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     

Thermal effects in the capillary rheometer

Summary The effect of viscous heating in a capillary rheometer is analysed for a power-law fluid by means of a perturbation expansion based upon a boundary-layer-core structure. This expansion is found to complement the eigenfunction series solution obtained by earlier investigators. A similar analysis is presented for the work-of-expansion effect. These two thermal effects are superimposed together with a third perturbation effect due to the pressure dependence of viscosity.On the basis of the present theory, earlier work in this area is discussed and, in some cases, apparent inaccuracies or inconsistencies are pointed out. A means is indicated for correcting data on the basis of the present theory.

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Zusammenfassung Es wird der Effekt der Erwärmung einer Potenzflüssigkeit infolge viskoser Reibung in einem Kapillar-Rheometer mittels einer Störungsrechnung untersucht, die auf der Unterteilung der Strömung in eine Grenzschicht und einen Kern basiert. Diese Störungsentwicklung ergänzt eine früher von anderen Autoren gefundene Reihenentwicklung mit Hilfe von Eigenfunktionen. Eine ähnliche Untersuchung wird für die thermische Ausdehnungsarbeit durchgeführt. Diese beiden thermischen Effekte sind zusammen einem dritten Störeffekt superponiert, der von der Druckabhängigkeit der Viskosität herrührt.Aufgrund der vorgelegten Theorie werden verschiedene auf diesem Gebiet früher durchgeführte Arbeiten diskutiert, und es werden in einigen Fällen offensichtliche Ungenauigkeiten und Folgewidrigkeiten aufgedeckt. Schließlich wird eine Methode zur Korrektur von Meßdaten mit Hilfe der vorliegenden Theorie angegeben.

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Nomenclature a tube radius - b ; evaluated atT ₀ andp = 0 when used in perturbation expansion - C _p specific heat - f - f ^* - h defined by eq. [15] - k thermal conductivity - L tube length - m defined by eq. [8] - m ₀ m(T₀, 0) - n power-law index - p pressure - Pe C _p W a/k Peclet number - Pr C _pa/k Prandtl number - Q volumetric flow rate - Q ₀ unperturbed value ofQ in specified-p formulation - r radial coordinate - Re W a/ _a Reynolds number - T temperature - T ₀ inlet temperature - u radial velocity component - u ₀ 0 unperturbed radial velocity - w axial velocity component - w ₀ /W(1 – ) unperturbed axial velocity - W Q/(a ²) average axial velocity - W ₀ Q ₀/(a ²) - z axial coordinate - (3n + 1)/n - ^* ; evaluated atT ₀ andp = 0 when used in perturbation expansion - 4^1-n - ^* - (n + 1)/n - ... shear rate - 4W/a apparent shear rate - p total pressure drop - T _a W ²/k characteristic temperature difference - T _b total bulk-temperature rise - ^* T - r/a - shear viscosity - _a m₀ - (1 –)/ ^1/3 - —p/z - ₀ ... unperturbed value of - z-averaged value of - µ ^{n + 1}/n - z/(a Pe) - _L L/(a Pe) - mass density - _w shear stress at wall - streamfunction - ^*T₀ (absolute temperature scale) - ( )₁ leading-order effect due to viscous heating - ( ) ₁ ^* leading-order effect due to work-of-expansion Note: in specified-p formulation,W gets replaced byW ₀ in definition of Pe, Re, and. With 7 figures and 7 tables     

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.

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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.

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

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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.     

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.     

Response of a viscous annular liquid layer in zero-gravity to different axial excitations of the rigid boundary plates

A cylindrical annular liquid layer between two plates and around a rigid center-core consisting of incompressible and viscous liquid is subjected to different axial excitations, such as one-sided, counter-directional and double-sided unequal excitations. The response of the free liquid surface, the velocity- and pressure-distribution has been determined.

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Zusammenfassung Eine zylindrische Flüssigkeitsschicht bestehend aus inkompressibler und viskoser Flüssigkeit wurde verschiedenen harmonischen Anregungsformen ausgesetzt. Dabei wurden die Fälle einseitiger, doppelseitiger entgegengesetzter und ungleicher doppelseitiger Anregung mit Phase behandelt. Die Vergrößerungsfunktionen für die freie Flüssigkeitsoberfläche, für die Geschwindigkeits- und Druckverteilung wurden bestimmt.

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List of symbols a radius of liquid layer - b radius of inner cylindrical core - (a–b) thickness of layer - e _r , e , k unit vectors in the radial, angular and axial direction resp. - h length of layer - I _m , K _m modified Bessel functions of first and second kind and order m - diameter ratio - p pressure - q 2na/h - q* na/h - r, , z cylindrical coordinates - complex frequency - S sa ²/ - t time - u, w velocity components in the radial- and axial direction - ₀ excitation amplitude - abbreviation - surface tension parameter - surface tension - dynamic viscosity - kinematic viscosity - density of liquid - free liquid surface elevation - dimensionless time - _rz shear stress - reduced forcing frequency - forcing frequency - stream function - _mn natural frequency of non-viscous liquid     

Rayleigh problem in slip flow with transverse magnetic field

An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.Nomenclature A constant - b characteristic length - B magnetic field vector - B ₀ magntidue of the applied magnetic field normale to the plate - B _x magnitude of the induced magnetic field parallel to the plate - C slip coefficient, (2–f)/f - C _f skin friction coefficient, - C _D average drag coefficient - erfc(_x) complementary error function, - E electric field vector - f Maxwell's reflection coefficient - H _a Hartmann number, (B ₀ ² b ²/)^1/2 - nondimensional magnetic parameter - J current vector - Kn=L/b Knudsen number - L mean free path - M Mach number - p constant parameter - P _m magnetic Prandtl number, Re _m/Re= ₀ - q velocity vector - Re Reynolds number, Ub/ - Re _m magnetic Reynolds number, ₀ Ub - t time - nondimensional time, tU/b - u velocity of the fluid parallel to the plate - nondimensional velocity, u/U - U velocity of the plate - Laplace transform of - x, y coordinates along and normal to the plate respectively - y nondimensional distance, y/b - Z nondimensional parameter, 1/Re ^1/2 Kn - ratio of specific heats - boundary layer thickness - velocity slip - viscosity - ₀ magnetic permeability - kinematic viscosity - nondimensional time parameter, ( /Re)^1/2/Kn - density - electrical conductivity     

Interaction of a vortex couple with a free surface

The characteristics of three-dimensional flow structures (scars and striations) resulting from the interaction between a heterostrophic vortex pair in vertical ascent and a clean free surface are described. The flow features at the scar-striation interface (a constellation of whirls or coherent vortical structures) are investigated through the use of flow visualization, a motion analysis system, and the vortex-element method. The results suggest that the striations are a consequence of the short wavelength instability of the vortex pair and the helical instability of the tightly spiralled regions of vorticity. The whirls result from the interaction of striations with the surface vorticity. The whirl-merging is responsible for the reverse energy cascade leading to the formation and longevity of larger vortical structures amidst a rapidly decaying turbulent field.List of symbols A _c Area of a vortex core (Fig. 6b) - AR Aspect ratio of the delta wing model - B base width of delta wing - b ₀ initial separation of the vortex couple - d ₀ depth at which the vortex pair is generated - c average whirl spacing in the x-direction - E energy density - Fr Froude number ( ) - g gravitational acceleration - L length of the scar band - L _ko length of the Kelvin oval - N _w number of whirls in each scar band - P _c Perimeter of a vortex core - q surface velocity vector - r _c core size of the whirl ( = 2A _c/P _c) - Re Reynolds number ( = ) - Rnd a random number - s inboard edge of the scar front (Fig. 6 a) - t time - u velocity in the x-direction - velocity in the y-direction - V _b velocity imposed on a scar by the vortex couple (Fig. 6 a) - V ₀ initial mutual-induction velocity of the vortex couple (=₀/2b ₀) - V _t tangential velocity at the edge of the whirl core - w width of the scar front (Fig. 6 a) - z complex variable - z _k position of the whirl center - half included angle of V-shaped scar band - wave number - _m initial mean circulation of the whirls - ₀ initial circulation of the vortex pair - _w circulation of a whirl - _min minimum survival strength of a whirl - t time step - gDz increment of z - gD change in vorticity - cut-off distance in velocity calculations - critical merging distance - curvature of the surface - wavelength - kinematic viscosity - angular velocity of a whirl core     

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.

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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.

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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     

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     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, 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 -{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 large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for 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, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

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      

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer turbulenten Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht und mit Meßwerten verglichen. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat aufgrund der radial ansteigenden Zentrifugalkräfte einen ausgeprägten Einfluß auf die Unterdrückung der turbulenten Bewegung. Dadurch verschlechtert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich zu.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 1: Effect of rotation on turbulent pipe flow

The effects of rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The theoretical results are compared with experimental findings. The flow is assumed to have a fully developed velocity profile. Rotation was found to have a very marked influence on the suppression of the turbulent motion because of radially growing centrifugal forces. Therefore, a remarkable decrease in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length increases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (30) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - Massenstrom - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _n Eigenfunktionen - Durchfluß-Reynoldszahl - Re v =D/ Rotations-Reynoldszahl - Ri Richardsonzahl - R Rohrradius - r Koordinate in radialer Richtung - dimensionslose Koordinate in radialer Richtung - T Temperatur - T Temperaturschwankung - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - v Geschwindigkeitsschwankung - turbulenter Wärmestrom - dimensionsloser Wandabstand - =1/6 Konstante - Integrationsvariable - Integrationsvariable - , ₁, ₂, dimensionslose Temperaturen - Wärmeleitzahl - _n Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential     

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     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

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.     

Verbessertes Diagramm zur Berechnung von Wärmeübertragern

Zusammenfassung Die bekannten Wärmeübertragerdiagramme, in denen die dimensionslosen Temperaturänderungen beider Stoffströme auf den Koordinatenachsen aufgetragen sind, werden modifiziert, um die Nachteile der bisherhigen Darstellung zu vermeiden. Diese neuen Diagramme werden für einige einfache Stromführungen angegeben und mit anderen gebräuchlichen Diagrammen verglichen. Anhand von Beispielen wird die Anwendung erläutert.

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Improved chart for heat exchanger design

The known heat exchanger charts with dimensionless temperature changes of both fluid streams as coordinate axes are modified to eliminate the disadvantages of the previous representation. Graphs are presented for some simple heat exchanger configurations. The new chart is compared to other usual charts. The application is illustrated by examples.

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Formelzeichen A Austauschfläche - F Korrekturfaktor für die logarithmische mittlere Temperaturdifferenz; F=I_M/I_LM - k Wärmedurchgangskoeffizient - NTU Anzahl der Übertragungseinheiten (number of transfer units);NTU _i = k A/ _i - P dimensionslose Temperaturänderung - Pe Pecletzahl - R Kapazitätsstromverhältnis;R ₁ = ₁/ ₂;R ₂ = ₂/ ₁ - Kapazitätsstrom - I_M mittlere Temperaturdifferenz - I_LM logarithmische mittlere Temperaturdifferenz - dimensionslose mittlere Temperaturdifferenz - I_M Temperatur Indizes 1, 2 Stoffstrom 1, 2 - am Eintritt - am Austritt Herrn Prof. Dr.-Ing. K. Stephan zum 60. Geburtstag gewidmet     

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

Summary A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

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Zusammenfassung Eine Zustandsgleichung vom Integraltyp mit einer deformationsabhängigen und faktorisierten Gedächtnisfunktion wird zur Beschreibung der Zeitabhängigkeit der Schubspannung, der ersten Normalspannungsdifferenz und, unter Verwendung des spannungsoptischen Gesetzes, auch des Auslöschungswinkels und der Strömungsdoppelbrechung einer Polystyrol-Schmelze bei Scherströmungen herangezogen. Die theoretischen Voraussagen werden mit Messungen verglichen. Die Nichtlinearität des viskoelastischen Verhaltens, repräsentiert durch die sogenannte Dämpfungsfunktion, wird durch eine einfache Exponentialfunktion mit nur einem Parametern angenähert. Die Dämpfungskonstanten kann, wie auch ein diskretes Relaxationszeitspektrum der Schmelze, aus der Frequenzabhängigkeit der Speicher- und Verlustmoduln bestimmt werden.

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a _i weight factor of thei-th relaxation time - a _T shift factor - C stress-optical coefficient - n flow birefringence in the shear flow plane - shear relaxation modulus - G() shear storage modulus - () shear loss modulus - H() relaxation time spectrum - h( _t,t ² ) damping function - M _w weight-average molecular weight - M _n number-average molecular weight - n damping constant - p ₁₂ shear stress - p ₁₁ –p ₂₂ primary normal stress difference - t current time - t past time - extinction angle - ( — _i) delta function - time and shear rate dependent viscosity - | ^*| absolute value of the complex viscosity - shear rate - _t,t relative shear strain between the statest andt - memory function - angular frequency - relaxation time - _i i-th relaxation time of the line spectrum - time and shear rate dependent primary normal stress coefficient - s steady-state value - t time dependence - ° linear viscoelastic behaviour With 6 figures and 1 table     

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     

Analyse und Berechnung der Strömungsvorgänge im zylindrischen Scherspalt von Plastifizieraggregaten

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

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

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

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984