Constitutive equations applied to dispersed systems

Summary It has been investigated whether constitutive equations, which have been proposed originally to describe the rheological behaviour of polymerlike materials, can be used to represent the rheology of dispersions. Such equations generally predict stresses that depend on both the shear () and a quantity ( ) which is the product of the shear rate ( ) and the time constant of the material ().The behaviour of dispersions depends in general on the concentration of the dispersed particles. The dissipative aspect of the rheological behaviour is almostNewtonian for very dilute dispersions while it becomes plastic for more densely packet dispersions. In the latter case the shear stress is practically independent of the shear rate at low shear rates. Such behaviour may be accounted for in the constitutive equations by assuming to be almost constant. This motivated us to choose the equation ofBogue where the relaxation time () depends on the shear rate ( ), according to 1/ = (1/ ₀) + a , where 1/ ₀ accounts for the viscous behaviour and a for the plastic behaviour.Comparing the actual rheological behaviour of dispersions of fat crystals in paraffin oil with the behaviour predicted by theBogue equation, it turns out that theBogue equation has some success in representing the stress overshoot in steady shear experiments. However, the predicted value of the normal stress for the concentrated dispersions is too low in comparison with the measured value. It is suggested that this discrepancy is due to the dilatant behaviour of these dispersions.Moreover, the values of the dynamic moduli measured in oscillatory shear are predicted incorrectly, due to considerable changes in particle network which already occur at very small deformations.With 10 figures     

Der Spannungs- und Verschiebungszustand drehsymmetrischer Membrane mit beliebig gekrümmter Meridiankurve

Zusammenfassung Die Einführung von Zylinderkoordinaten (x, r, ) in die Gleichgewichtsbedingungen der Schnittkräfte bzw. in die Beziehungen zwischen Verzerrung und Verschiebungen am differentialen Schalenabschnitt ermöglicht die Berechnung des Spannungs- und Verschiebungszustandes von drehsymmetrischen Membranen mit beliebig gekrümmter Meridiankurve auf die Integration einer einfachen, linearen partiellen Differentialgleichung zweiter Ordnung für eine charakteristische FunktionF bzw. zurückzuführen. Eine geschlossene Lösung und damit eine Darstellung der Schnittkräfte und Verschiebungen durch explizite Formeln ist bei harmonischer Belastung cosn für zwei Funktionsgruppen=x ² und=x ^–3 möglich. Im Sonderfall der drehsymmetrischen und der antimetrischen Belastung mitn=0 undn=1 gelten die Gleichungen der Schnitt- und Verschiebungsgrößen für eine beliebige Meridianfunktion=(). Die Betrachtungen der Randbedingungen offener Schalen bei harmonischer Belastung geben über die infinitesimalen Deformationen einer drehsymmetrischen Membran mit überall negativer Krümmung Aufschluß.     

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

Wärmeübergang in Rohren bei aufgeprägter,periodischer Kompression/Expansion

Zusammenfassung Der Aufsatz behandelt den Wärmeübergang in einem kompressiblen Fluid, welchem eine periodische Kompression/Expansion aufgeprägt wird. Der interne instationäre Energietransport wird zunächst durch eine partielle Differentialgleichung beschrieben. Die numerische Lösung dieser Gleichung unter Variation der Prozeßparameter dient als Grundlage zur Anpassung eines Wärmeübergangskoeffizienten. Um die periodischen Verläufe des Wärmestroms und der Fluidtemperatur wiedergeben zu können, wird dabei das Konzept der komplexen Nusselt-Zahl angewendet. Es ergeben sich schließlich Korrelationen für die komplexe Nusselt-Zahl sowohl für laminare als auch für turbulente Strömungsverhältnisse.

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Heat transfer in pipes under periodical compression and expansion

The paper deals with the heat transfer in a compressible fluid, which is periodically compressed and expanded. First the internal instationary energy transport is described by a partial differential equation. The numerical solution of this equation with varying process parameters is the basis for fitting a heat transfer coefficient. Here the concept of the complex Nusselt-number is used in order to reproduce the periodical changes of the heat flux and the fluid temperature. The investigations finally lead to correlations for the complex Nusselt-number for laminar and also turbulent flow conditions.

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Formelzeichen a Temperaturleitfähigkeit - c _p mittlere spez. isobare Wärmekapazität - c _v mittlere spez. isochore Wärmekapazität - d Durchmesser - q Wärmestromdichte - Wärmestrom - r radiale Ortskoordinate - R Radius - t Zeit - T Temperatur - u axiale Strömungsgeschwindigkeit - v radiale Strömungsgeschwindigkeit - V Volumen - zeitlich gemitteltes Volumen - x axiale Ortskoordinate Griechische buchstaben Wärmeübergangskoeffizient - Isotropenexponent - Wärmeleitfähigkeit - kinematische Viskosität - Dichte - Phasenverschiebung - Winkelgeschwindigkeit Indizes c komplex - eff effektiv - im Imaginäranteil - ink inkompressibel - m über den Querschnitt gemittelt - re Realanteil - t turbulent - W Wand Dimensionslose Kennzahlen Nu= · d/ Nusselt-Zahl - Pe = · d ² /a kinematische Péclet-Zahl - Pr=/a Prandtl-Zahl - Q*=( ) dimensionsloser Wärmestrom - Re=u _m · d/v Reynolds-Zahl     

Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

The dynamics of an analytic reversible vector field (X,) is studied in with one real parameter close to 0; X=0 is a fixed point. The differential D_x (0,0) generates an oscillatory dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a slow dynamics which changes from a hyperbolic type—eigenvalues are —to an elliptic type—eigenvalues are —as passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.     

Homologies of Intersections of Pseudomanifolds with Local Homotopic Conditions on Links of Strata

In terms of local homotopic properties of the links of strata of an n-dimensional PL-pseudomanifold X, we obtain a sufficient condition for the natural homomorphisms of the jth intersection homology groups with perversity multiindices and to be isomorphisms for all j i, where i < n – 3.     

Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients

The paper concerns the asymptotic behavior as of solutions u(t, x) of the Cauchy problem for the inhomogeneous equation , x>0, with a mass-preserving boundary condition at x=0. It is assumed that F C ¹[0, 1] F(0)=F(1)=0, and that one of the following conditions holds: (i) F has just one zero in (0, 1) and the derivative F is negative at 0 and 1, (ii) F is zero in (0, ) and positive in (, 1) with 0<<1, (iii) F is positive in (0, 1) and c ₀ ² >2F(0) (the pushed case). Here c ₀ is the minimum of possible speeds of travelling waves for the equation with k(x) replaced by zero. Under a natural restriction imposed on k(x) we prove, for a wide class of initial functions, that u(t, x) approaches w(x + m(t)) as uniformly in x>0, where w is a travelling wave with the speed c ₀ and m is a solution of dm/dt = c ₀-k(m). The result has immediate applications to the multidimensional non-linear diffusion equations arising in population genetics, combustion theory, etc.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

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On a Boundary-Value Problem of Periodic Type for First-Order Linear Functional Differential Equations

We establish unimprovable sufficient conditions for the unique solvability of the boundary-value problem

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Nonlinear rheology of a concentrated spherical silica suspension:

Time-dependent nonlinear flow behavior was investigated for a model hard-sphere suspension, a 50 wt% suspension of spherical silica particles (radius = 40 nm; effective volume fraction = 0.53) in a 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The suspension had two stress components, the Brownian stress ^B and the hydrodynamic stress ^H After start-up of flow at various shear rates , the viscosity growth function ₊ (t, ) was measured with time t until it reached the steady state. The viscosity decay function _– (t, ) was measured after cessation of flow from the steady as well as transient states. At low where the steady state viscosity ( ) exhibited the shear-thinning, the _– (t, ) and ⁺ (t, ) data were quantitatively described with a BKZ constitutive equation utilizing data for nonlinear relaxation moduli G (t, ). This result enabled us to attribute the thinning behavior to the decrease of the Brownian contribution ^B = ^B / (considered in the BKZ equation through damping of G (t, )). On the other hand, at high where ( ) exhibited the thickening, the BKZ prediction largely deviated from the ₊ (t, ) and ₊ (t, ) data, the latter obtained after cessation of steady flow. This result suggested that the thickening was due to an enhancement of the hydrodynamic contribution ^H = ^H / (not considered in the BKZ equation). However, when the flow was stopped at the transient state and only a small strain (<0.2) was applied, ^H was hardly enhanced and the _– (t, ) data agreed with the BKZ prediction. Correspondingly, the onset of thickening of ₊ (t, ) was characterized with a -insensitive strain ( 0.2). On the basis of these results, the enhancement of ^H (thickening mechanism) was related to dynamic clustering of the particles that took place only when the strain applied through the fast flow was larger than a characteristic strain necessary for close approach/collision of the particles.     

Comparison of one-dimensional models and a 1-D/2-D model for closed-loop thermosyphons with vertical heat transfer sections

The results of a comparison between traditional one-dimensional (1-D) models and a 1-D/2-D model for closed-loop thermosyphons with vertical heat transfer sections are reported in this paper. Attention is limited to problems in which the flow is laminar. For cases where heat losses from the insulated portions of the loop are negligible,St _m=o, it is shown that traditional 1-D models can significantly overpredict the average fluid velocity in the loop for high power inputs (highGr _m). Local results of two-dimensional numerical simulations in the heated and cooled sections reveal that this discrepancy arises because the 1-D models do not account for mixed-convection effects which distort the velocity and temperature profiles from their fully developed forced convection shapes. Furthermore, for cases whereSt _m o, predictions of heat losses (or gains) produced by the 1-D models are handicapped by inaccuracies in the corresponding temperature predictions inside the loop.

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Vergleich eindimensionaler Modelle mit einem 1-D/2-D Modell zur Berechnung geschlossener Thermosiphonkreisläufe mit vertikalen Wärmeübergangsabschnitten

Zusammenfassung In dieser Arbeit werden die Ergebnisse eines Vergleichs zwischen traditionellen eindimensionalen (1-D) Modellen und einem 1-D/2-D Modell zur Berechnung geschlossener Thermosiphonkreisläufe mit vertikalen Wärmeübergangsabschnitten mitgeteilt. Die Untersuchung beschränkt sich auf reine Laminarströmungen. In Fällen, wo die Wärmeverluste an den isolierten Abschnitten des Kreislaufs vernachlässigt werden können (St _m=o), zeigt sich, daß traditionelle 1-D Modelle die mittlere Strömungsgeschwindigkeit im Kreislauf bei hohem Leistungseintrag (Gr _m hoch) signifikant überbewerten. Örtliche Ergebnisse zweidimensionaler Simultationsrechnungen für die beheizten bzw. gekühlten Abschnitte zeigen, daß diese Widersprüche im Unvermögen der 1-D Modelle begründet liegen, Mischkonvektionseffekte berücksichtigen zu können. Diese deformieren die reiner Zwangskonvektion in ädaquate Form der Geschwindigkeits- und Temperaturfelder. Ferner leidet im FalleSt _m o die Berechnung von Wärmeverlusten oder -gewinnen nach den 1-D Modellen unter der ungenauen Kenntnis der Temperaturverteilung im Inneren des Kreislaufs.

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Nomenclature f Fanningfriction factor [= _w/(V ²)/2] - g acceleration due to gravity - Gr _q Grashof number - Gr _m modified Grashof number - L total length of the closedloop - P _w power input - Pr Prandtl number [=C _p/k] - q heat flux - R radius of curvature of the 180° bends, Fig. 1 - Re Reynolds number - Re _ref reference Reynolds number - r _i internal radius of the pipe [=D/2] - St _m modified Stanton number - T area-weighted mean cross-sectional temperature - T _a ambient temperature - T _b bulk temperature - T _w mean wall temperature in the cooled section - U overall heat loss coefficient (W/m–°C) - V average velocity in the loop - V nondimensional average velocity [=V/V _ref] - V _ref reference velocity - X parameter used in Eq. (3) - volumetric thermal expansion - z height difference between the middle of the heated and cooled sections, Fig. 1 - length parameter - dimensionless ambient temperature - modified heat loss coefficient This research was financially supported by the Natural Sciences and Engineering Research Council of Canada, in the form of a Post-Graduate Scholarship granted to M. A. Bernier and through individual operating grants awarded to Prof. B. R. Baliga.     

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     

Messung des ersten Normalspannungskoeffizienten von Kunststoffschmelzen im Bogenspalt

Zusammenfassung Bei einer stationären Schichtenströmung in einem Bogenspalt (azimutale Druckströmung im Ringspalt) bildet sich zwischen Innen- und Außenwand eine Druckdifferenz aus, deren Größe ein Maß für den 1. Normalspannungskoeffizienten der elastischen Flüssigkeit im Spalt ist. Die Strömung läßt sich zur Messung des 1. Normalspannungskoeffizienten verwenden. Der Schergeschwindigkeitsbereich der Messung liegt, wie bei der Kapillarrheometrie zur Bestimmung der Viskosität, zwischen 1 und 1000 s^–1. Die Auswertung der Messungen ist wegen des inhomogenen Scherfeldes relativ kompliziert. In der Arbeit wird ein besonders wirkungsvolles numerisches Auswerteverfahren hergeleitet und auf bestehende Messungen angewendet. Eine Besonderheit des Auswerteverfahrens ist die Freiheit der Wahl des Approximationsansatzes für die Viskositätskurve, während analytische Verfahren meist an einen bestimmten Ansatz gebunden sind. Außerdem braucht, im Gegensatz zu anderen derartigen Verfahren, die Position des schubspannungsfreien Stromfadensr ₀ nicht bestimmt zu werden.

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Summary The stress in steady viscometric flow of molten polymers is determined by the viscosity and by the two normal stress coefficients ₁ and ₂. The paper describes a method of measuring ₁ by means of steady circumferential shear flow in an annulus. The cylinders are stationary and the fluid flows due to a circumferential pressure gradient. The radial normal stresses at the outer and at the inner wall are different from each other. The pressure-differencep is a measure for the 1. normal stress coefficient of the viscoelastic fluid. Due to the inhomogeneous shear field, the evaluation of ₁ fromp measurements is quite complicated. A powerful numerical method of evaluation has been developed and applied to existing data. The method is not restricted to a special empirical formula for the flow curve (as an analytical method would be) and does not require the knowledge of the positionr ₀ of the stress-free stream line.

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a Pa s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten, s. Gl. [8] - AR _i — Koeffizient des Druckgefälles in-Richtung (Programm PFEIL) - AU _i — Koeffizient für Integration nach Simpson-Regel (Programm PFEIL) - b s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten - B _i — Koeffizient auf der rechten Seite des linearen Gleichungssystems (Programm PFEIL) - c — Exponent des Ansatzes des 1. Normalspannungskoeffizienten - CL _i CM _i CR _i — Koeffizienten der dimensionslosen Geschwindigkeit in dem linearen Gleichungssystem (Programm PFEIL) - F ₁,F ₂,F ₃ — Ableitungen der Summe der Fehlerquadrate nacha, b undc - G _k — Gewichtsfaktor - h m Spaltweite,r _a –r _i - H — dimensionslose Spaltweite, (r _a –r _i )/r _a - l m Länge des Bogenspaltes, 0,75(r _a +r _i ) - m — Exponent des Potenzansatzes, s. Gl. [13] - n — Dämpfungskonstante - N ₁ Pa 1. Normalspannungsdifferenz, _rr – - N ₂ Pa 2. Normalspannungsdifferenz - p Pa Druck - p Pa Druckgradient in-Richtung - P — dimensionsloser Druckgradient in-Richtung, s. Gl. [14] - p, p _k Pa Normalspannungsdifferenz zwischen Innen- und Außenwand im Bogenspalt, (– p + _rr ) _a – (–p + _rr ) _i - Q — Summe der Fehlerquadrate - r, R= r/r _a m, — Radiusvektor (Koordinate in Gradientenrichtung) - r ₀,R ₀=r ₀/r _a m, — Radius des neutralen Fadens - R — dimensionslose radiale Schrittweite - T, °C Temperatur bzw. Bezugstemperatur - v ms^–1 Geschwindigkeitskomponente in-Richtung - V ,V _,i — dimensionslose Geschwindigkeitskomponente in-Richtung - V _a ,V _k — dimensionslose Geschwindigkeit an der Außen- bzw. Innenwand - v _r ,v _z ms^–1 Geschwindigkeitskomponenten inr-undz-Richtung - ms ^–1 mittlere Geschwindigkeit in-Richtung - z m Koordinate in der indifferenten Richtung - K^–1 Temperaturkoeffizient der Viskosität - s^–1 Schergeschwindigkeit - s^–1 kritische Schergeschwindigkeit der Viskositätskurve, s. Gl. [13] - s^–1 Bezugsschergeschwindigkeit, - — dimensionslose Schergeschwindigkeit - — dimensionslose kritische Schergeschwindigkeit, - Pa s Viskosität - ₀ Pa s Nullviskosität - Pa s Bezugsviskosität, - — Radienverhältnis,r _i /r _a - ₁ Pa s ² 1. Normalspannungskoeffizient - Pa s² mittlerer 1. Normalspannungskoeffizient - ₂ Pa s² 2. Normalspannungskoeffizient - — Koordinate in Strömungsrichtung - Pa Spannung - a an der Außenwand - i, an der Innenwand - i laufender Index inr-Richtung - k Nummer des Meßpunktes - n Anzahl der Meßpunkte - n _i nord für Programm PFEIL - s _i süd für Programm PFEIL Mit 9 Abbildungen und 2 Tabellen     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

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.     

On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

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.     

Multiaxial elongation of polyisobutylene with various and changing strain rate ratios

The multiaxial elongational rheometer equipped with rotary clamps is modified such that in addition to simple, equibiaxial and multiaxial elongations also tests with new modes of elongation can be performed. As an example, polyisobutylene is elongated with a ratio of the principal strain rates of and magnitudes of the maximum strain rate , 0.04 and 0.08 s^–1. As a test result, the first elongational viscosityµ ₁ (t) is obtained which follows closely the linear viscoelastic shear viscosity . In contrast, the second elongational viscosityµ ₂ (t) remains below . By means of a further modification of the rheometer, the test modes can be varied during the deformation period. This allows one to investigate the influence of a well-defined rheological pre-history on the following rheological behaviour. As an example a variation ofm = 0.5 2 was performed. The measured normal-stress differences superpose from the single steps of deformation similar to the linear viscoelastic prediction.Dedicated to Prof. F. R. Schwarzl on the occasion of his 60th birthday