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     

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     

Concentration and velocity measurements in the flow of droplet suspensions through a tube

Two optical methods, light absorption and LDA, are applied to measure the concentration and velocity profiles of droplet suspensions flowing through a tube. The droplet concentration is non-uniform and has two maxima, one near the tube wall and one on the tube axis. The measured velocity profiles are blunted, but a central plug-flow region is not observed. The concentration of droplets on the tube axis and the degree of velocity profile blunting depend on relative viscosity. These results can be qualitatively compared with the theory of Chan and Leal.List of symbols a particle radius,m - a/R, non-dimensional particle radius - c volume concentration of droplets in suspension, m³/m³ - c ₅ stream-average volume concentration of droplets in suspension, - D 2 R, tube diameter, m - L optical path length, m - L _ij path length of laser beam through thej-th concentric layer when the beam crosses the tube diameter at the point on the inner circumference of thei-th layer, m - N exponent in Eqs. (3) and (4) - Q volumetric flowrate of suspension, - R tube radius, m - Re _{S S} D/µ, flow Reynolds number - r radial position (r = 0 on a tube axis), m - r r/R, non-dimensional radial position - v velocity of suspension, m/s - v v/v _S , non-dimensional velocity - v ₀ centre-line velocity of suspension (r = 0), m/s - v _S Q/ R ², stream-average velocity of suspension, m/s - x streamwise position (x = 0 at tube inlet), m - x x/D, non-dimensional streamwise position - _c density of continuous phase, kg/m³ - _d density of dispersed phase, kg/m³ - _s stream-average density of suspension, kg/m³, equals density when homogenized - _d - _c, phase density difference, kg/m³ - µ_c viscosity of continuous phase, Pa · s - µ_d viscosity of dispersed (droplet) phase, Pa · s - µ_d/_c, viscosity ratio - interfacial tension, N/m This work was financially supported by the National Science Foundation (USA) through an agreement no. J-F7F019P, M. Sklodowska-Curie fund     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

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)     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Begrenzungen für die Anwendbarkeit von Preston-Rohren in Kanalströmungen

Zusammenfassung Abweichungen der Geschwindigkeitsverteilung in Wandnähe von einem universellen Gesetz verursachen Fehler bei der Bestimmung der Wandschubspannung mittels Preston-Rohren, die in der Kreisrohrströmung geeicht wurden. Mit Hilfe des in [3] abgeleiteten Geschwindigkeitsgesetzes und geeigneter Annahmen über die Meßfehler von Pitot-Sonden werden für die Kanalströmung die möglichen Abweichungen von der Preston-Rohr-Eiehkurve und die daraus resultierenden Fehler der zu messenden Wandschubspannung berechnet und in Abhängigkeit eines die örtliche Geometrie kennzeichnenden Parameters dargestellt.

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Limitations on the use of preston tubes in channel flow

Deviations of the velocity distribution near a wall from a universal law cause errors in wall shear stresses determined by means of the Preston tube calibrated in pipe flow. Using the relationship for the velocity distribution presented in [3] and suitable assumptions on Pitot tube errors possible deviations from the Preston tube calibration curve and resulting errors in wall shear stress are calculated and presented as a function of a local geometry parameter for channel flow.

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Bezeichnungen a Profillänge zwischen benetzter Wand und Geschwindigkeitsmaximum - C _I Konstante im Geschwindigkeitsgesetz - c_W Korrekturfaktor für Wandeinfluß bei Pitot-Rohr-Messungen - c_T Korrekturfaktor für den Einfluß der turbulenten Geschwindigkeitsschwankungen bei Pitot-Rohr-Messungen - c _S Korrekturfaktor für den Scherströmungseinfluß bei Pitot-Rohr-Messungen - d Preston-Rohr-Durchmesser - die örtliche Geometrie kennzeichnender Parameter - P dynamische Druckanzeige des Preston-Rohres - r _m radiale Koordinate des Ortes maximaler Geschwindigkeit - R Krümmungsradius des benetzten Umfanges - Re Reynoldszahl - t Konstante im Gesehwindigkeitsgesetz - U örtliche Strömungsgeschwindigkeit - Geschwindigkeitsparameter - u _p ⁺ der Preston-Rohr-AnzeigeP entsprechender Geschwindigkeitsparameter - x ^* dimensionsloser Parameter für die dynamische Druckanzeige des Preston-Rohres - y Wandabstand - Wandabstandsparameter - dem Mittelpunkt des Preston-Rohres entsprechender Wandabstandsparameter - y ^* dimensionsloser Parameter für die Wandschubspannung - Konstante im Geschwindigkeitsgesetz - kinematische Zähigkeit - Dichte - Wandschubspannung     

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     

Regularity for the stationary Navier-Stokes equations in bounded domain

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain ^N, 5N . If u, p satisfy the additional conditions

A generalized rheological model of thixotropic materials

Summary A generalization of the rheological model of thixotropic materials, presented previously, was carried out. In the generalized rheological equation of state the yield stress depending on the structural parameter was introduced. In the generalized rate equation the difference in the destruction and recovery rates of the material structure was taken into account. A procedure leading to the determination of nine rheological parameters of the generalized model was worked out. The model was checked experimentally for a thixotropic paint.

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Zusammenfassung Eine früher dargestellte Theorie thixotroper Stoffe wird verallgemeinert, wobei eine von dem Strukturparameter abhängige Fließspannung eingeführt wird. Weiterhin wird der Unterschied zwischen der Zerstörungs-und der Wiederaufbaugeschwindigkeit der Stoffstruktur berücksichtigt. Eine Methode zur Bestimmung der neun benötigten Stoffparameter wird ausgearbeitet. Das Modell wird am Beispiel einer thixotropen Farbe experimentell geprüft.

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Notation a rheological parameter in eq. [26], s^–1 - A rheological parameter in eq. [16] - b rheological parameter in eq. [26] - c function in eq. [21] - averaged value of functionc in eq. [28] - c function in the rate equation [23], defined by eq. [21] - G function [1] defining material of the rate type - h function [2] determining the state of thixotropic fluid - k rheological parameter in the Herschel-Bulkley equation [17] or, in special case, in eq. [8], Ns ⁿ /m² - K function in eq. [18], Ns ^m /m² - m rheological parameter in eq. [18] or, in special case, in eq. [10] - n rheological parameter in the Herschel-Bulkley model [17] or, in special case, in eq. [8] - s rheological parameter in eq. [16] - t time, s - x arbitrary real variable - rheological parameter in eq. [9], s - shear rate, s^–1 - structural parameter, defined by eq. [2] - substantial derivative of structural parameter, s^–1 - _e function [6] describing the equilibrium curve in the coordinate system ( ) - ₀ initial value of structural parameter (att = 0) - natural time function of the thixotropic material, defined by eq. [22] - shear stress, N/m² - substantial derivative of shear stress, N/m² s - _e function describing equilibrium flow curve in the coordinate system ( ) - ₀ equilibrium yield stress, defined by eq. [12], N/m² - _y function of structural parameter describing the yield stress - function in eq. [11] Notation used in the algorithm:(Appendix) i,j,k integer - k _e (i) ordinal number of the experimental point at which the line of _i = const intersects the equilibrium flow curve - l _i number of the experiments of the type stepchange of the shear rate - l _j number of experimental points in one experiment of the type step-change of the shear rate - n _e number of experimental points on the equilibrium flow curve - n _k number of experimental points on the line of constant - n _y number of lines of constant - t(j) measured time interval (from the moment of the step-change of shear rate) - abscissa of the experimental point of ordinal numberk on the line of _i = const, in the coordinate system ( ) - abscissa of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system ( ) - shear rate at which the experiment of the type step-change of shear rate was carried out - _e (i) ordinate of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system ( ) - _y (i) value of yield stress at = _i - _s (i,j) experimental value of shear stress at constant value of shear rate (2i) for time intervalt(j) - (i,k) ordinate of the experimental point of ordinal numberk on the line of _i = const, in the coordinate system ( ) - ₀ the admissible value of the difference between the experimental and theoretical value of shear stress With 4 figures and 1 table     

Bifurcation Phenomena from Standing Pulse Solutions in Some Reaction-Diffusion Systems

Bifurcation phenomena from standing pulse solutions of the problem is considered. (>0) is a sufficiently small parameter and is a positive one. It is shown that there exist two types of destabilization of standing pulse solutions when decreases. One is the appearance of travelling pulse solutions via the static bifurcation and the other is that of in-phase breathers via the Hopf bifurcation. Furthermore which type of destabilization occurs first with decreasing is discussed for the piecewise linear nonlinearities f and g.     

Oscillation Types and Bifurcations of a Nonlinear Second-Order Differential-Difference Equation

This paper considers the second-order differential difference equation

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

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

Control of unstable oscillations in a confined and bounded injecting wall channel

The analysis of a confined flow field generated through two separated injecting walls was carried out by studying the effect of a mass flow rate difference between the two walls. Such a parameter has been found to play a major role in unstable flow field behaviour. Concretely speaking, we have identified two vortex shedding phenomena, i.e. the main flow and the wall vortex shedding phenomena. Results clearly show that when the mass flow rate is increased at the second injecting wall, wall vortex coherence is enhanced and impinging of such structures forces a coupling phenomenon to develop between flow field dynamics and acoustics. On the other hand, only a 15% mass flow rate difference of the first injecting block is sufficient to prevent such coupling between acoustics and vortex shedding phenomenon. Consequently, the resonance phenomenon is pronouncedly weakened and significant oscillation reduction is achieved.Nomenclature sound velocity (m/s) - nth longitudinal acoustic mode (Hz) - h_t height of the nozzle throat (m) - h_c channel height (m) - l length between the edge of the second injecting block and the nozzle location (m) - L channel length (m) - P mean pressure at the front-head (Pa) - P fluctuating pressure (Pa) - q_m, q₁, q₂ total, first and second injecting block mass flow rate (kg/s) - S_x normalised power spectral density of the x fluctuations (Hz^-1) - s=w h_c characteristic surface area (m²) - T temperature of the flow (K) - u, v longitudinal and lateral velocity component (m/s) - u_X longitudinal mean velocity at X location (m/s) - u_m maximum longitudinal velocity (m/s) - u, v longitudinal and lateral fluctuating velocity (m/s) - characteristic acoustic velocity (m/s) - v_w wall injection velocity (m/s) - w channel width (m) - X, Y, Z non-dimensional axis normalised respectively by l, h_c and w - dynamic viscosity (kg/ms) - density (kg/m³) - time delay (s)Dimensionless parameters turbulence intensity - Mach number - Re_c= M a h_c/ Reynolds number - Re_w= v_w h_c/ wall injection Reynolds number - correlation coefficient of pressure and velocity fluctuations - normalised longitudinal velocity component - parameter of unbalanced mass flow rate between the two injecting blocks - specific heat ratio - coherence function     

An analytical solution for the temperature profiles during injection molding,including dissipation effects

An analytical solution is obtained for the stationary temperature profile in a polymeric melt flowing into a cold cavity, which also takes into account viscous heating effects. The solution is valid for the injection stage of the molding process. Although the analytical solution is only possible after making several (at first sight) rather stringent assumptions, the calculated temperature field turns out to give a fair agreement with a numerical, more realistic approach. Approximate functions were derived for both the dissipation-independent and the dissipation-dependent parts which greatly facilitate the temperature calculations. In particular, a closed-form expression is derived for the position where the maximum temperature occurs and for the thickness of the solidified layer.The expression for the temperature field is a special case of the solution of the diffusion equation with variable coefficients and a source term.Nomenclature a thermal diffusivity [m²/s] - c specific heat [J/kg K] - D channel half-height [m] - L channel length [m] - m 1/ - P pressure [Pa] - T temperature [°C] - T _W wall temperature [°C] - T _i injection temperature [°C] - T _A Br independent part of T - T _B Br dependent part of T - T ^core asymptotic temperature - v _z() axial velocity [m/s] - W channel width [m] - x cross-channel direction [m] - z axial coordinate [m] - (x) gamma function - (a, x) incomplete gamma function - M(a, b, x) Kummer function - small parameter - () temperature function - thermal conductivity [W/mK] - viscosity [Pa · s] - ₀ consistency index - power-law exponent - density [kg/m] - similarity variable Dimensionless variables Br Brinkman number - Gz Graetz number -      

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

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 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     

A Helmholtz/Weyl Decomposition in Energy Space

For a bounded region in a Helmholtz/Weyl decomposition of the Sobolev space is given,with orthogonality with respect to the strain-energy inner product of elasticity (anisotropic or isotropic).