Dehnungsverhalten von Polyäthylen-Schmelzen

Zusammenfassung In einem Dehnungsrheometer werden Spannungs-Dehnungs-Diagramme von Polyäthylen-Schmelzen bei 150 °C und bei konstanter Dehnungsgeschwindigkeit gemessen ( zwischen 0,001 und 1 sec^–1). Weiterhin wird der reversible (elastische) Dehnungsanteil bestimmt. Messungen mit einem Dehnungstester für Kunststoff-Schmelzen ergänzen die Ausführungen.Die Ergebnisse zeigen deutlich, daß bei Dehnung mit zunehmender Verformungsgeschwindigkeit die Dehnungsviskosität nicht abnimmt, im Gegensatz zu dem bekannten strukturviskosen Verhalten bei Scherung.Bei Dehnungen bis zu=1 kann das Verhalten unabhängig von beschrieben werden, wenn als viskoelastische Materialfunktion die Dehnungs-Spannviskosität betrachtet wird. In diesem Bereich von gilt dabei die BeziehungT(t)=3 _s (t) mit _s (t) als zeitabhängige Scherviskosität im linear-viskoelastischen Bereich.Bei größeren Dehnungen und nicht zu kleinen Dehnungsgeschwindigkeiten zeigt verzweigtes Polyäthylen eine zusätzliche starke Spannungszunahme. In dem Bereich dieser zusätzlichen Verfestigung ist das Verhalten im wesentlichen eine Funktion der Dehnung und fast unabhängig von . Die zusätzliche Verfestigung scheint eine Folge der Verzweigungsstruktur des verzweigten Polyäthylens zu sein, da bei Linear-PE ein derartiger Verlauf des Spannungs-Dehnungs-Diagramms nicht beobachtet wird.Die Betrachtung des reversiblen Dehnungsanteils _R zeigt bei der ausführlich untersuchten Schmelze I (verzweigtes PE) drei verschiedene Bereiche: Unterhalb einer Grenzdehnungsgeschwindigkeit ist _R =0, unterhalb einer Versuchszeitt ^** ist _R =. Im dazwischenliegenden Bereich treten elastische und viskose Dehnungsanteile auf,= _R + _V , wobei für niedrige gilt, daß _R lg . Die Grenze wird der Frequenz der thermisch aktivierten Platzwechsel zugeordnet,t ^** erscheint als Zeit, innerhalb der die Verhakungen wie fixierte Vernetzungen wirken.In dem Anhang wird der Einfluß der Grenzflächenspannung zwischen PE-Schmelze und Silikonöl auf die Ergebnisse der Dehnungsversuche diskutiert.

---

Summary Stress-strain relations for different PE melts are measured at 150 °C in an extensional rheometer under the condition of a constant extensional strain rate ( between 0,001 and 1 sec^–1). Further, the recoverable (elastic) portion _R of the total strain ( in Hencky's measure) is determined and additional measurements with a tensile tester for polymer melts are described.The results show clearly that in extension there is no decrease of the tensile viscosity with increasing deformation rate, in contrast to the well-known pseudoplastic behaviour in shear. Within total strains<1 the tensile behaviour can be described independently from by means of a viscoelastic material function called stressing viscosity . In this range of the relation _T (t)=3 _s (t) holds, where _s (t) is the stressing viscosity in shear in the linear viscoelastic range. For larger tensile strains and not too small branched PB shows a remarkable increase in stress. This work-hardening behaviour is mainly a function of and almost independent from . This additional hardening seems to be due to the branches in branched PE, because linear PE does not show this effect.The discussion of the recoverable tensile strain _R gives three regions of tensile rate: Below a critical there is _R =0. At times shorter thant ^** the equation _R = is valid. Within these limits both elastic and viscous portions of the total strain= _R + _V exist. may correlate with the frequency of the thermally activated position changes of the molecular segments.t ^** is assumed to be the time for the entanglements to act as fixed cross-links.In the appendix the influence of the interface tension between PE melt and silicone oil on the results of the tensile experiments is discussed.

---

---

Vorgetragen auf der Deutschen Rheologen-Tagung, Berlin, 11.-13. Mai 1970.

---

An der Weiterentwicklung des Dehnungsrheometers, an der Durchführung und Auswertung der Messungen waren die HerrenB. Kienle, F. Landmesser, M, Reuther undF. Scherr beteiligt. Herr Dr.F.Ramsteiner und HerrH. Schroeck haben sich um die Herstellung der Stränge aus Linear-PE bemüht. Herr Dr.W. Ball besorgte die GPC-Messungen und Herr Dr.P. Simak die Ultrarot-Untersuchung. Den vorgenannten Herren sei für ihre Hilfe beim Zustandekommen dieser Arbeit gedankt. Herrn Dr.H. Baur danke ich für wertvolle Diskussionen.     

Buoyancy- and pressure-driven motion in a vertical porous layer: Effects of quadratic drag

The seepage velocity arising from pressure and buoyancy driving forces in a slender vertical layer of fluid-saturated porous media is considered. Quadratic drag (Forcheimer effects) and Brinkman viscous forces are included in the analysis. Parameters are identified which characterize the influence of matrix permeability, quadratic drag and buoyancy. An explicit solution is obtained for pressure-driven flow which illustrates the influence of quadratic drag and the strong boundary layer behavior expected for low permeability media. The experimental data of Givler and Altobelli [2] for water seepage through a high porosity foam is found to yield good agreement with the present analysis. For the case of buoyancy-driven flow, a uniformly valid approximate solution is found for low permeability media. Comparison with the pressure-driven case shows strong similarities in the near-wall region.Nomenclature B function of - d layer thickness - D discriminant defined by Equation (9) - modified Darcy number - F Forcheimer constant - g gravitational acceleration - k porous matrix permeability - m parameter defined by Equation (11) - p pressure - p modified pressure - pressure gradient - R buoyancy parameter - T ₀ nominal layer temperature - u seepage velocity - dimensionless seepage velocity - _c composite approximation - _i boundary layer velocity - _o outer or core flow approximation - _m midplane velocity - U matching velocity - V cross-sectional average velocity - w variable defined by Equation (12) - x, z Cartesian coordinates - , dimensionless Cartesian coordinates - inertia parameter - T layer temperature difference - larger root of cubic given by Equation (8) - fluid dynamic viscosity - _e effective viscosity of fluid saturated medium - variable defined by Equation (18) - 0 fluid density - smaller root of cubic given by Equation (8) - variable defined by Equation (18) - stretched inner coordinate - porosity - function of      

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

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.

---

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.

---

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     

Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment

In this paper we examine the closure problem associated with the volume averaged form of the Stokes equations presented in Part II. For both ordered and disordered porous media, we make use of a spatially periodic model of a porous medium. Under these circumstances the closure problem, in terms of theclosure variables, is independent of the weighting functions used in the spatial smoothing process. Comparison between theory and experiment suggests that the geometrical characteristics of the unit cell dominate the calculated value of the Darcy's law permeability tensor, whereas the periodic conditions required for thelocal form of the closure problem play only a minor role.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface associated with the local closure problem, m² - A _p surface area of a particle, m² - b vector used to represent the pressure deviation, m^–1 - B ⁰ B+I, a second order tensor that maps v _m ontov - B second-order tensor used to represent the velocity deviation - d _p 6V _p/A_p, effective particle diameter, m - d a vector related to the pressure, m - D a second-order tensor related to the velocity, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor calculated on the basis of a spatially periodic model, m² - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p characteristic length for the volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - characteristic length (pore scale) for the-phase - _i i=1, 2, 3 lattice vectors, m - weighting function - m(-y) , convolution product weighting function - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - n unit normal vector pointing from the-phase toward the -phase - p pressure in the-phase, N/m² - p _m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p – p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function - r position vector, m - r position vector locating points in the-phase, m. - V averaging volume, m³ - B volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v _m superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - v traditional superficial volume averaged velocity, m/s - v – v _m , spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the -phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * , weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

On the prediction of riblet performance with engineering turbulence models

The paper reports the outcome of a numerical study of fully developed flow through a plane channel composed of ribleted surfaces adopting a two-equation turbulence model to describe turbulent mixing. Three families of riblets have been examined: idealized blade-type, V-groove and a novel U-form that, according to computations, achieves a superior performance to that of the commercial V-groove configuration. The maximum drag reduction attained for any particular geometry is broadly in accord with experiment though this optimum occurs for considerably larger riblet heights than measurements indicate. Further explorations bring out a substantial sensitivity in the level of drag reduction to the channel Reynolds number below values of 15 000 as well as to the thickness of the blade riblet. The latter is in accord with the trends of very recent, independent experimental studies.Possible shortcomings in the model of turbulence are discussed particularly with reference to the absence of any turbulence-driven secondary motions when an isotropic turbulent viscosity is adopted. For illustration, results are obtained for the case where a stress transport turbulence model is adopted above the riblet crests, an elaboration that leads to the formation of a plausible secondary motion sweeping high momentum fluid towards the wall close to the riblet and thereby raising momentum transport.Nomenclature c _f Skin friction coefficient - c _f Skin friction coefficient in smooth channel at the same Reynolds number - k Turbulent kinetic energy - K ⁺ k/ _w - h Riblet height - S Riblet width - H Half height of channel - Re Reynolds number = volume flow/unit width/ - Modified turbulent Reynolds number - R _t turbulent Reynolds numberk ²/ - P _k Shear production rate ofk, _t (U _i /x _j + U _j /x _i ) U _i /x _j - dP/dz Streamwise static pressure gradient - U _i Mean velocity vector (tensor notation) - U Friction velocity, _w/ where _w=–H dP/dz - W Mean velocity - W _b Bulk mean velocity through channel - y ⁺ yU /v. Unless otherwise stated, origin is at wall on trough plane of symmetry - Kinematic viscosity - _t Turbulent kinematic viscosity - Turbulence energy dissipation rate - Modified dissipation rate – 2(k ^1/2/x _j )² - Density - _k , Effective turbulent Prandtl numbers for diffusion ofk and      

An experimental investigation of separating flow on a convex surface

The mean and turbulent characteristics of an incompressible turbulent boundary layer developing on a convex surface under the influence of an adverse pressure gradient are presented in this paper.The turbulence quantities measured include all the components of Reynolds stresses, auto-correlation functions and power spectra of the three components of turbulence. The results indicate the comparative influence of the convex curvature and adverse pressure gradient which are simultaneously acting on the flow. The investigation provides extensive experimental information which is much needed for a better understanding of turbulent shear flows.Nomenclature a, b constants in equation for velocity defect profile (Fig. 6) - c _f skin-friction coefficient (= _w/F _1/2 U ₁ ² ) - E(k ₁) one-dimensional wave number spectra - f frequency in Hz - G Clauser's equilibrium parameter = (H–1)/H(c _f /2) - H shape parameter (= ₁/ ₂) - k ₁ wave number (=2f/U) - L _u, L _v, L _w length scales of u, v and w fluctuations - p _s static pressure on the measurement surface - p _w reference tunnel wall static pressure - q ² total turbulent kinetic energy - R radius of curvature of the convex surface - R() auto-correlation function - T _u, T _v, T _w time scales of u, v and w fluctuations - U local mean velocity - U ₁ local free stream velocity - U _* friction velocity - u, v, w velocity fluctuations in x, y and z directions respectively - X streamwise coordinate measured along the surface from A (Fig. 1b) - x streamwise coordinate measured along the surface reckoned from station 9 - y coordinate normal to the surface - z spanwise coordinate - ₁/ _w · dp/dx - - boundary layer thickness - ₁ displacement thickness - ₂ momentum thickness - ₃ energy thickness - kinematic viscosity - density - time delay - _w wall shear stress     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Rotationsviskosimetrie mit kugelförmigen Spindeln

Zusammenfassung Die Rheodynamik der stationären viskometrischen Drehströmung um eine rotierende Kugel wird mit Methoden der Variationsrechnung untersucht. Neben iterativen numerischen Lösungsmethoden, die zu exakten Resultaten führen, wird auch eine approximative Ein-Gradienten-Lösung konstruiert, die durch Quadraturen dargestellt wird. Ausgehend von dieser analytischen Approximation werden einfache Methoden zur Auswertung von Experimentaldaten vorgeschlagen, die mit Hilfe von Eintauch-Rotationsviskosimetern mit kugelförmigen Meßspindeln gewonnen wurden.

---

Summary The rotational viscometric flow around a rotating sphere has been studied by variational methods. The exact numerical, as well as an approximate analytical solutions are given. Employing the analytical approximation, a simple method of evaluating viscometric data from immersional (portable) viscometers with a rotating sphere is proposed.

---

A Achsenschnitt durch den Bereich der Strömung - B - b, c anpaßbare empirische Konstanten - C Kalibrierungsoperator - D Schergeschwindigkeit der viskosimetrischen Strömung - D _ij Komponenten des Deformationsgeschwindigkeitstensors - D _I, _I Stoffkonstanten der VF des Ellis-Modells - g metrischer Koeffizient - H() Funktional der Ein-Gradienten-Approximation, Gl. [27] - J[] energetisches Potential - J _a[] Ein-Gradienten-Approximation fürJ - K Konsistenzkoeffizient, Parameter der VF des Potenzmodells - m Parameter des Ellis-Modells - M Drehmoment - n Parameter des Potenzmodells - n, n Differentialindices der VF, Gl. [20c, d] - n*,n** Differentialindices der RC, Gl. [9], [13] - r, , z polare Zylinderkoordinaten - R Spindelhalbmesser - rheometrischer Operator - S Spindeloberfläche - U(D) energetische Funktion nachBird, Gl. [20e] - v _i physikalische Komponenten der Geschwindigkeit - Z() transformierte VF, Gl. [20f] - (n) durch Gl. [35] definierte Funktion - k Verhältnis der Radien von Spindel und Wand - ( durch Gl. [43] definierte Funktion - natürliche (Radial-)Koordinate - Schubspannung der viskosimetrischen Strömung - _ij Komponenten des Spannungstensors - _S() Spannungsprofil an der Spindeloberfläche - _M Maximalspannung an der Spindeloberfläche - mittlere Spannung an der Spindeloberfläche, Gln. [3], [22] - natürliche (Meridional-) Koordinate - Winkelgeschwindigkeit in der Flüssigkeit - Winkelgeschwindigkeit der Spindelrotation - ( rheometrische Charakteristik Mit 4 Abbildungen und 3 Tabellen     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

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     

The investigation on the properties and structures of starting vortex flow past a backward-facing step by WBIV technique

An experimental investigation of a starting vortex flow around a backward-facing step was conducted in a water channel. The properties and structures of the flow were investigated by qualitative flow visualization using the hydrogen bubble method and by quantitative velocity and vorticity measurements using White-light Bubble Image Velocimetry (WBIV) — a newly developed PIV method. Some invariant properties and 4-stage structures of starting vortex flow were observed.List of symbols a flow acceleration during starting stage - h height of backward-facing step - d _v dimensionless vortex size - t time - t dimensionless time - U free uniform velocity - u, v streamwise and spanwise velocity components respectively - Re Reynolds number based on a and h - x, y streamwise and spanwise coordinates respectively in flow field - x _c , y _c dimensionless vortex center position - vorticity - ov dimensionless vorticity - _max maximum vorticity - ov _max dimensionless maximum vorticity - circulation - dimensionless circulation - kinematic viscosity This work was supported by the CNSF Grant 1939 100-1-3     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Eine Methode der thermischen Berechnung von Gegen- und Gleichstrom-Spiralwärmeaustauschern

Zusammenfassung Eine neue Methode zur thermischen Berechnung wird entwickelt, die die Berücksichtigung der tatsächlichen Stromführung erlaubt. Es wird eine spezielle Spirale als Erzeugende der zylindrischen Oberfläche im geometrischen Modell eines Spiralwärmeaustauschers (SWA) betrachtet. Anstelle einer archimedischen Spirale wird die Erzeugende durch eine Spirale dargestellt, die auf der Basis der Evolvente eines gleichseitigen regelmäßigen Vielecks entsteht. Das Ergebnis ist, daß der SWA hypothetisch in viele Sektionen aufgeteilt ist. Die Folge dieser Änderung sind konstante Koeffizienten in einem linearen gewöhnlichen Differentialgleichungssystem. Die Austrittstemperaturen werden aus diesem System mit Hilfe der Laplace-Transformierten für jede Sektion getrennt bestimmt und durch Randbedingungen miteinander verknüpft.

---

Method of thermal calculation for rating countercurrent and cocurrent spiral heat exchangers

A new method of thermal calculation is developed which takes the actual flow arrangement into account. A special spiral is considered as a trace of the cylindrical surface in the geometrical model of a Spiral Heat Exchanger (SHE). Instead of an Archimedic spiral, the trace is formed on the other spiral, which is drawn by involution of a regular manysided figure. As a result SHE is hypothetically divided into many sections. The consequences of this change are constant coefficients in a linear ordinary differential equation system. The sought temperatures are determined from this system by using Laplace's transform for each section separately and coupled by boundary conditions.

---

Bezeichnungen b, b Kanalbreite (Kanalhöhe),b gemessen entlang dem Radius, der vom Anfang der Spirale ausgeht, undb gemessen senkrecht zu einer der beiden den Kanal begrenzenden Kurven - C _ijl Konstante Gl. (13) - E Einheitsmatrix - F(), F _i () verallgemeinerte Temperaturfunktion (Vektor und seine Komponenten) - Laplace-Transformierte vonF - h ₀ Höhe des SWA - k Wärmedurchgangskoeffizient - L Operator der Laplace-Transformation - m ₀ Anzahl der Seiten des regelmäßigen Vielecks - M Matrix im Gleichungssystem (8), Gl. (9) - n Anzahl der Kanäle - r, =r/b Radius, reduzierter Radius, reduzierter Krümmungs-radius, Koeffizienten im Gleichungssystem (6), (7) - r _m Radiusvektor der Kanäle im Sektorm mit den Komponenten ₁= ₁/b, ₂=r ₂/b ₃=r ₃/b, ..., _m =r _m /b - R = _I / _II Quotient der WärmekapazitätenW - s Bildvariable der Laplace-Transformation - t Fluidtemperatur - dimensionslose Temperatur des Fluids I - , Parameter im MatrixM (), Gl. (9) - dimensionslose Temperatur des Fluids II - Radius der archimedischen Spirale - Winkel - , geometrische Koordinaten - dimensionslose Temperaturänderungen (Betriebscharakteristik des Wärmeaustauschers) - Anzahl der Ubertragungseinheiten - = verallgemeinerter Winkel - )I betr. Fluid I - )II betr. Fluid II - ein betr. Eintritt des Fluids Verzeichnis der Abkürzungen SWA Spiralwärmeaustauscher - S-A archimedische Spirale - S-ERV Spirale aus der Evolvente eines regelmäßigen Vielecks     

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     

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.

---

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.

---

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