On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

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     

Zum Mischen rheologisch inhomogener Stoffsysteme auf Einschneckenmaschinen

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

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

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

Verallgemeinerter Ausdruck für den Einfluß temperaturabhängiger Stoffwerte auf den turbulenten Wärmeübergang

Zusammenfassung Semiempirische Beziehungen für den turbulenten Wärmeübergang müssen bei starker Temperaturabhängigkeit der Stoffwerte einen entsprechenden KorrekturfaktorK _s in bezug auf den quasi-isothermen Wärmeübergang enthalten. Nach einer kritischen Übersicht der bis heute empfohlenen Ausdrücke für den KorrekturfaktorK _s , die früher für Heizung und Kühlung gesondert, dann wieder für beide Arten des Wärmeüberganges in einer einzigen Formel und neuerdings wieder gesondert aufgestellt wurden, und zwar für Flüssigkeiten und Gase immer getrennt, wird auf Grund der dimensionslos geschriebenen Differentialgleichungen für die Bewegung, die Energie und die Randbedingungen gezeigt, von welchen Größen der Korrekturfaktor abhängt. Für den allgemeinsten Fall, den mit nichtlinearen Temperaturcharakteristiken der Stoffwerte wirdK _s von sechs schon normierten Variablen abhängen. Es müssen folgende Einschränkungen gemacht werden: Ausschluß flüssiger Metalle, einphasige Strömungen in geraden Kanälen, ohne Dissipationswärme, ohne Strahlung und natürliche Konvektion. Es wird eine einzige Beziehung für nicht lineare Temperaturcharakteristiken hergeleitet, die nicht mehr zwischen Gasen und Flüssigkeiten, Heizung und Kühlung unterscheiden muß und sieben Variable enthält. Die Resultate der bis heute bekannten Beziehungen mit nur einer Variablen stimmen mit denen hier abgeleiteten gut überein.

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A generalized expression for the influence of temperature-dependent physical properties on turbulent heat transfer

Semiempirical relations for turbulent heat transfer with considerable influence of temperature on physical properties must contain a corrective factorK _s in relation to quasiisothermal heat transfer. After a critical review of previously recommended expressions for this corrective factorK _s (which was in former times taken into account by different expressions for heating and cooling, then combined in one formula for both kinds of heat transfer and recently separated again, all the time allowing for fluids and gases in different expressions), the author shows from the dimensionless equations of motion, energy and of boundary conditions the variables on which the corrective factor depends. For the general case of non-linear temperature characteristics of physical propertiesK _s depends on six dimensionless variables. The following limitations of validity are necessary: onephase currents (fluid metals excluded) in straight channels without viscous dissipation, without radiation and natural convection. One single relation for nonlinear temperature-characteristics, containing seven variables, is derived, making no difference between gases and liquids or heating and cooling. Results of the relations known until now, which contain only one variable, agree well with those derived here.

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Bezeichnungen

Stoffwerte Dichte - c _p spezifische Wärme bei konstantem Druck - Wärmeleitfähigkeit - a Temperaturleitfähigkeit - dynamische Zähigkeit - =/ kinematische Zähigkeit - spezifische Quellwärmeleistung - Pr=v/a Prandtlzahl (als mit der Temperaturleit-fähigkeit normierte kinematische Zähigkeit) Geometrische Daten und Zeit D charakteristische Länge: Rohrdurchmesser oder hydraulisch gleichwertiger Durchmesser - L Länge des Rohres oder Kanals - t Zeit Strömungstechnische Daten Geschwindigkeitsvektor des Fluids - w _x ,w _y ,w _z Komponenten von inx-, y- bzw.z-Richtung - w Absolutwert einer charakteristischen Geschwindigkeit - b Beschleunigungsvektor - b ₀ Absolutwert eines charakteristischen Beschleunigungsvektors - p Druck Temperaturen und Wärmeströme Celsiustemperatur - T Kelvintemperatur - charakteristischer Temperaturunterschied des Wärmeüberganges - q Wärmestromdichte - Wärmeübergangskoeffizient Normierte Größen =b/b₀ normierter Beschleunigungsvektor - ^*=/ normierte Temperatur - normierter Temperaturunterschied - normierter Wärmeleitfähigkeitsunterschied - normierter Unterschied der Prandtlzahl - normierter Unterschied der kinematischen Zähigkeit - =p/₀ w ² normierter Druck - =wt/D normierte Zeit - normierter Geschwindigkeitsvektor - dimensionslos geschriebener Tensor der Deformationsgeschwindigkeit     

Laminar source flow between parallel porous disks

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.Nomenclature a half distance between disks - radial coordinate - r dimensionless radial coordinate, /a - axial coordinate - z dimensionless axial coordinate, /a - radial coordinate of a point in the flow - R dimensionless radial coordinate of a point in the flow, /a - velocity component in radial direction - u =a/, dimensionless velocity component in radial direction - velocity component in axial direction - v = a/}, dimensionless velocity component in axial direction - static pressure - p = (a ²/ ²), dimensionless static pressure - =p(r, z)–p(R, z), dimensionless pressure drop - V magnitude of suction or injection velocity - Q volumetric flow rate of the source - Re source Reynolds number, Q/4a - reduced Reynolds number, Re/r ² - critical Reynolds number - R _w wall Reynolds number, Va/ - viscosity - density - =/, kinematic viscosity - shear stress at upper disk - ₀ = (a ²/ ²), dimensionless shear stress at upper disk - shear stress ratio, ₀/( ₀)_inertialess - u = , dimensionless average radial velocity - u/u, ratio of radial velocity to average radial velocity - dimensionless stream function     

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     

Sensitivity of three-segment electrodiffusion probes to eddy shedding

The influence of eddy shedding on the instantaneous readings of a three-segment cylindrical electrodiffusion velocity probe was investigated in an immersed jet with a very low turbulence intensity, = 1.2%. The velocity fluctuations measured by the three-segment probe were smaller than 2.6%, and the maximum error in the flow angle estimation was 2. Vortices with the Strouhal frequency were detected by a simple electrodiffusion probe placed downstream of the three-segment probe, but no peaks with this frequency were found on the frequency spectra of the three-segment probe. From the probe response to a stepwise change of the polarization voltage the characteristic times of the transient process were estimated. List of symbols a parameter in Eq. (1) [A s^b m^-b] - A amplitude gain - b parameter in Eq. (1) - c parameter in Eq. (3) [A s^–1/2] - d probe diameter [m] - f frequency [s^–1] - f _s recording frequency [s^–1] - G power spectrum - I _k relative current through k-th segment, Eq. (2) - i total current [A] - i _k current through k-th segment [A] - N number of data samples - Re Reynolds number, - Sr Strouhal number, - t time [s] - t ₀ characteristic transient time [s] - v jet velocity [m s^-1] - v time mean value of velocity [m s^-1] - v _x, y velocity components measured by probe [m s^-1] - var variance, var - dynamic viscosity [Pa s] - density [kg m^-3] - relative deviation, [%] - flow angle, see Fig. 1 - dimensionless frequency For the financial support of this work we express our thanks to the DFG, Bonn. The assistance of Dr. Ondra Wein and Dr. Pavel Mitschka is greatly appreciated.     

Nonlinear Taylor stability of viscoelastic fluids

Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.Nomenclature a amplitude of the fundamentals - A _ij ⁽¹⁾ , A _ij ⁽²⁾ first and second Rivlin-Ericksen tensors - d r ₂–r ₁ - D d/dx - E - F - g _ij metric tensor - G torque on the inner cylinder in the supercritical regime - h height of the cylinders - k ₀ /d ² - k ₁ /d ² - I ₁ - I ₂ - I ₃ - I ₄ - r ₁, r ₂ radii of inner and outer cylinders respectively - r ₀ 1/2(r ₁+r ₂) - R Reynolds number ₁ r ₁ d/ ₀ - R _c critical Reynolds number - T Taylor number r _{1 1} ² d ^{3 2}/ ₀ ² *) - T _c critical Taylor number - u ₁, v ₁, w ₁ Fundamentals of the disturbance - u _i , v _i , w _i , (i>1) harmonics - mean velocity (not laminar velocity) - u –u ₁/ar _{1 1} - v v ₁/Rar _{1 1} - x (r–r ₀)/d - , material constants - 0 viscosity - wave number d - density - ₁ angular velocity of inner cylinder - tilde denotes complex conjugate     

Turbulent wall pressure fluctuations over wavy surfaces

Measurements of the spectral characteristics of the wall pressure fluctuations produced by a turbulent boundary layer flow over solid sinusoidal surfaces of moderate wave amplitude to wave-length ratios have been obtained. The wave amplitudes were sufficiently small so that the flow remained attatched. The results show that the root mean square pressure level reaches a maximum on the adverse pressure gradient side of the wave at a position somewhat before the trough. Spectral analysis of the pressure fluctuations in narrow frequency bands reveals considerable differences in low and high frequency behavior. At low frequencies, the peak fluctuation amplitude was found at the trough whereas at high frequencies, the peak occurs just after the crest and a minimum is found at the trough. Pressure fluctuations having streamwise correlation lengths on the order of or larger than the wavelength of the surface do not return to their equilibrium (crest) amplitudes as they travel the length of a wave. Pressure fluctuations having streamwise correlation lengths about one order of magnitude less than a wavelength return exactly to their equilibrium amplitudes. Two-point correlation measurements show a decrease in longitudinal coherence on the adverse pressure gradient side of the wave at low frequencies and a considerable increase over a broad frequency range on the positive pressure gradient side. No change is found in the lateral coherence.List of symbols C _f skin friction coefficient - C _p pressure coefficient - C _n Fourier amplitudes of the pressure coefficient - C _dp pressure drag coefficient - d pinhole diameter - f frequency - h half the crest to trough distance - h ₊ nondimensional wave amplitude = - k _n wavenumber = - k fundamental wavenumber = - l _p pressure correlation length - p _s mean surface pressure - P ambient pressure - p fluctuating pressure - p ² mean square pressure - q dynamic head = 1/2 U ² - R space-time correlation - P Reynolds number based on wavelength = - R Reynolds number based on momentum thickness = - t time - R free stream velocity - U mean streamwise velocity - U _e streamwise velocity at the edge of the boundary layer - u ^* friction velocity = - x streamwise coordinate - y wall-normal coordinate - z spanwise coordinate - ⁺ non-dimensional wavelength = ^*) - phase of the cross-spectral density - ^* boundary layer displacement thickness - _long longitudinal coherency - _lat lateral coherency - wavelength of wavy surface - v kinematic viscosity - radian frequency = 2 f - spectral or cross-spectral density - _n phase of the Fourier series - density - time delay - _w wall shear stress - boundary layer momentum thickness     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

LDV measurements of a turbulent air-solid two-phase flow in a 90° bend

Simultaneous measurements of the mean streamwise and radial velocities and the associated Reynolds stresses were made in an air-solid two-phase flow in a square sectioned (10×10 cm) 90° vertical to horizontal bend using laser Doppler velocimetry. The gas phase measurements were performed in the absence of solid particles. The radius ratio of the bend was 1.76. The results are presented for two different Reynolds numbers, 2.2×10⁵ and 3.47×10⁵, corresponding to mass ratios of 1.5×10^–4 and 9.5×10^–5, respectively. Glass spheres 50 and 100 m in diameter were employed to represent the solid phase. The measurements of the gas and solid phase were performed separately. The streamwise velocity profiles for the gas and the solids crossed over near the outer wall with the solids having the higher speed near the wall. The solid velocity profiles were quite flat. Higher negative slip velocities are observed for the 100 m particles than those for the 50 gm particles. At angular displacement =0°, the radial velocity is directed towards the inner wall for both the 50 and 100 m particles. At =30° and 45°, particle wall collisions cause a clear change in the radial velocity of the solids in the region close to the outer wall. The 100 m particle trajectories are very close to being straight lines. Most of the particle wall collisions occur between the =30° and 60° stations. The level of turbulence of the solids was higher than that of the air.List of symbols D hydraulic diameter (100 mm) - De Dean number,De = - mass flow rate - number of particles per second (detected by the probe volume) - r radial coordinate direction - r _i radius of curvature of the inner wall - r ₀ radius of curvature of the outer wall - r ^* normalized radial coordinate, - R mean radius of curvature - Re Reynolds number, - R _r radius ratio, - U ,U _z mean streamwise velocity - U _r ,U _y mean radial velocity - U _b bulk velocity - , _z rms fluctuating streamwise velocity - _r , _y rms fluctuating radial velocity - -r shear stress component - z-y shear stress component - x spanwise coordinate direction - x ^* normalized spanwise coordinate, - y radial coordinate direction in straight ducts - y ^* normalized radial coordinate in straight ducts, - z streamwise coordinate direction in straight ducts - z ^* normalized streamwise coordinate in straight ducts, Greek symbols streamwise coordinate direction - kinematic viscosity of air     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Nichtkorrekte Aufgaben in der Rheometrie

Zusammenfassung Unter Verwendung des Begriffs der nichtkorrekt gestellten Aufgabe wird eine theoretische Begründung für die schlechte Berechenbarkeit von Funktionen gegeben, die aus meßfehlerbehafteten Daten über die Lösung der inversen Aufgabe berechnet werden. Die wichtigsten nichtkorrekten Aufgaben der Rheometrie werden angegeben sowie die Variante eines Regularisierungsverfahrens (Tichonovsche Regularisierung) vorgestellt, die numerisch stabile Lösungen nichtkorrekter Aufgaben zuläßt. Dabei wird festgestellt, daß die Güte der Lösung u.a. von der Form des stabilisierenden Funktionals und der Anzahl sowie der Art der Nebenbedingungen beeinflußt wird, es aber keine allgemeinen Regeln zur Formulierung der Restriktionen oder der Auswahl des Gütekriteriums gibt.

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The most important ill-posed problems in rheometry are discussed. One version of a regularization method (Tichonov-regularization), which gives stable solutions of such problems, is described. It is shown that the quality of the solution depends on the form of the stabilizing functional and on the quantity and the type of constraints. There are, however, no general rules for formulating the restrictions or selecting the performance criteria. The method is demonstrated for the determination of the relaxation function of several rheological models.

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, ^T, der zu adjungierte, transponierte bzw. inverse Operator - ₀ Anfangsviskosität - Deformationsgeschwindigkeit eines newtonschen Fluids - Deformationsgeschwindigkeit beim Anlaufversuch - Schubspannung - _w Schubspannung an der Wand (z. B. einer Kapillaren) - t, t, s Zeit     

Velocity and turbulence measurements in combustion systems

A laser-Doppler velocimeter is used in the measurement of high-temperature gas flows. A two-stage fluidization particle generator provides magnesium oxide particles to serve as optical scattering centers. The one-dimensional dual-beam system is frequency shifted to permit measurements of velocities up to 300 meters per second and turbulence intensities greater than 100 percent.Exiting flows from can-type gas turbine combustors and burners with pre-mixed oxy-acetylene flames are described in terms of the velocity, turbulence intensity, and temperature profiles.The results indicate the influence of the combustion process on turbulence.List of Symbols A exit area of combustor or burner (m²) - A/F mass air-fuel ratio - D exit diameter of combustor or burner (m) - M mass flow rate of gases (kg/s) - N _D number of Doppler bursts used in each velocity measurement - Q volumetric flow rate at T _r (m³/s) - R exit radius of combustor or burner (m) - R _1/2 distance from centerline to radius where the velocity is one-half of the local centerline velocity (m) - Re exit Reynolds number based on cold flow, QD/A - r distance from centerline of flow (m) - T temperature (°C) - T _CL centerline temperature (°C) - T _r inlet (cold) air temperature of combustor or burner (°C) - T.I. turbulence intensity, - mean velocity (m/s) - U _i instantaneous velocity individually realized by LDV (m/s) - mean velocity at centerline of flow (m/s) - mean square velocity fluctuation (m²/s² - x distance along centerline downstream of exit (m) - absolute viscosity at T _r (kg/(ms)) - density at T _r (kg/m³)     

Velocity field in isothermal turbulent bubbly gas-liquid flow through a pipe

Velocity field was measured by laser Doppler velocimetry in isothermal, turbulent bubbly gas-liquid flow through a 26.6 mm inner diameter vertical pipe. The measurements were made about 33 diameters downstream from the pipe entrance, gas injection being just upstream of the entrance. The gas phase radial distribution at the measurement plane exhibited influence of the injection device in that higher gas fraction existed in the central region of the pipe. For comparison, velocity field was also measured in isothermal, turbulent single-phase liquid flow through the same pipe at the same axial plane. Measured were the radial distributions of liquid mean axial and radial velocities, axial and radial turbulent intensities, and axial Reynolds shear stress. The radial distributions of gas bubble mean axial velocity and axial velocity fluctuation intensity were also measured by LDV. A dualsensor fiberoptic probe was used at the same time to measure the radial distributions of gas fraction, bubble mean axial velocity and size slightly downstream of the LDV measurement plane.List of Symbols an average gas bubble diameter - f, f _TP friction factor, friction factor for gas-liquid flow - k _L liquid turbulent kinetic energy - , gas, liquid mass flow rate - R inner radius of pipe - r, {sitR}^* radial coordinate; nondimensional radial coordinate (=r/R) - Re _L liquid Reynolds number - U _G mean axial velocity of gas bubble - U _L mean axial velocity of liquid - U _LO mean axial velocity for flow at the total mass velocity with properties of the liquid phase - u _L ⁺ nondimensional mean axial velocity of liquid in wall coordinate - friction velocity - axial velocity fluctuation intensity of liquid - axial velocity fluctuation intensity of gas bubbles - V_L mean radial velocity of liquid - v _L radial velocity fluctuation intensity of liquid - (uv)_L single-point cross-correlation between axial and radial velocity fluctuations of liquid ( axial Reynolds shear stress) - T _in mean liquid temperature at test section inlet - x flow quality - y normal distance from wall - y ⁺ nondimensional normal distance from wall in wall coordinate (=yu/v_L) - _G gas phase residence time fraction - _L rate of dissipation in the liquid - _L Kolmogorov length scale in the liquid - _L liquid kinematic viscosity - _L characteristic turbulence length scale in the liquid - _G, _L density of gas, liquid - _m gas-liquid mixture density This work was partly supported by National Science Foundation, Thermal Transport and Thermal Processing Program, Chemical and Thermal Systems Division, under Grant No. CTS-9411898.     

Rheological characteristics of glass-fibre-filled polypropylene melts

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

The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

Stoffaustausch zwischen zwei fluiden Phasen,von denen sich eine als Strahl in der anderen bewegt

Zusammenfassung Der Übergang eines Stoffes zwischen zwei fluiden Phasen wird betrachtet, von denen sich einer als Strahl in der anderen bewegt. Die Geschwindigkeit der laminar strömenden Phase wird durch eine Gleichung ausgedrückt, die Geschwindigkeitsprofile zwischen der Kolben- und der Rohrströmung kontinuierlich beschreibt. Der Transport des Stoffes im Strahl durch Diffusion in radialer und durch Konvektion in axialer Richtung wird für den isothermen, stationären Fall untersucht. Die das Problem beschreibende Differentialgleichung wird anscheinend erstmals geschlossen gelöst. Die Lösungen beinhalten konfluente hypergeometrische Funktionen. Berechnet werden Eigenwerte, Koeffizienten, örtliche und mittlere Konzentrationsfelder sowie Stoffübergangszahlen.

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Mass transfer between two fluids, one of the two fluids is moving as jet within the other

The mass transfer between two fluids is calculated, one of the two fluids is moving as a jet within the other. The velocity of the laminar flowing phase is expressed by an equation, which describes continously the velocity profiles from plug flow to tubular flow. For the isothermal, stationary state the transport of substance i by radial diffusion and by axial convection is investigated. It appears to be that the differential equations describing the problem are solved rigorously for the first time. The solutions contain confluent hypergeometrical functions. Results include eigenvalues, coefficients, local and mean concentration fields, mass transfer numbers.

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Verwendete Zeichen und ihre Bedeutung a - A, A_n Koeffizienten - B, B_n Koeffizienten - c Konzentration, Konstante im Anhang - C_r=0 Mittenkonzentration - c₀ Konzentration in Phase I bis z=0 - c_II Konzentration in Phase II - ¯c mittlere Konzentration, definiert in Gl. (35) - C Koeffizient, definiert in Gl. (A 21) - D Diffusionskoeffizient - Da Damköhlerzahl - E Funktion, gegeben durch Gl. (A 12) - f, f(R) Funktion f von R - f_n, f_n (R) Funktionswerte - g, g(Z) Funktion g von Z - g_n, g_n (Z) Funktionswerte - h(Z) Funktion h von z - H_q Koeffizienten, gegeben durch Gl. (A 10) - j Massenstromdichte - J _k , J_q Besselfunktion der Ordnungk, q - k definiert durch Gl. (A 9) - n laufende Zahl - m laufende Zahl - p laufende Zahl - Pe=Re·Sc Pecletzahl - q laufende Zahl - Q_n Koeffizienten, definiert in Gl. (31) - r radiale Koordinate - r₀ Radius - R r/r₀ - Re=u₀r₀/ Reynoldszahl - S=2r₀z Zylinderfläche - Sc=/D Schmidtzahl - Sh=2r₀ /D Sherwoodzahl - Sherwoodzahl, definiert in Gl. (52) - Sh_u Sherwoodzahl, definiert in Gl. (54) - Sh_z Sherwoodzahl, definiert in Gl. (40) - Sherwoodzahl, definiert in Gl. (45) - t R² - u Geschwindigkeit - u₀ maximale Geschwindigkeit - v - Volumenstrom - w Variable - x Variable - y abhängige Variable - z axiale Koordinate, Lauflänge - Z z/r₀ - Z_Pe dimensionslose Lauflänge, definiert durch Gl. (34) - a_n Koeffizienten, definiert durch Gl. (A 19) - Stoffübergangskoeffizient - Stoffübergangskoeffizient, definiert in Gl. (48) - _u Stoffübergangskoeffizient, definiert in Gl. (49) - _z Stoffübergangskoeffizient, definiert in Gl. (38) - Stoffübergangskoeffizient, definiert in Gl. (44) - definiert in Gl. (A 21) - Gammafunktion - c Konzentrationsdifferenz - m Stoffmenge - Zahl zwischen Null und Eins - laufende Zahl - kinematische Zähigkeit - (v) (t) - konfluente hypergeometrische Funktion - (t) - konfluente hypergeometrische Funktion - , _n Eigenwerte Hochzeichen - * kennzeichnet asymptotische Lösungen