Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

Temperature and concentration dependent viscosity and gelation temperature of ABA triblock copolymer solutions

In solutions of ABA-triblock copolymers in a poor solvent for A thermoreversible gelation can occur. A three-dimensional dynamic network may form and, given the polymer and the solvent, its structure will depend on temperature and polymer mass fraction. The zero-shear rate viscosity of solutions of the triblock-copolymer polystyrene-polyisoprene-polystyrene in n-tetradecane was measured as a function of temperature and polymer mass fraction, and analyzed; the polystyrene blocks contained about 100 monomers, the polyisoprene blocks about 2000 monomers. Empirically, in the viscosity at constant mass fraction plotted versus inverse temperature, two contributions could be discerned; one contribution dominating at high and the other one dominating at low temperatures. In a comparison with theory, the contribution dominating at low temperatures was identified with the Lodge transient network viscosity; some questions remain to be answered, however. An earlier proposal for defining the gelation temperature T _gel is specified for the systems considered, and leads to a gelation curve; T _gel as a function of polymer mass fraction.Mathematical symbols {} functional dependence; e.g., f{x} means f is a function of x - p _log logarithm to the base number p; e.g., ¹⁰log is the common logarithm - exp exponential function with base number e - sin trigonometric sine function - lim limit operation - – in integral sign: Cauchy Principal Value of integral, e.g., - derivative to x - partial derivative to x Latin symbols dimensionless constant - b constant with dimension of absolute temperature - constant with dimension of absolute temperature - B dimensionless constant - c mass fraction - dimensionless constant - constant with dimension of absolute temperature - d ^* dimensionless constant - D{0} constant with dimension of absolute temperature - e base number of natural (or Naperian) logarithm - g distribution function of inverse relaxation times - G relaxation strength relaxation function - h distribution function of relaxation times reaction constant enthalpy of a molecule - H Heaviside unit step function - i complex number defined by i ² = –1 - j{0} constant with dimension of viscosity - j index number - k Boltzmann's constant - k _H Huggins' coefficient - m mass of a molecule - n number - N number - p index number - s entropy of a molecule - t time - T absolute temperature Greek symbols as index: type of polymer molecule - as index: type of polymer molecule - shear as index: type of polymer molecule - shear rate - small variation; e.g. T is a small variation in T relative deviation Dirac delta distribution as index: type of polymer molecule - difference; e.g. is a difference in chemical potential - constant with dimension of absolute temperature - (complex) viscosity - constant with dimension of viscosity - [] intrinsic viscosity number - inverse of relaxation time - chemical potential - number pi; circle circumference divided by its diameter - mass per unit volume - relaxation time shear stress - angular frequency     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G() andG(), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersg _i , _i ,i = 1,2, N, of the discrete spectrum to obtain a best fit ofG, G, and it was found to be essential that bothg _i and _i are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesg _i,0, _i,0,N ₀. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ ₀,, J _d,i and _i of the creep complianceJ(t) can then readily be calculated using the Laplace transform.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

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.     

Rheological models for gypsum plaster pastes

Shear stress and shear rate data obtained for gypsum plaster pastes were correlated by means of different rheological models. The pastes were prepared from a commercial calcium sulfate hemihydrate at various water/plaster ratios ranging from 100/150 to 100/190. The tests were performed at 25°C using a rotating coaxial cylinder viscosimeter. The measurements were accomplished by applying a step-wise decreasing shear rate sequence. Discrimination among the models was made: (1) on the basis of the fitting goodness; (2) by checking the physical meaning of the calculated parameters; (3) on the basis of the stability of the parameters and of their prediction capacity beyond the limits of the experimental data. In the light of above, the Casson model seemed to be most effective for application to gypsum plaster pastes. K Consistency - n Power-law index - N Number of experimental data - P Number of parameters - Shear rate (s^–1) - ₀ Viscosity (Pa · s) - _d Dispersing medium viscosity (Pa · s) - _p Plastic viscosity (Pa · s) - Viscosity at zero shear rate (Pa · s) - Viscosity at infinite shear rate (Pa · s) - [] Intrinsic viscosity - ² Variance - Shear stress (Pa) - ₀ Yield stress (Pa) - Solid volume fraction - _m Maximum solid volume fraction     

Interfacial effects on the melt state behavior of polyethylene-EPDM-calcium carbonate composites

Dynamic mechanical spectra of various composites of high density polyethylene (PE), ethylene propylene diene rubber (EPDM), and calcium carbonate were obtained at 190°C with a parallel plate instrument. Interfacial effects were found to have a significant influence on the dynamic mechanical behavior of these composites.Composites of calcium carbonate in PE displayed prominent particle—particle interaction effects. This resulted in a greatly enhanced dynamic shear modulusG _d due to the filler addition. Treatment of the calcium carbonate with gamma-aminopropyltriethoxysilane (-APS) or gamma-methacryloxypropyltrimethoxysilane (-MPS) significantly reduced the particle—particle interactions. Solution deposition of EPDM or EPDM grafted with maleic anhydride (EPDM-MA) on the calcium carbonate, before incorporation into a PE composite, also had a significant effect on the composite properties. Comparison of data from composites treated with EPDM vs. EPDM-MA suggested the presence of an interaction between the calcium carbonate surface and the maleic anhydride modification. This conclusion was further supported by solid state proton NMR relaxation model experiments which showed significant immobilization of the EPDM-MA chains on the filler surface. The treatment of calcium carbonate with-APS or-MPS before incorporation into multicomponent polyethylene-rubber-filler composites also had a significant influence on the dynamic mechanical properties of the resultant composites. There is evidence for a reaction between-APS and EPDM-MA during processing on the roll mill.     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Experimental studies of suspended ferromagnetic fibers in a magnetic field

The analysis of the rotation of a ferromagnetic ellipsoid suspended in a Newtonian fluid and subjected to a uniform magnetic field is extended to include a long, slender cylindrical fiber which is magnetically saturated. Experimental observations of rotating nickel cylinders with aspect ratiosL/D ranging from 5 to 40 agree with the theoretical predictions that: (1) the proper magnetoviscous time constant for the motion is _MV = _s/µ ₀ M _s ² , (2) larger fiber aspect ratios result in considerably longer orientation times; and (3) the strength of the applied external field has only a slight effect on the overall fiber rotation, and has no effect on the maximum angular velocity achieved. Quantitative agreement of theory and experiments is obtained for fibers withL/D 20; for the shorter fibers, the theory tends to overpredict the fiber rotation rate by as much as 30%. D diameter of the cylinder - D ^P (r) position-dependent demagnetization tensor, implicitly defined in eq. (2.5) - D _xx,D _yy,D _zz volume-averaged demagnetizing factors for an ellipsoid equivalent to a uniformly magnetized cylinder, defined in eq. (2.6) - H _i ;H _i magnetic field inside a ferromagnetic body; magnitude ofH _i - H ₀;H ₀ magnetic field applied by external sources; magnitude ofH ₀ - k geometric parameter in the hydrodynamic resistance of a body rotating in a Newtonian fluid, eq. (2.2) - L length of the cylinder - L ^(h);L _z ^(h) hydrodynamic torque exerted on a rotating body; thez-component ofL ^(h) on the cylinder - L ^(m);L _z ^(m) magnetic torque exerted on a magnetic body in a magnetic field, eq. (2.4); thez-component ofL ^(m) on the cylinder - M the magnetization of a magnetic material - M _s the saturation magnitude ofM, approached by all ferromagnetic materials asH _i becomes large - r position vector of a point within a ferromagnetic body - V volume of a magnetic particle - x, y, z rectangular coordinate axes fixed in the cylinder according to figure 1 - angle of inclination of the axis of the cylinder with respect toH ₀ - shear rate - small parameter of slender body theory,=1/ln (2L/D) - _s constant viscosity of the suspending fluid - µ ₀ the magnetic permeability of free space,µ ₀=4 · 10^–7 H/m - _MV the magnetoviscous time constant, a characteristic time for a process involving a competition of viscous and magnetic stresses - ₁ the first normal-stress coefficient - ; _z angular velocity of a rotating body; angular velocity of a cylinder about thez-axis, _z =– d/dt     

Zum Mischen rheologisch inhomogener Stoffsysteme auf Einschneckenmaschinen

Zusammenfassung Das Mischen von Stoffen mit unterschiedlichen rheologischen Eigenschaften in Schneckenmaschinen ist in der Kunststoffauf- und -verarbeitung eine Standardaufgabe. Trotzdem gibt es hierfür kein zufriedenstellendes mathematisch-physikalisches Modell. Daher werden zunächst einfache Mischmodelle diskutiert. Auf der Basis dieser Modelle wird dann unter Berücksichtigung der Besonderheiten des Plastifizierextruderprozesses eine Mischgütebeziehung mathematisch formuliert. Die experimentelle Überprüfung erfolgt mit Hilfe der Grauwertanalyse extrudierter Zweistoffsysteme, bei denen ein Stoff mit Ruß eingefärbt war. Da der Mischprozeß hochgradig stochastisch ist, streuen die Meßergebnisse. Unter Berücksichtigung dieses Tatbestandes ist der theoretische Ansatz zufriedenstellend.

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Mixing of polymer resins with different rheological properties is a usual demand in plastics processing using screw extruders. A mathematical model describing this processing problem sufficiently is not known, however. Therefore, simple mixing models will be discussed. Based on these, a concept for the calculation of mixing homogeneity will be presented, including the particular requirement of the plasticating screw process. An experimental investigation utilizes the grey-value analysis of extruded two-component materials, which in one phase is carbon-black filled. Considering the fact that the mixing process is highly random, the theoretical model leads to a good level of aggreement with the scattering measurement data.

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b Schneckenkanalbreite - B Bandbreite der Grauwerte - c Konstante - mittlere Konzentration, bezogen auf die Grauwertbandbreite - h Höhe, Gangtiefe, Schneckenkanalhöhe - h ₀ Gangtiefe der Einzugszone - h ₁ Gangtiefe der Ausstoßzone - L Länge - gemittelte Schmelzebettlänge - n Exponent des Potenzfließgesetzes - s Standardabweichung der Grauwerte bezogen auf die Grauwertbandbreite - S Standardabweichung der Grauwerte - t Verweilzeit - t ₁ kürzeste Verweilzeit - mittlere Verweilzeit - ₀ Umfangsgeschwindigkeit - mittlere Geschwindigkeit - V Volumenstrom - w Dicke eines Kontrollelements - w Ausstreichdicke eines Kontrollelements - x Koordinate - Mittelwert der Grauwerte - y Koordinate - Scherdeformationswinkel - Scherdeformation - mittlere Scherdeformation - Schergeschwindigkeit - Viskosität - ₁ dimensionslose kürzeste Verweilzeit - dimensionsloser Volumenstrom - _LSM laminarer Schermischgrad - _{LSM, the} theoretischer laminarer Schermischgrad - _{LSM, exp} experimenteller laminarer Schermischgrad - ² Varianz der Verweilzeit im Schmelzebett - Schubspannung - Gangsteigungswinkel der Schnecke - ø Volumenanteil - dimensionslose Kennzahl     

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.     

Evaluating Herschel-Bulkley fluids with the back extrusion (annular pumping) technique

A mathematical model was developed to describe the behavior of Herschel-Bulkley fluids in a back extrusion (annular pumping) device. A technique was also developed to determine the rheological properties (yield stress, flow behavior index, and consistency coefficient) of these fluids. Mathematical terms were expressed in four dimensionless terms, and graphical aids and tables were prepared to facilitate the handling of the expressions.Nomenclature a radius of the plunger, m - dv/dr shear rate, s^–1 - F force applied to the plunger, N - F _b buoyancy force, N - F _cb force corrected for buoyancy, N - F _T recorded force just before the plunger is stopped, N - F _Te recorded force after the plunger is stopped, N - g acceleration due to gravity, m/s² - H(t) momentary height between plunger and container bottom, m - K a/R, dimensionless - L length of annular region, m - L(t) depth of plunger penetration, m - n flow behavior index, dimensionless - p static pressure, Pa - P _L pressure in excess of hydrostatic pressure at the plunger base, Pa - p ₀ pressure at entrance to annulus, Pa - P pressure drop per unit of length, Pa/m - Q total volumetric flow rate through the annulus, m³/s - r radial coordinate, measured from common axis of cylinder forming annulus, m - R radius of outer cylinder of annulus, m - s reciprocal of n, dimensionless - t time, s - T dimensionless shear stress, defined in Eq. (3) - T ₀ dimensionless yield stress, defined in Eq. (4) - T _w dimensionless shear stress at the plunger wall - _p velocity of plunger, m/s - velocity, m/s - mass density of fluid, kg/m³ - Newtonian viscosity, Pa s - P p ₀ –p _L , Pa - consistency coefficient, Pa sⁿ - value of where shear stress is zero - _–, ₊ limits of the plug flow region (Fig. 1) - r/R - shear stress, Pa - _y yield stress, Pa - _w shear stress at the plunger wall, Pa - dimensionless flow rate defined in Eq. (24) - dimensionless velocity defined by Eq. (5) - _–, ₊ dimensionless velocity outside the plug flow region - _max dimensionless maximum velocity in the plug flow region - _p dimensionless velocity at the plunger wall     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

Rheological characteristics of glass-fibre-filled polypropylene melts

The rheological properties of glass fibre-filled polypropylene melts have been investigated. A high pressure capillary rheometer has been used for the experimental study. The effect of shear rate, temperature, and fibre concentration on the melt viscosity and viscoelastic properties have been studied. An equation has been proposed to correlate the melt viscosity with shear rate, temperature and fibre content. A master curve relation on this basis has been brought out using the shift factora _T . a _T shift factor (=/ _r ) - A _i coefficients of the polynomical of eq. (1) (i = 0, 1, 2, ,n) - B constant in the AFE equation (eq. (2)) (Pa s) - B constant in eq. (3) - D extrudate diameter - d capillary diameter - activation energy at constant shear rate (kcal/mole) - E activation energy at constant shear stress (kcal/mole) - T melt temperature (K) - X fraction glass fibre by weight - shear rate (s^–1) - shear viscosity (Pa s) - normal stress coefficient (Pa s²) - ₁ – ₂ first normal-stress difference (Pa) - shear stress (Pa) - r at reference temperature     

Pure liquids described as concentrated cluster dispersions

The Fulcher-Tammann-Hesse-Vogel equation, ln = A + B/(T – T ₀ ), is shown to be equivalent to the general viscosity-composition relationship, ln _r =k f /(1 – f ), for binary mixtures. The Cailletet-Mathias law of the Rectilinear Diameter is rearranged to represent a density mixture formula for two components. Temperature-independent viscosities and densities can then be calculated for dense, solid cluster fractions, dispersed in a low-density, low-viscosity non-clustered continuous phase. The cluster fraction decreases with temperature. The value ofT ₀ is shown to be related to the liquid- or solid-like behavior of the clusters. For liquids with a vapor pressure < 1 mm Hg at the melting point, the calculated cluster volume fraction suggests close packing of clusters, ranging in shape from monodisperse spheres to polydisperse non-spherical particles. Examples are given for molecular liquids, molten metals, and molten salts. The size of the clusters is estimated from the heat of evaporation.     

Analyse und Berechnung der Strömungsvorgänge im zylindrischen Scherspalt von Plastifizieraggregaten

Zusammenfassung Dieser Aufsatz zeigt eine Möglichkeit auf, zylindrische Scherteile einer Plastifiziereinheit, auf der strukturviskose Materialien verarbeitet werden, approximativ zu berechnen. Es ist möglich, den Volumenstrom und Druckabfall, die mittlere Schergeschwindigkeit, Scherdeformation und Schubspannung im Scherspalt zu approximieren. Durch diese Gleichungen wird eine Abschätzung der Verteil- und Zerteilvorgänge im Scherelement möglich.

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A method is described for approximating the flow in cylindrical shearing gaps of plasticating extruder, which is applicable to shear thinning materials. It is possible to calculate the through-put and pressure drop as well as the shear rate, strain and shear stress in the gap. With these equations the distribution and separation process in shearing gaps can be evaluated.

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D Zylinderdurchmesser - d ₁ Schnecken-Kerndurchmesser der Meteringzone - d _s Durchmesser des zylindrischen Scherteils - K Konstante im Potenzfließgesetz - K _0T Koeffizient des Potenzfließgesetzes - L ₁ Länge der Anlaufschräge - L _s Länge des zylindrischen Scherteils - n Fließindex - n ₀ Drehzahl - p Druckabfall über der Scherteillänge - s Scherspalthöhe - T _M Massetemperatur - ₀ Umfangsgeschwindigkeit - _0x Geschwindigkeitskomponente inx-Richtung - _x, _z Geschwindigkeit inx- bzw.z-Richtung als Funktion der Koordinatey - Volumenstrom - x, z Ortskoordinaten - Exponent des Potenzfließgesetzes - Schergeschwindigkeit - mittlere Schergeschwindigkeit - Viskosität - dimensionslose Höhe - Dichte der Schmelze - Schubspannung - _yx, _yz Schubspannungskomponenten - _xx, _zz Normalspannungskomponenten - _ps dimensionsloser Druckgradient - dimensionsloser Volumenstrom - _x, _z dimensionslose Geschwindigkeit inx- bzw.z-Richtung