Modell zur Beschreibung des Fließens wandgleitender Substanzen durch Düsen

Zusammenfassung Zur Analyse des Fließens einer direkt an der Düsenwand gleitendenOstwald-deWaele-Flüssigkeit (Potenzgesetz) wird ein Modell entwickelt, das die rheologischen Vorgänge tribologisch, d. h. analog derCoulombschen Reibung fester Körper beschreibt.Es zeigt sich, daß in der Düse zwei Bereiche zu unterscheiden sind: ein Haftbereich in der Nähe des Düseneinlaufs und ein am Düsenaustritt liegender Gleitbereich. Die Länge des Gleitbereichs, der Verlauf des Drucks und der Schubspannung längs der Düse sowie die Änderung des Geschwindigkeitsprofils im Gleitbereich werden ermittelt.Überschreitet die Wandschubspannung einen kritischen Betrag, so entsteht am Düsenende ein labiler Bereich, in dem der Betrag der Wandschubspannung sprunghaft auf einen kleineren Wert sinken kann. Der von verschiedenen Autoren gefundene Sprung in der Fließkurve bestimmter Polymerschmelzen kann damit grundsätzlich erklärt werden.

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Summary Starting from theCoulomb Friction Law for solids, a theoretical model is developed for the pressure flow of a viscous power-law fluid with slip at the wall.It is shown that two flow regions exist in the die: a first region at the upstream part of the die, where the fluid sticks to the wall; and a second region at the downstream part of the die, where the fluid slips at the wall. The length of the slip region, the development of pressure and shear stress along the die as well as the change of the velocity distribution are given for the slip region.For shear stresses above a critical value, an instability region is found at the exit of the die. In this region, a sudden decrease of shear stress can occur. This seems to explain the discontinuity in the flow curve reported by several investigators.

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F Querschnittsfläche der Kapillaren - Volumendurchsatz - K _R Reibkraft - L Düsenlänge - m Stoffwert (Fließexponent) - N Normalkraft - p hydrostatischer Druck - p _L Druck am Düsenende - p ₁ Druck an der Übergangsstelle Haften-Gleiten - p ₀ Druck vor der Düse - p _0H Druck vor der Düse im Falle des Wandhaftens - r Radius - R Düsenradius - v _g Gleitgeschwindigkeit - v _z Strömungsgeschwindigkeit inz-Richtung - z Koordinate in Strömungsrichtung - z ₁ Längskoordinate der Übergangsstelle Haften-Gleiten - Schergeschwindigkeit - Stoffwert - Viskosität - µ Gleitkoeffizient - µ _H Haftkoeffizient - Dichte - dimensionsloser Radiusr/R - _rz Schubspannung in der Flüssigkeit - _rz ^(R) Wandschubspannung in der Flüssigkeit - ₀ Stoffwert - _wg Wandschubspannung im Falle des Gleitens - _wH Haftschubspannung an der Wand Auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologen in Berlin vom 28.–30. April 1975.Mit 10 Abbildungen     

Instability of Sedimenting Bidisperse Particle Gas Suspensions

A linear stability analysis for a sedimenting bidisperse gas-solid suspension (or gas fluidized bed) is performed. Mass, momentum and energy conservation equations for each of the two species are derived using constitutive equations that are valid at high Stokes numbers, (St₁ 1). The homogeneous suspension becomes unstable at sufficiently large St₁ to waves of particle volume fraction with the wave number in the vertical direction. Numerical calculations of the growth rate in an unbounded suspension indicate that the marginal stability limits are controlled by the small wave number (k 1) behavior. Depending on the Stokes number and the volume fractions ₁ and ₂ of the two species, the suspension becomes unstable due to O(k) or O(k²) contributions to the growth rate. The O(k) term corresponds to an instability due to kinematic waves similar to that predicted for bidisperse suspensions of particles in viscous liquids [22]. The O(k²) contribution represents an instability to dynamic waves similar to that obtained from an analysis of averaged equations for monodisperse fluidized beds [4].     

Transition,turbulence and oscillating flow in a pipe a visual study

Transitional and turbulent oscillatory flow in a rigid pipe with long entry sections was investigated using flow visualization to establish the existence of coherent structures. Flow tracer and high speed motion pictures were used. The simple harmonic motion of a scotch yoke and flywheel linked to a piston and cylinder provided the flow driving force. The camera was convected with the flow by attaching it through a gearing system to the scotch yoke.List of symbols A cross sectional area of flow - C, K constants - D pipe diameter - N _Re,ave Reynolds number based on average velocity (DU _ave /v) - N _Re,p Reynolds number based on maximum oscillatory velocity (DU _max /v) - Reynolds number based on maximum oscillatory velocity and Stokes (boundary) layer thickness (U _max /v) - R pipe radius - U instantaneous velocity in the flow direction - short-term average instantaneous velocity - U ^* friction velocity (U _ave (f/2)^1/2) - U _amp amplitude parameter (U _max /U _ave ) - U _ave average velocity - U _s steady velocity - U _t instantaneous oscillatory velocity - U _max maximum oscillatory velocity ( X _max /T) - u _r , u _z deviations from _r, and _z - y radial coordinate from wall (R–r) - y ⁺ dimensionless radial coordinate from wall (y U*/v) - frequency parameter [R (/v) ^1/2] - Stokes (boundary) layer thickness [C (2 v/)^1/2] - normalized time into cycle - fluid viscosity - v fluid kinematic viscosity (/) - density - angular frequency (2/T) - - overbar, average - sub-c critical value     

Nonlinear viscoelastic properties and change in entanglement structure of linear polymer

Simultaneous measurements of stress relaxation and differential dynamic modulus were made at 268 K over a time scale of 10 to 10⁴⁵ s for nearly monodisperse polybutadiene (M _w =2.2x10⁵, 1,2-structure 70%, M _e =3600) and also one having coarse cross-linking (M _c =29000). Static shear strain ranged from 0.1 to 2.0. In a long-time region (t> _k ), the relaxation modulus G (; t) could be expressed by the product G (0; t) h (y). The observed h() agreed well with the Doi-Edwards theory without use of IA approximation. Both the cured and uncured samples showed initial drop of the differential storage modulus G (), ; t) followed by gradual recovery, but did not attain the value before shearing G (, ; t) for the uncured sample showed smaller values than that for the cured one in the whole measured time scale at the higher strain, confirming the two origins of nonlinear viscoelasticity of well entangled polymer; induced chain anisotropy and induced decrement in entanglement density. G (, ; t) curves for the cured sample agreed well with the BKZ predictions. But the curves for the uncured sample agreed well with the BKZ prediction only at the time scale of t< _k . BKZ prediction showed significant upward deviations at t> _k . Such the differences are discussed in terms of the two origins.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

An asymptotic treatment of the transient development of axisymmetric surface waves

A linearized theory is developed for the derivation of an asymptotic solution of the initial value problem of axisymmetric surface waves in an infinitely deep fluid produced by an arbitrary oscillating pressure distribution. An asymptotic treatment of the problem is presented in detail to obtain the solution for the surface elevation for sufficiently large time. Finally, the main prediction of this analysis for some particular pressure distributions of physical interest is exhibited.Nomenclature R, , Y cylindrical polar coordinates - frequency - g acceleration due to gravity - density of fluid - T time - (R, Y; T) velocity potential - E(R, T) vertical surface elevation - P(R, T) applied surface pressure - r, y nondimensional cylindrical polar coordinates, - p(r, t) nondimensional surface pressure - t nondimensional time, T - (r, y; t) nondimensional velocity potential, - (r, t) nondimensional vertical surface elevation, - (k) Hankel transform of a function p(r) with respect to r - I ₁ transient wave integral - I ₂ steady state wave integral     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

Pulsatile flow in a tube with wall injection and suction

This paper studies similarity solutions for pulsatile flow in a tube with wall injection and suction. The Navier-Stokes equations are reduced to a system of three ordinary differential equations. Two of the equations represent the effects of suction and injection on the steady flow while the third represents the effects of suction and injection on pulsatile flow. Since the equations for steady flow have been studied previously, the analysis centers on the third equation. This equation is solved numerically and by the method of matched asymptotic expansions. The exact numerical solutions compare well with the asymptotic solutions.The effects of suction and injection on pulsatile flow are the following: a) Small values of suction can cause a resonance-like effect for low frequency pulsatile flow. b) The annular effect still occurs but for large injection or suction the frequency at which this effect becomes dominant depends on the cross-flow Reynolds number. c) The maximum shear stress at the wall is decreased by injection, but may be increased or decreased by suction.Nomenclature a radius of the tube - a ₀ ² i ² - A₀, B₀, C₀, D₀, E₀ constant coefficients appearing in the expression for pressure - b a non-dimensionalized length - b ₀ ² i ^{2 2} - b _k complex coefficients of a power series - B - C ₁, C ₂, D complex constants - d - D _1,2 - f() F(a ^1/2)/aV - f ₀,f ₁,... functions of order one used in asymptotic expansions of f() - F(r) rv _r - g() - G(r) a steady component of velocity in axial direction - h() 4/C₀ a ² H(a ^1/2) - h ₀,h ₁,h ₂,...;l ₀,l ₁,l ₂,... functions of order one used in asymptotic expansions for h() in outer regions - H(r) complex valued function giving unsteady component of velocity - H ₀, H ₁, H ₂, ... K ₀, K ₁, K ₂, ...; L ₀, L ₁, L ₂, ... functions of order one used in asymptotic expansions for h() in inner regions - i - J ₀, J ₁, Y ₀, Y ₁ Bessel functions of first and second kind - k - K Rk/2b ² - O order symbol - p pressure - p ₁(z, t) arbitrary function related to pressure - r radial coordinate - r ₀ (1+16 ^{4 4})^1/4 - R Va/, the crossflow Reynolds number - t time - u() G(r)/V - v _r radial velocity - v _z axial velocity - V constant velocity at which fluid is injected or extracted - z axial coordinate - ² a ²/4 - 4.196 - small parameter; =–2/R (Sect. 4); =–R/2 (Sect. 5); =2/R(Sect. 6) - r ²/a ² - * 0.262 - Arctan (4 ^{2 2}) - , inner variables - kinematic viscosity - b - * zero of g() - density - (r, t) arbitrary function related to axial velocity - frequency     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Aspect ratio effect on flow-induced forces on circular cylinders in a cross-flow

Finite-span circular cylinders with two different aspect ratios, placed in a cross-flow, are investigated experimentally at a cylinder Reynolds number of 46,000. Simultaneous measurements of the flow-induced unsteady forces on the cylinders and the stream velocity in the wake are carried out. These results together with mean drag measurements along the span and available literature data are used to evaluate the flow mechanisms responsible for the induced unsteady forces and the effect of aspect ratio on these forces. The coherence of vortex shedding along the span of the cylinder is partially destroyed by the separated flow emanating from the top and by the recirculating flow behind the cylinder. As a result, the fluctuating lift decreases drastically. Based on the data collected, it is conjectured that the fluctuating recirculating flow behind the cylinder is the flow mechanism responsible for the unsteady drag and causes it to increase beyond the fluctuating lift. The fluctuating recirculating flow is a direct consequence of the unsteady separated flow. The unsteady forces vary along the span, with lift increasing and drag decreasing towards the cylinder base. When the cylinder span is large compared to the wall boundary layer thickness, a submerged two-dimensional region exists near the base. As the span decreases, the submerged two-dimensional region becomes smaller and eventually vanishes. Altogether, these results show that fluctuating drag is the dominant unsteady force in finite-span cylinders placed in a cross-flow. Its characteristic frequency is larger than that of the vortex shedding frequency.List of symbols a span of active element on cylinder, = 2.5 cm - C _D local rms drag coefficient, 2D/ U ² da - C _L local rms lift coefficient, 2l/ U ² da - C _D local mean drag coefficient, 2D/ U ² da - C _D spanwise-averaged C _D for finite-span cylinder - (C _D ) _2D spanwise-averaged mean drag coefficient for two-dimensional cylinder - C _p pressured coefficient - -(C _p ) _b pressure coefficient at = - d diameter of cylinder, = 10.2 cm - D fluctuating component of instantaneous drag - D local rms of fluctuating drag - D local mean drag - E _D power spectrum of fluctuating drag, defined as - E _L power spectra of fluctuating lift, defined as - f _D dominant frequency of drag spectrum - f _L dominant frequency of lift spectrum - f _u dominant frequency of velocity spectrum - h span of cylinder - H height of test section, = 30.5 cm - L fluctuating component of instantaneous lift - L local rms of fluctuating lift - R _Du () cross-correlation function of streamwise velocity and local drag, - R _Lu () cross-correlation function of stream wise velocity and local lift, - Re Reynolds number, U d/y - S _L Strouhal number based on f _L ,f _L d/U - S _D Strouhal number based on f _D ,f _D d/U - S _u Strouhal number based on f _u , f _u d/U - t time - u fluctuating component of instantaneous streamwise velocity - U mean streamwise velocity - mean stream velocity upstream of cylinder - x streamwise distance measured from axis of cylinder - y transverse distance measured from axis of test section - z spanwise distance measured from cylinder base - angular position on cylinder circumference measured from forward stagnation - kinematic viscosity of air - density of air - time lag in cross-correlation function - _D normalized spectrum of fluctuating drag - _L normalized spectrum of fluctuating lift     

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

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Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

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Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Flow structure in a flooded ball bearing

The flow structure in the confined space between the outer ring, the cage and the balls of a bearing is investigated using a large scale model allowing to perform visualizations, by tracer and dot-paint techniques, and velocity measurements, by Laser Doppler Velocity (LDV), through the transparent rotating outer ring. The visualization results show, in the region between two consecutive balls, the existence of a reversed flow on the cage surface resulting from the aspiration and blowing effect of the rotation of the balls in their cage housings. Systematic measurements of azimuthal velocities in different cross-sections of the gap confirmed the qualitative visualziation findings in laminar flow. For turbulent flow the results show that the extension of the reversed flow region is reduced and the reversed velocities are proportionally smaller as compared to the laminar case.List of symbols R radial position - R _b radius of the balls - R _c radius evaluated at the external surface of the cage - R _e radius evaluated at the inner wall of the outer cylinder - R _i radius evaluated at the outer wall of the inner cylinder - R _m radius of the center of the balls - Re ₀ Reynolds number in the space between the fixed inner cylinder and the rotating outer cylinder: Re ₀ = _e R _e(R _e - R _i)/v - Re ₁ Reynolds number in the space between the inner and outer cylinders: Re ₁ = 2_e R _e(R _e - R _i)/v - Re Reynolds number in the outer cylinder/cage gap: Re = _e R _e(R _e - R _c)/v - U axial velocity - V azimuthal velocity - V _e azimuthal velocity of the internal wall of the outer cylinder - V _i azimuthal velocity of the external wall of the inner cylinder - Z axial position - azimuthal position - kinematic viscosity - _i angular velocity of the inner cylinder - _e angular velocity of the outer cylinder - _c angular velocity of the balls about the axis of the bearing - _r angular velocity of the balls about their center This work was performed as part of a research effort aimed at investigating the many aspect of ball bearings flooded in cryogenic liquids and supported financially by the Centre National d'Etudes Spatiales (CNES) la Société Européenne de Propulsion (SEP) and the Centre National de la Recherche Scientifique (CNRS). The authors wish to deeply thank the many individuals, and in particular Dr. G. Jeanblanc from CNES and Mrs. Pierre and Moëllo from SEP, for their continuous encouragement.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables     

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

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Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

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a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

Rheological properties of polyethylene oxide solutions

Summary Measurements were made on solutions of Polyethylene oxide (WSR-301) varying in concentration from .0511 g/dl to 4.014 g/dl, prepared from two samples of dry material of different ages (I, II), using aWeissenberg Rheogoniometer with cone-and-plate and parallel-plate geometries, and also using capillary viscometers. Steady shear data were obtained for eight decades of strain-rates (10^–3 <k < 10⁵ sec^–1), and oscillatory data for over four decades of frequency (10^–3 <f < 10¹ Hz). Results are presented for the shear-dependent viscosity,(k), normal stress differences, ₁(k), ₂(k), and the complex viscosity, ^*(f). It was found that characteristic fluid times obtained from continuum arguments correlated the experimental(k), (f) andG(f) data.Using the ^*(f) data, the stress relaxation function,(), was calculated, from which the second-order fluid coefficients and ₀ were obtained, and compared to the directly measured values.Evidence is given to show that the sign of ₁(k) varies both with concentration and strainrate.Using solutions prepared from sample II, correlations with the material properties of solutions of sample I were found which indicated the effect of aging on stored dry samples.

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Zusammenfassung Es wurden Viskositätsmessungen an Polyäthylenoxid-(WSR-301)-Lösungen mit Konzentrationen zwischen 0,0511 g/dl und 4,014 g/dl ausgeführt, die aus zwei Proben von Trockenmaterial verschiedenen Alters (I, II) genommen waren. Verwendet wurde einWeissenberg-Rheogoniometer mit Kegel-Platte- und Parallel-Platten-Geometrie sowie verschiedene Kapillarviskosimeter. Werte für die stationäre Scherung wurden über acht Dekaden der Schergeschwindigkeit (10^–3 <k < 10⁵ s^–1) erhalten, solche für periodische Beanspruchung über mehr als vier Dekaden der Frequenz (10^–3 <f < 10 Hz). Es werden die Werte der Scherviskosität(k), der Normalspannungsdifferenzen ₁(k) und ₂(k) sowie der komplexen Viskosität ^*(f) mitgeteilt. Man findet, daß die experimentell ermittelten Werte von(k), (f) undG(f) mit Hilfe charakteristischer Zeitkonstanten, die man aus kontinuumsmechanischen Überlegungen gewinnt, korreliert werden können.Aus dem Verlauf von ^*(f) wurde die Spannungsrelaxationsfunktion() berechnet, woraus sich die Koeffizienten zweiter Ordnung und ₀ bestimmen lassen. Diese wurden mit den auf direkte Weise gewonnenen Werten verglichen. Es wird nachgewiesen, daß das Vorzeichen von ₁(k) sowohl bei der Veränderung der Konzentration als auch der Deformationsgeschwindigkeit wechselt.Durch Vergleich der an den Proben I und II erhaltenen Ergebnisse wird auf Alterungserscheinungen bei der trocken gelagerten Probe geschlossen.

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With 19 figures and 1 table     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

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     

Magnetohydrodynamic natural convection heat transfer from radiate vertical porous surfaces

Magnetohydrodynamic natural convection heat transfer from radiate vertical surfaces with fluid suction or injection is considered. The nonsimilarity parameter is found to be the conductive fluid injection or suction along the streamwise coordinate = V{4x/² g(T _w – T )}^1/4. Three dimensionless parameters had been found to describe the problem: the magnetic influence number N = B ² _y /V ², the radiation-conduction parameter R _d = k _R /4aT ³ , and the Gebhart number Ge _x = gx/cp to represent the effect of the viscous dissipation. It is found that increasing the magnetic field strength causes the velocity and the heat transfer rates inside the boundary layer to decrease. Its apparent that increasing the radiation-conduction parameter decreases the velocity and enhances the heat transfer rates. The Gebhart number, i.e, the viscous dissipation had no effect on the present problem.Nomenclature a Stefan-Boltzmann constant - B _y Magnetic field flux density Wb/m² - Cf _x Local skin friction factor - c _p Specific heat capacity - f Dimensionless stream function - Ge _x Gebhart number, gx/cp - g Gravitational acceleration - k Thermal Conductivity - L Length of the plate - N Magnetic influence number, B ² _y /V ² - p Pressure - Pr Prandtl number - q _r Radiative heat flux - q _w (x) Local surface heat flux - Q _w (x) Dimensionless Local surface heat flux - R _d Planck number (Radiation-Conduction parameter), k _R /4aT ³ - T Temperature - T Free stream temperature - T _w Wall temperature - u, v Velocity components in x- and y-directions - V Porous wall suction or injection velocity - V _w Porous wall suction or injection velocity - x, y Axial and normal coordinates - Thermal diffusivity Greek symbols _R Roseland mean absorption coefficient, 4/3R _d - Coefficient of thermal expansion - Nonsimilarity parameter, V{4x/² g(T _w – T )}^1/4 - Peseudo-similarity variable - Dimensionless temperature - _w Ratio of surface temperature to the ambient temperature, T _w /T - Dynamice viscosity - Kinemtic viscosity - Fluid density - Electrical conductivity - _w Local wall shear stress - Dimensional stream function     

On-line measurement of elasticity and viscosity in flowing polymeric liquids

A new slit-die rheometer (the Stressmeter) for on-line and sample measurement of the viscosity, , and the first normal stress difference, N ₁, in steady shear flow for molten polymers and other high-viscosity liquids is described. Two liquid-filled transverse slots, located in one die wall near the center station, give pressures P ₂ and P ₃ from whose difference the wall shear stress is calculated. In the other die wall at a location opposite the center of the P ₂ slot is a flush-mounted transducer, giving a pressure P ₁. N ₁ is calculated from the hole pressure P ^* = P ₁–P ₂. A metering pump, used to measure the flow rate Q, is supplied with melt either from an extruder (online mode) or from a pressurized sample cylinder (sample mode). The wall shear rate is calculated from Q and ; the Weissenberg-Rabinowitsch correction and a new small-viscous-heating-correction algorithm (affecting ) are used. Viscous heating corrections are small; entrance and exit errors are negligible. The instrument is tested by comparing its results with those obtained from cone-plate and capillary rheometers. Measurement ranges extend to = 200 kPa, = 3000 s^–1, and temperature = 250°C.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland