On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - 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 REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - 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 - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average     

Evaluation of the performance characteristics of a thermal transient anemometer

The Goertler instability of a hypersonic boundary layer and its influence on the wall heat transfer are experimentally analyzed. Measurements, made in a wind tunnel by means of a computerized infrared (IR) imaging system, refer to the flow over two-dimensional concave walls. Wall temperature maps (that are interpreted as surface flow visualizations) and spanwise heat transfer fluctuations are presented. Measured vortices wavelengths are correlated to non-dimensional parameters and compared with numerical predictions from the literature.List of symbols c _p Specific heat coefficient at constant pressure of the free stream - F Input (true) image - F ₀ Fourier number - Restored image - G Recorded (degraded) image - G Goertler number based on the boundary layer thickness, as defined by Eq. (3) - H System transfer function - M Mach number - Pr Prandtl number - p ₀ Stagnation pressure - Exchanged net heat flux - Convective heat flux - Radiative heat flux - r Recovery factor - Re _m Unit Reynolds number - Re _x Local Reynolds number based on the distance from the leading edge - Re Local Reynolds number based on the boundary layer thickness - Curvature radius - St Stanton number, as defined by Eq. (7) - T _aw Adiabatic wall temperature - T _w Wall temperature - T ₀ Stagnation temperature - t Time - V Free stream velocity - x Streamwise spatial coordinate - y Normal-to-wall spatial coordinate - z Spanwise spatial coordinate - Thermal diffusivity coefficient - Disturbance wavenumber - Non dimensional wavenumber - Boundary layer thickness - Goertler number based on the vortices wavelength - Vortices wavelength - Free stream density - Disturbance total amplification, as defined by Eq. (3) - Disturbance (spatial) growth rate - Non-dimensional growth rate - Perturbation amplitude of a generic quantity - Perturbation amount     

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     

Quantitative liquid jet instability measurement system using asymmetric magnification and digital image processing

A new technique for measuring the growth of instabilities on the surface of liquid jets flowing into gas is demonstrated. A collimated beam of white light illuminates the jet from behind, forming a shadow image. A pair of cylindrical lenses are arranged to provide different magnifications in the streamwise and cross-stream directions. A number of streamwise diameters and one cross-stream diameter are thus captured with maximum resolution in a single image on a charge-coupled device (CCD) electronic camera. A short-duration spark is used to freeze the jet motion. A mask representing the theoretical edge-response of the imaging system is digitally convolved with the cross-stream gray scale data to obtain sub-pixel resolution of the jet edge profile. The method is demonstrated using the well-known capillary jet instability and a ratio of streamwise to cross-stream magnifications of 40. Well-resolved single images show the development of the instability from small perturbations through the formation of the first drop. The system forms an accurate automated method of measuring the development of liquid jet instabilities. It can readily be applied to practical problems including liquid jet atomization.List of symbols a undisturbed jet radius - k nondimensional wavenumber (= 2a/) - Q gas-to-liquid density ratio - r ₀ mean jet radius, from initial region of image - R Reynolds number (= 2Ua/) - U mean jet velocity - We Weber number - z streamwise coordinate, origin at jet orifice - temporal growth rate - _s measured spatial growth rate - nondimensional temporal growth rate - r absolute value of height of peaks or troughs relative to r ₀ - r ₁ height of first extremum in a particular record - instability wavelength - liquid viscosity - liquid density - surface tension of liquid-gas interface     

Rheological properties of reactive polymer melts.

A new method for describing the rheological properties of reactive polymer melts, which was presented in an earlier paper, is developed in more detail. In particular, a detailed derivation of the equation of a first-order rheometrical flow surface is given and a procedure for determining parameters and functions occurring in this equation is proposed. The experimental verification of the presented approach was carried out using our data for polyamide-6.Notation E Dimensionless reduced viscosity, eq. (34) - E ₀ Newtonian asymptote of the function (36) - E power-law asymptote of the function (36) - E _{= 1} the value ofE at = 1 - k degradation reaction rate constant, s^–1 - k ₁ rate constant of function (t), eq. (26), s^–1 - k ₂ rate constant of function (t), eq. (29), s^–1 - K(t) residence-time-dependent consistency factor, eq. (22) - M _w weight-average molecular weight - M _x x-th moment of the molecular weight distribution - R gas constant - S _x M _x /M _w - t residence time in molten state, s - t _j thej-th value oft, s - T temperature, K - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9vqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B4E!\[\dot \gamma \] shear rate, s^–1 - _i thei-th value of , s^–1 - _r =1 the value of at = 1, s^–1 - ^* reduced shear rate, eq. (44), s^–1 - dimensionless reduced shear rate, eq. (35) - viscosity, Pa · s - shear-rate and residence-time dependent viscosity, Pa · s - zero-shear-rate degradation curve - degradation curve at - _{t0 (t)} zero-residence-time flow curve - Newtonian asymptote of the RFS - instantaneous flow curve - power-law asymptote of the RFS - _0,0 zero-shear-rate and zero-residence-time viscosity, Pa · s - _E=1 value of viscosity atE=1, Pa · s - ^* reduced viscosity, eq. (43), Pa · s - zero-residence-time rheological time constant, s - density, kg/m³ - (t),(t) residence time functions     

Strömungstechnische Gesichtspunkte der Flußbelastung durch Abwasser aus Kläranlagen

Zusammenfassung Für ein im Durchlauf betriebenes System bestehend aus einem Fluß (Vorfluter) und den angeschlossenen Kläranlagen wird eine Methode zur Bestimmung der Vorfluterbelastung durch die eingeleiteten Klärwässer angegeben. Die Methode erfaßt mit Rücksicht auf die Anwendung des Verursacherprinzips im Gewässerschutz die Belastung durch jede Kläranlage für sich, und zwar in Abhängigkeit von der Wasserführung, den Emissionsraten der betreffenden Kläranlage und dem Selbstreinigungsvermögen von den organischen Stoffen aus der betreffenden Kläranlage. Die abhängigen Veränderlichen sind mit der Fließgeschwindigkeit gewichtete Mittelwerte von Schmutzstoffdichten über den Vorfluterquerschnitt. Im Falle konstanter Vorflutertemperatur und zeitunabhängiger Struktur der Klärwässer ergeben sich beispielsweise für die abhängigen Veränderlichen einfache analytische Darstellungen, welche sich als spezielle Formen des -Theorems erweisen. Es wird gezeigt, bei einem unendlich langen Vorfluter mit konstantem Volumenstrom stromabwärts der Klärwassereinleitungen stimmen die erwähnten gewichteten Mittelwerte mit den entsprechenden ungewichteten stromabwärts der Klärwassereinleitungen überein. Die entwickelte Methode kann leicht erweitert werden, um den Sauerstoffschwund im Vorfluter durch jede Kläranlage für sich zu bestimmen.

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Fluid mechanical aspects of river pollution by effluents from waste treatment plants

The pollution of a river by effluent inflows from waste treatment plants is modeled under steady-state conditions. With respect to modern policies of environmental protection the method describes the river pollution by each plant separately, depending on the flow conditions, the emission rates of the plant and the microbiological decomposition of the biodegradable matter from the plant. Each dependent variable is a weighted cross-sectional mean of a density of organic matter. If the water temperature is constant and the composition of each effluent is independent of time the method gives simple analytic expressions for the dependent variables, which prove to be special versions of the -theorem. It is shown for an infinitely long river of constant volume rate of flow downstream of the effluent inflows: the weighted means mentioned agree with the corresponding nonweighted downstream of the effluent inflows. The present paper can easily the extended to determine the oxygen deficit in the river due to each plant.

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Bezeichnungen a Anzahl der Kläranlagen - D(t_b) Kennzahl, Einführung in 4.3 - ^eA Emissionsrate der abbaubaren or ganischen Verschmutzung aus der -ten Kläranlage - ^eU Emissionsrate der nichtabbaubaren organischen Verschmutzung aus der -ten Kläranlage - Vorfluterquerschnitt, Einführung in Gl. (4) - F Flächeninhalt von - dF Betrag eines Flächenelements, Einführung in Gl. (6) - J_A Diffusionsstromdichten, Einführung in Gl. (2) bzw. Gl. (3) - L Anzahl der Stromstrecken - M Gesamtmasse der abbaubaren or- ganischen Verschmutzung in den N Teilchen, Einführung in Gl. (17) - N Anzahl der verschmutzten Flußwasserteilchen, welche die -te Nahfeldvermischungszone während des Zeitintervalles t_a t_b für immer verlassen - P(x, t, x, t_c) Teilchendichte, Einführung in Gl. (11) und Gl.(12) - Q Selbstreinigungsvermögen, Einführung in Gl.(26) - t Zeitpunkt, Einführung in Gl.(11) - t, t_b Intervallgrenzen, Einführung in 4.1 - t_c Zeitpunkt, Einführung in Gl.(11) - t Zeitdifferenz, Einführung im Anschluß an Gl.(10) - t^* charakteristische Zeit, Einführung in 4.3 - Strömungsgeschwindigkeit Komponente von ¯b in Richtung der zu Tal weisenden Oberflächennormalen eines Vorfluterquerschnitts, Einführung in Gl. (5) und Gl. (6) - Volumenstrom, Einführung in Gl. (7) - x Ortsvektor - x Ortsvektor eines bestimmten markierten Teilchens zur Zeit t_c, Einführung in Gl.(11) - x längs der Stromachse gemessene Längenkoordinate - x x-Koordinate des Vorfluterquerschnitts durch x - x,x₊₁ x-Koordinaten der Vorfluterquerschnitte, welche die -te Stromstrecke stromaufwärts bzw. stromabwärts begrenzen. Einführung in 4.2. - transformierte Variable, Einführung in Gl.(65) - Zeitvariable - (t_b) Kennzahl, Einführung in 4.3. - Masse der abbaubaren organischen Verschmutzung in dem markierten Teilchen, Einführung in Gl.(14) - , Integrationsvariablen, Einführung in Gl.(38) bzw. Gl.(28) - _A durch die -te Kläranlage bedingte Dichte der abbaubaren organischen Verschmutzung - _U durch die -te Kläranlage bedingte Dichte der nichtabbaubaren organischen Verschmutzung - Mittelwerte von bzw· , Einführung in Gl.(31) bzw. Gl.(8) - _m -Wert zu einem Maximum, Einführung in Gl.(31) - Verhältnis zweier Mittelwerte, Einführung in Gl.(64) - stochastischer Mittelwert einer Zufallsgröße Y - Y Schwankung einer Zufallsgröße Y um den stochastisehen Mittelwert - Mittlung über den Vorfluterquerschnitt Der saubere Vorfluter sei definiert durch Standardwerte für Mindestanforderungen an die Flußwasserqualität. Vorschläge für solche Standardwerte werden in jüngster Zeit unter Berücksichtigung des Umweltschutzes ausführlich diskutiert ([1]; [2], S.- K 13 -).     

Hydro-mechanical aspects of the sand production problem

This paper examines the hydro-mechanical aspect of the sand production problem and sets the basic frame of the corresponding mathematical modelling. Accordingly, piping and surface erosion effects are studied on the basis of mass balance and particle transport considerations as well as Darcy's law. The results show that surface erosion is accompanied by high changes of porosity and permeability close to the free surface. Quantities which can be measured in experiment, like the amount of produced solids or fluid discharge, can be used in an inverse way to determine the constitutive parameters of the problem.Notation dV Volume element - dV _s Volume of solids pt - dV _v Volume of voids - dV _ff Volume of fluid phase - dV _fs Volume of fluidized-particles - Volume of mixture - dM _s Mass of solids - dM _ff Mass of fluid phase - d _{M fs} Mass of fluidized-particles - Mass of mixture - s Density of solids - f Density of fluid - ff Density of fluid phase - fs Density of fluidized-particles - Density of mixture - _i ^ff Velocity of fluid - _i ^fs Velocity of fluidized-particles - _i ^s Velocity of solids - Velocity of mixture - q ^ff Volume-discharge of fluid - q ^fs Volume-discharge of fluidized-particles - Volume-discharge of mixture - m ^ff Mass-discharge of fluid - m ^fs Mass-discharge of fluidized-particles - Mass-discharge of mixture - _er Rate of mass-eroded - _dep Rate of mass-deposited - Mass generation term - dS _i Surface element - Pore-surface element - D _IJ Tensor of mechanical dispersion - x _i Location - t Time - Porosity - c Transport concentration - c _cr Critical value ofc - p Fluid-pressure - k Permeability coefficient - _k Kinematic viscosity - Spatial frequency of erosion starter points     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

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 - -      

Flow of viscoelastic fluid between confocal elliptic cylinders

Summary Using elliptic coordinates, the flow pattern of a fluid of grade four between two elliptic tubes is determined. A comparison between the position of the maximum of the axial velocity in the present case and in the case of two concentric circular tubes shows a basic difference. In the elliptic case the maximum is shifted towards the external wall, while in the case of concentric circular tubes the shift is in the direction of the internal wall. The secondary flow shows dissymmetry with reference to the intermediate line , which itself lies nearer to the external wall.

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Zusammenfassung Unter Benutzung elliptischer Koordinaten wird die Strömung zwischen zwei elliptischen Rohren bestimmt. Ein Vergleich zwischen der Lage des axialen Geschwindigkeitsmaximums im vorliegenden Fall und im Fall zweier konzentrischer Kreisrohre ergibt einen grundsätzlichen Unterschied: Das Maximum ist im elliptischen Fall zur äußeren Wand hin verschoben, während die Verschiebung im Fall der konzentrischen Kreisrohre zur inneren Wand hin erfolgt. Die Sekundärströmung ist unsymmetrisch relativ zur mittleren Stromlinie , die selbst näher zur äußeren Wand liegt.

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A planar domain representing the annular region - vector inx ₁,x ₂-plane - x _i rectangular coordinates - rectangular unit vectors - , elliptic coordinates - ₁, ₂ ellipses representing respectively the internal and external tubes - = ₂ – ₁ annular widthy = ( – ₁)/ - µ 1^st grade material constant - _i 2^nd grade material constants - _i 3^rd grade material constants - _i 4^th grade material constants - I unit tensor - T _E extra stress (T + pI) - V potential of body forces - material density = (p/) + V = –ax ₃ + () - a specific driving force - arbitrary scalar function - A _k Rivlin-Eriksen tensors - S stress scalar defined onA - t stress vector defined onA - P stress tensor defined onA - v axial velocity - v _i i ^th term in the approximation ofv - u velocity vector perpendicular to the axis 4( ₃ + ₄ + ₅ + 1/2₆) –2/µ(2 ₁ + ₂)( ₂ + ₃) - T stress tensor - p arbitrary hydrostatic pressure - u _i i ^th term in the approximation ofu - stream function definingu - _i i ^th term in the approximation of With 8 figures and 1 table     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Eine stationäre Meßmethode zur Bestimmung der Relaxationszeit viskoelastischer Fluide bei räumlicher Beanspruchung

Zusammenfassung Es wird gezeigt, daß bei Kenntnis der Fließkurve viskoelastischer Flüssigkeiten allein aus der Drehmomentkennlinie des stationär betriebenen Kugel-Kugel-Rheometers eine Relaxationszeit der räumlichen Beanspruchung bestimmt werden kann. Ausgehend von derColeman-Nollschen Entwicklungsschreibweise der rheologischen Zustandsfunktion wird das Geschwindigkeitsfeld als Potenzreihe der Kreisfrequenz bis zur 3. Ordnung bestimmt und zur Drehmomentbeziehung integriert.Messungen an einigen Versuchssubstanzen bestätigen die Tauglichkeit der entwickelten Methode.Häufig verwendete Formelzeichen –a N/m² isotroper Druckanteil - m/s Geschwindigkeitsvektor - e ₁₄ Integrationskonstanten - f _i() Geometriefunktionen - m vektorielle Feldfunktion - ms vektorielle Feldfunktion - ms² vektorielle Feldfunktion - k _i() Geometriefunktionen - t ₀ s Relaxationszeit der räumlichen Beanspruchung - m/s Geschwindigkeitsvektor erster Ordnung - m/s Geschwindigkeitsvektor zweiter Ordnung - m/s Geschwindigkeitsvektor dritter Ordnung - D 1/s Deformationsgeschwindigkeitstensor - 1/s², 1/s³ korotationale, zeitliche Ableitung vonD - 1 Einheitstensor - M Nm Antriebsmoment der rotierenden Kugel - M _i Nm Teilmomente - R m Kugelradius - R _G m Hohlkugelradius - S N/m² Spannungstensor - W 1/s Rotationsgeschwindigkeitstensor - ₁ N s/m² Stoffparameter 1. Ordnung - ₂, ₃ N s²/m² Stoffparameter 2. Ordnung - ₄, ₅, ₆ N s³/m² Stoffparameter 3. Ordnung - RadienverhältnisR/R _G - ₀ N s/m² Anfangsviskosität - kg/m³ Dichte der Flüssigkeit - 1/s Kreisfrequenz der rotierenden Kugel Vorgetragen auf dem 6. Internationalen Rheologie-Kongreß in Lyon-Frankreich vom 4.–8. September 1972.Jetzt: BASF-AG, LudwigshafenMit 4 Abbildungen     

Zur analytischen Beschreibung des Füllstoffeinflusses auf das Fließverhalten von Kunststoff-Schmelzen

From literature some representative equations have been compiled describing the influence of filler on the viscosity of polymer melts. By application of these on the experimental results obtained from GF-SAN it was found that the relative viscosity _R , i.e. the ratio of the viscosities of the filled and unfilled melt, shows a pronounced dependence on the shear rate but not on the shear stress. Defining _R with constant and not with constant (as it is usually done), an analytical approach is possible independent of Further the influence of pressure, temperature and filler content on the zero-shear viscosity of filled polymer melts may be expressed by a modified Arrhenius equation.

     

Singular solutions of some quasilinear elliptic equations

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

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

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

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

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.     

Geometric Analysis of One Singular Cauchy Problem

We consider the Cauchy problem , x(0) = 0, where a ₀₀₀ = 0, a ₀₀₁ = 0, and a ₀₀₂ = 0, and prove the existence of continuously differentiable solutions x(0,] with required asymptotic properties.     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function