoelastic behavior of the Hookean dumbbell with internal viscosity

The linear viscoelastic modulus G(t) predicted by the analytical formulations of Schieber (1993), Wedgewood (1993), Dasbach et al. (1992), and Booij and van Wiechen (1970) for the free-draining Hookean dumbbell with internal viscosity (IV) are compared with exact analytical results and exact numerical results obtained from Brownian dynamics simulations. All of these analytical formulations employ the Booij and van Wiechen expression for the IV force, thereby eliminating errors associated with linearization of the deformational velocity, however the theories differ in the approximations employed to solve configuration moment equations. Comparison with the exact G(t) results provides a means of testing these approximations. The approximate theories all correctly predict the singular part of G(t) at t = 0, providing correct predictions of , however deviations from the exact G(t) are seen in all cases for t > 0.     

Normal stress measurements

An apparatus designed to measure the dynamic viscoelastic response of polymer melts is described. Dynamic elasticity (G) and viscosity () can be measured over a frequency range 10^–2-10^–3 Hz and at temperatures up to 350 °C. The sample under test is held in a cone and plate assembly. A small strain is introduced by driving the plate with a variable speed synchronous motor and off-centre cam at low frequencies and by an electromagnetic vibrator at high frequencies. The amplitudes of the cone and plate are detected using the optical lever principle and photocell strips. The phase difference between the cone and plate is measured from a recorder trace at low frequencies and by direct reading on a meter at high frequencies.Results are described of measurements on silicone fluids, and on commercial grades of polyethylene with different molecular weight distributions and degrees of branching.A discussion is given of the correlation between the dynamic viscosity measurements and those taken under steady-flow conditions.Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968. — Original paper published in J. Sci. Instruments Series 2,1, 1102–1112 (1968).     

Numerical modeling of perturbation propagation in a supersonic boundary layer

The growth of two-dimensional disturbances generated in a supersonic (M = 6) boundary layer on a flat plate by a periodic perturbation of the injection/suction type is investigated on the basis of a numerical solution of the Navier-Stokes equations. For small initial perturbation amplitudes, the second-mode growth rate obtained from the numerical modeling coincides with the growth rate calculated using linear theory with account for the non-parallelism of the main flow. Calculations performed for large initial perturbation amplitudes reveal the nonlinear dynamics of the perturbation growth downstream, with rapid growth of the higher multiple harmonics.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 33–44. Original Russian Text Copyright © 2004 by Egorov, Sudakov, Fedorov.     

Planar oscillations and dissipative heating of viscoelastic plates with a piezoeffect

Planar oscillations of thin piezoplates are important within the context of using this type of piezoelements as resonator frequency filters, frequency stabilizers, elements of piezotransformers, and other technological devices. In the publications currently known one usually considers piezoplates with elastic material behavior and linear governing equations. By their mechanical nature, however, a number of piezoelements, particularly piezoceramics, are viscoelastic, which, depending on the loading conditions, can lead to substantial dissipative heating of the piezoelement and confine its operation [3]. The use of piezopolymers and their composites raises particularly important issues of dissipative heating. At the present time the behavior of a piezoelement including heating can be described by the theory of thermoelectroviscoelasticity (TEVE) [2, 3], including the interaction between electromechanical and thermal fields. The complexity of TEVE problems leads to the necessity of using numerical methods to solve them, with the finite element method (FEM) being widely used in recent years. The present study is devoted to stating and solving TEVE problems concerning thin piezoceramic plates by the FEM. We treat a thin piezoceramic plate, confined by an arbitrary contour L and polarized across its thickness. A harmonic potential difference e^it is supplied to electrodes located on the smooth boundaries of the plate. Convective heat exchange with the surrounding media of temperatures T _k ^s and T^s is implemented at the contour surfaces and boundaries free of electrodes. The heat transfer coefficients equal, respectively, _k ^T and ^T. The initial plate temperature is T₀. The smooth boundary are free of mechanical loading. The mechanical forces at the contour surfaces are distributed symmetrically with respect to the mean plane of the plate.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev (Ukraine). Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 69–76, February 1994.     

The cantilever strip plate under torision, bending or flexure at infinity

A homogeneous, isotropic plate occupies the region 0x ₁, |x ₂|a, |x ₃|h, where the ratio h/a is sufficiently small so that the classical theory of thin plate bending applies. The short end of the plate at x ₁=0 is clamped while the long sides are free. This cantilever plate is now loaded at x ₁=+ by an applied twisting moment, by a bending moment or by flexure. Despite the fundamental nature of these problems, and the long history of thin plate theory, no solutions are to be found in the existing literature that will determine (for instance) the important unknown resultants V ₁, M ₁₁ at the clamped end x ₁=0. The main reason for this is that this combination of boundary conditions leads to severe oscillating singularities of the field in the corners (0, ±a). The fact that such singularities must exist is widely known, but we present here for the first time a method of solution that takes these singularities fully into account.Our numerical results show that the values of M ₁₁, V ₁ on x ₁=0 bear little resemblance to those of the corresponding Saint-Venant solutions, which do not fully satisfy the boundary conditions at the clamped end. Indeed, significantly large values of these resultants were found at points far enough from the corners so as to be relevant in actual engineering applications. Also of interest are certain weighted integrals of M ₁₁, V ₁ which we calculate. These constants determine the effect of the clamping at large distances (greater than 4a, say) from the cla,ped end. At such distances, the effect of the clamping is merely to impose an additional rigid body deflection on the plate.Finally, we consider the plate of finite length. Provided that the aspect ratio is 2 or more, we give accurate approximate solutions for the torsion, bending or flexure of a finite plate clamped at both ends.     

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

Self-similarity of nonstationary boundary-layer motions

In this paper we show that problems concerning the development of a boundary layer on a semi-infinite plate when the outer flow speed is of the form U = (1 + ct)^b a, and on a cylinder when the outer flow speed has the forms U = ctx^m and U = (1 + ct)^b ax^m, are self-similar. We present the results of numerical calculations for various values of , b, and m. We consider the problem of a stepwise nonstationary heating of a plate, impulsively set into motion in an incompressible fluid; we show that this problem is self-similar and obtain its solution numerically.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 122–125, July–August, 1975.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

An alternative approach to the linear theory of viscoelasticity and some characteristic effects being distinctive of the type of material

In the introduction some postulates on which the linear theory of viscoelasticity is based are recalled, and the postulate of passivity is substituted by a stronger postulate called detailed passivity.Next, a symmetric formulation of this theory is presented which is founded in a well-balanced way on the limiting properties of elasticity and viscosity. This leads to the introduction of the basic functions of creep compliance J ⁺(t) and stressing viscosity ⁺(t) associated to one another, whereas the basic functions retardation fluidity ⁺(t) and relaxation modulus G ⁺(t) emerge as their time derivatives. Correspondingly, four complex basic functions are defined as their Carson transforms.In addition to the proper retardation and relaxation terms, these basic functions contain the non-disappearing constants of either instantaneous compliance J ₀ or instantaneous viscosity ₀ and also of either ultimate fluidity or ultimate modulus G . Therefrom ensues a classification of linear viscoelastic materials into four types: instantaneous elasticity or viscosity is allowed to combine with ultimate viscosity or elasticity. The latter alternative, signifying fluidlike or solidlike materials, leads, of course, to a quite different behavior in many situations; however, remarkable distinctive features are associated to the first one as well.A few respective examples are outlined: 1) propagation of shear waves in a half-space with periodic and step-shaped excitation, 2) dissipation of work in a torsional vibration damper, and 3) shear flow between two parallel porous plates with injection and suction.Finally, materials with viscous initial behavior are defended against the notion that they be of no or almost no real significance.Delivered as a Plenary Lecture at the Fourth European Rheology Conference, Seville (Spain), 4–9 September 1994. The herein only outlined topics are taken from a recently pulished monograph (Geisekus, 1994) in which complete derivations of the results and more detailed discussions are given.Dedicated to Professor K. Walters on the occasion of his 60th birthday.     

Conversion of oscillatory shear data from highly filled polymers by direct numerical integration

A procedure was developed to enable the direct numerical integration of the Fourier integral transform equation relating G(t) to G() by considering integration limits that vary as a function of time and which define a range of discrete sub-intervals within the complete frequency domain data set. The method provides results that are in very close agreement to results determined from a relaxation spectrum. However, at low values of time the solution to the variable limit integral transform is sensitive to the absence of a contribution beyond the upper experimental limit of the frequency domain data. G(t) results determined from the conversion of shifted master G() experimental data using the variable limit integral transformation and the relaxation spectrum compared favourably with actual shifted master G(t) experimental data. The former curves were characterised by the same form and trend as the experimental results, confirming that the underlying viscoelastic behaviour is well represented. While the variable limit Fourier integral transform procedure provides a good approximation to relaxation spectrum results, the latter is clearly the more robust method of converting data from the frequency to the time domain.It was observed that the time-temperature superposition procedure used in the construction of shifted master curves can magnify potential differences between the shifted G(t) values determined from the conversion of G() data and the actual experimental G(t) results, when compared to data that has not been shifted to a master curve.     

Über eine neue Form der Reynoldsschen Gleichung zur Schmier-Hydrodynamik

Übersicht Mit Substitutionen für die Spalthöhe und den hydrodynamischen Druck wird eine neue Differentialgleichung abgeleitet, welche an die Stelle der Reynoldsschen Gleichung tritt. Gegenüber dieser hat sie den Vorteil, nicht nur spezielle Lösungen — die mit dem physikalischen Geschehen meist nicht übereinstimmen-zu geben, sondern für alle praktisch wichtigen Fälle geschlossene Lösungen zu ermöglichen, sofern geeignete Ansätze für die Spaltkontur gewählt werden. Solche Ansätze lassen sich entsprechend dem Aufbau der neuen Differentialgleichung unschwer in einem Konturen-Katalog zusammenfassen, der für alle Bedürfnisse der Praxis ausreicht. Zwei Beispiele, deren Lösungen bekannt sind (die unendlich breite Platte und der unendlich breite Kreiszylinder), sollen der Einführung in die neue Methode dienen und die Genauigkeit aufzeigen. Im Hauptteil wird das bisher ungelöste Problem Kugel auf Ebene behandelt und der Verlauf des hydrodynamischen Druckes im Kugelspalt angegeben.

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Summary By substitutions for the distance across the gap and for the hydrodynamic pressure a new differential equation is derived which replaces the Reynolds equation. Compared to the latter the new equation has the advantage that it has analytical solutions not only in special cases which usually do not correspond to practical problems, but also in all essential practical cases if suitable expressions are chosen for the gap profile. Corresponding to the structure of the new differential equation such profile expressions can easily be listed up in a profile-catalogue which meets all practical requirements. Two examples with known results (the infinitely wide plate and the infinitely long cylinder) are presented for the purpose of introduction into the new method and for proof of accuracy. In the main part the problem of a sphere on a plate is treated which is still unsolved, and the hydrodynamic pressure profile between the two bodies is calculated.

     

Asymptotic analysis of the nonlinear equations for an infinite,rubber-like slab under an equilibrated vertical line load

An infinite slab of incompressible Rivlin-Saunders material of constant thickness 2H is subject to an equilibrated, radially varying, vertical body force, comprising a concentrated, downward line load and a smooth, upward, exponentially distributed load with a characteristic decay length R. The deformation is axisymmetric and described by three stretches and a shear strain (or, equivalently, four strains) and a rotation which satisfy three relatively simple compatibility conditions. Force equilibrium is satisfied identically by the introduction of three stress functions. The incompressibility constraint is used to eliminate the normal stretch. With the introduction of stress-strain relations, the field equations are reduced to a set of seven, first-order, quasilinear partial differential equations. The loads, the radial distance, and the unknowns are scaled by the small parameter =H/R. As 0, 11 possible sets of field equations are found, including linear plate theory, von Kármán plate theory, Föppl membrane theory, large-strain membrane theory, and Wu's large-stretch (asymptotic) membrane theory. Notably absent as limiting cases are thick plate theories.This work was supported by the National Science Foundation under grant MSM-8618657-02.     

Heat transfer enhancement in a channel with a built-in rectangular cylinder

A two dimensional numerical investigation of the unsteady laminar flow pattern and forced convective heat transfer in a channel with a built-in rectangular cylinder is presented. The channel in the entrance region has a length to plate spacing of ten. The computations were made for several Reynolds number and two rectangular cylinder aspect ratios. Hydrodynamic behavior and heat transfer results are obtained by solution of the complete Navier-Stokes and energy equation. The results show that these flow exhibits laminar self-sustained oscillations for Reynolds numbers above the critical one. This study show that oscillatory separated flows result in a significant heat transfer enhancement but also in a significant pressure drop increase.

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Erhöhung des Wärmeübergangs in einem Spaltkanal mit quer eingebautem Rechteckprisma

Zusammenfassung Es wird eine zweidimensionale numerische Untersuchung des instationären Wärmeübergangs und Druckverlustes im laminar durchströmten Spaltkanal mit quer eingebautem Rechteckprisma dargelegt und zwar für verschiedene Reynoldszahlen und zwei Prismenabmessungen. Als Lösung der Navier-Stokes- und der Energiegleichung resultieren selbsterregt oszillieren de Strömungs- und Temperaturfelder, verbunden mit starker Erhöhung des Wärmeübergangs und des Druckverlustes.

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List of symbols C _f skin friction coefficient, Eq. (11) - C _D drag coefficient, Eq. (11) - D drag [N/m] - f _app apparent friction factor, Eq. (10) - h cylinder height [m] - H channel height [m] - k thermal conductivity of cylinder [W/mK] - k ₀ thermal conductivity of air [W/mK] - l cylinder length [m] - L channel length [m] - Nu Nusselt number, Eq. (7) - P dimensionless pressure - Pr Prandtl number of air - Re Reynolds number, Eq. (6) - t time [s] - T temperature [K] - T _b bulk temperature [K], Eq. (8) - U, V dimensionless velocity components - X, Y dimensionless coordinates Greek symbols thermal diffusivity [m²/s] - velocity factor, Eq. (11) - dimensionless temperature, Eq. (5) - fluid density [kg/m³] - kinematic viscosity [m²/s] - dimensionless time, Eq. (5) - difference     

The dynamic performance of the Weissenberg Rheogoniometer

The dynamic performance of a standard Model R18 Weissenberg Rheogoniometer has been studied in detail. The Rheogoniometer was carefully calibrated and used to measure accurately the rheological behaviour of a highly nonlinear viscoelastic polymer solution (1% polyacrylamide in 50% glycerol/water).In this paper the elaborate procedures that were used to calibrate the electronic signal processing equipment are described. The various static and dynamic calibration/correction factors are defined and incorporated into a computer implemented calculation scheme for evaluating the linear dynamic properties from the raw digital transfer function analyser readings.The linear dynamic properties of the polymer solution are presented together with the corresponding steady shearing properties. Both cone and plate and parallel plates geometries were used and good agreement was obtained over the wide range (six decades) of frequencies and shear rates employed.Fluid inertia effects were found to become important when the modified Reynolds number,Re _c ² orRe(H/R)², exceeded a value of about 0.1. These effects had a strong influence on the phase angle() which could readily be detected by varying the gap angle/width. The Walters-Kemp equations were found to give consistently accurate values for the linear dynamic properties for modified Reynolds numbers up to 11.6 which was the highest reached.     

Linear viscoelasticity,simple and planar melt extension of linear polybutadienes with bimodal molar mass distributions

Shear oscillations, simple and planar elongations have been performed with anionically polymerized polybutadienes (PB) and their blends at room temperature. The PB components were of different molar mass averages and of narrow molar mass distributions; the blends had bimodal molar mass distributions and are represented by the weight ratio w of the high molecular component. The crossover G() = G() obtained from oscillatory measurements shows correlations with molecular parameters. For the zero shear viscosity the well-known relation ₀ M _w ^3.4 is found. The recoverable equilibrium shear compliance J _e ⁰ is nearly the same for the components; for the blends it strongly depends on w with a pronounced maximum at small w. In elongation outside the linear region strain hardening is found; its magnitude depends on M _w of the components, the composition w of the blend, the mode of elongation (simple or planar), and the elongational strain rate. The hardening revealed in the increase of the elongational viscosity above the linear viscoelastic limit increases as a function of w up to a maximum similar to J _e ⁰ such that, for both properties, the molecular processes may be the same. The elongational viscosity µ₂ (from the lateral stress in planar elongation) is above the linear viscoelastic limit for bimodal and below this limit for conventional broad molar mass distributions. In general, it can be stated that with a more narrow molar mass distribution of linear polymers the elongational behavior of the melts comes closer to the linear viscoelastic limit.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.Extended version of a paper presented at the Annual Conf. German Soc. of Rheology, Berlin, May 13–15, 1991.     

The load-bearing capacity of a continuous-flow squeeze film of liquid

A new form of squeeze film system is described in which the movement of one plate towards the other is simulated by the continuous volume generation of liquid over the plate area. The liquid exudes from 1580 holes distributed uniformly over the lower plate surface. An advantage of the system is that there are no moving parts, but it is important to evaluate the device using Newtonian liquids in order to compare the load bearing capacity with that predicted by equations developed for orthodox squeeze film systems. Liquid maldistribution is shown to be a problem which may be solved in various ways, one of which is to ensure that the pressure drop through the plate is high relative to that in the squeeze film.Results obtained using Newtonian liquids make satisfactory comparison with theoretical predictions, though liquid inertia probably makes a lower contribution to load bearing than is the case for an orthodox squeeze film. Liquid maldistribution is allowed for on a theoretical basis or corrected by the use of a distributor plate placed below the perforated surface.Preliminary tests using viscoelastic solutions (based on polyacrylamide of high molecular weight) suggest that the load bearing properties of the squeeze film are significantly enhanced. A load 600 per cent greater than the theoretical load is obtained in one case, the suggestion being made that this is due to stress of viscoelastic origin.Nomenclature D Exit diameter of holes in spinnerette - F ₁ to F ₆ Vertical force on top plate due to flow in squeeze film, defined by (1), (8), (11), (12), (13) and (14) respectively - h Plate separation - h _L Distance of distributor plate from lower surface of spinnerette (function of r) - I ₀ Modified Bessel function of first kind, order 0 - I ₁ Modified Bessel function of first kind, order 1 - K ₀ Modified Bessel function of second kind, order 0 - L Length of hole, based on diameter D, giving same pressure drop as actual spinnerette holes - dm/dt Mass flowrate of liquid - N Total number of holes in spinnerette (1580) - p Isotropic pressure in squeeze film - P _RES Isotropic pressure in reservoir behind lower plate of spinnerette - p–P _RES - (dp/dr)_s Pressure gradient in squeeze film - (dp/dr)_L Pressure gradient in lower film below spinnerette when distributor plate is used - Q Total liquid volume flowrate - q _s Volume flowrate through squeeze film at radius r - q _L Volume flowrate through lower film at radius r - r radial coordinate - R radius of upper disc - - v Velocity of upper disc relative to lower one (simulated by Q/R ² in continuous flow system) - V _R Average radial liquid velocity at radius R - V _S Liquid exit velocity from single hole - V _r V V _z Point velocity components in r, and z directions respectively - z Axial coordinate - Parameter in (8) (3ND ⁴/32LR ² h ³) - Viscosity of liquid - Density of liquid - _rz Shear stress