Elastohydrodynamic lubrication in a plane slider bearing

Summary The present work deals with the case of a two-dimensional slider bearing with a rigid pad and an elastic bearing. Fluid viscosity is assumed to be only a pressure function. We determined the bearing deformation, the pressure distribution and the load capacity at different values of the inclination angle of the slider, with a numerical integration of the system consisting of the elasticity and Reynolds equations. The results show that, with an iso-viscous fluid, bearing elasticity causes a load capacity decrease. Instead bearing elasticity together with the variation of fluid viscosity due to pressure causes a load capacity greater than that of the iso-viscous case (=0).

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Sommario Il presente lavoro studia il problema della coppia prismatica lubrificata con pattino rigido di allungamento infinito e cuscinetto deformabile; si suppone che la viscosità del fluido sia funzione della sola pressione. Il sistema di equazioni, costituito dall'equazione di Reynolds e dall'equazione dell'elasticità, è stato risolto numericamente, determinando la deformazione del cuscinetto, andamento della pressione e la capacità di carico per diversi valori dell'inclinazione del pattino. I risultati dimostrano che, con fluido isoviscoso, la deformabilità del cuscinetto determina una riduzione della capacità di carico. Se si considera, invece, effetto combinato dell'elasticità del cuscinetto e della variazione della viscosità del fluido, la capacità di carico risulta maggiore di quella che si ottiene con fluido isoviscoso (=0).

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Nomenclature /L - /L - x/L - x/L - - ¯C CZ/h ₁ - E elasticity modulus - h film thickness - H elastic deformation of the bearing - h ₁ minimum film thickness - h ₂ inlet thickness - inclination of the pad - h Z/h ₁ - HZ/h ₁ - L pad length - viscosity - ₀ viscosity with no over-pressure - p over pressure - p P _ec-P _rc where:ec=elastic caserc=rigid case - P h ₁ ² /6₀VL - h ₂/h ₁=1+L/h ₁ - FV bearing velocity - W load capacity per unit width - Wh ₂ ¹ /6₀ VL ² - Z E h ₃ ¹ /12 ₀ VL ² A first version of this paper was presented at the 7th National AIMETA congress, held at Trieste, October 2–5, 1984. This work was supported by C.N.R.     

Mixed convection flow in narrow vertical ducts

The mixed convection flow in a vertical duct is analysed under the assumption that , the ratio of the duct width to the length over which the wall is heated, is small. It is assumed that a fully developed Poiseuille flow has already been set up in the duct before heat from the wall causes this to be changed by the action of the buoyancy forces, as measured by a buoyancy parameter . An analytical solution is derived for the case when the Reynolds numberRe, based on the duct width, is of 0 (1). This is extended to the case whenRe is 0 (^–1) by numerical integrations of the governing equations for a range of values of representing both aiding and opposing flows. The limiting cases, || 1 andR=Re of 0 (1), andR and both large, with of 0 (R ^1/3) are considered further. Finally, the free convection limit, large with R of 0 (1), is discussed.

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Mischkonvektion in engen senkrechten Rohren

Zusammenfassung Mischkonvektion in einem senkrechten Rohr wird unter der Voraussetzung untersucht, daß das Verhältnis der Rohrbreite zur Länge, über welche die Wand beheizt wird, klein ist. Es wird angenommen, daß sich bereits eine voll entwickelte Poiseuille-Strömung in dem Rohr eingestellt hat, bevor Antriebskräfte, gemessen mit dem Auftriebsparameter , aufgrund der Wandbeheizung die Strömung verändern. Es wird eine analytische Lösung für den Fall erhalten, daß die mit der Rohrbreite als charakteristische Länge gebildete Reynolds-ZahlRe konstant ist. Dies wird mittels einer numerischen Integration der wichtigsten Gleichungen auf den FallRe =f (^–1) sowohl für Gleich- als auch für Gegenstrom ausgedehnt. Weiterhin werden die beiden Grenzfälle betrachtet, wenn || 1 undR=Re konstant ist, sowieR und beide groß mit proportionalR ^1/3. Schließlich wird der Grenzfall der freien Konvektion, großes mit konstantem R, diskutiert.

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Nomenclature g acceleration due to gravity - Gr Grashof number - G modified Grashof number - h duct width - l length of the heated section of the duct wall - p pressure - Pr Prandtl number - Q flow rate through the duct - Q ₀ heat transfer on the wally=0 - Q ₁ heat transfer on the wally=1 - Re Reynolds number - R modified Reynolds number - T temperature of the fluid - T ₀ ambient temperature - T applied temperature difference - u, velocity component in thex-direction - v, velocity component in they-direction - x, co-ordinate measuring distance along the duct - y, co-ordinate measuring distance across the duct - buoyancy parameter - ₀ modified buoyancy parameter, ₀=R ^–1/3 - coefficient of thermal expansion - ratio of duct width to heated length, =h/l - (non-dimensional) temperature - _w applied temperature on the wally=0 - kinematic viscosity - density of the fluid - ₀ shear stress on the wally=0 - ₁ shear stress on the wally=1 - stream function     

Experimentelle und numerische Untersuchung laminarer,axisymmetrischer Freistrahlen mit und ohne Auftrieb

Zusammenfassung Das Verhalten axisymmetrischer, laminarer und inkompressibler Freistrahlen mit und ohne Auftrieb in einer homogenen Umgebung wird experimentell und numerisch untersucht. Die dazu erstellte Versuchsanlage wird kurz beschrieben. Charakteristische Grö\en von Fluidstrahlen ohne Auftrieb lassen sich unter Beachtung der beschreibenden Kenngrö\en parameterfrei darstellen. Auftriebsbehaftete Fluidstrahlen werden durch drei Parameter, die Reynoldszahl, die Grashofzahl und die Prandtlzahl vollständig beschrieben. Die Einflüsse der einzelnen Grö\en werden anhand der numerischen Lösung diskutiert, welche ihrerseits mit asymptotischen Verfahren kontrolliert wird. Die experimentellen Ergebnisse stimmen mit den berechneten Werten sehr gut überein. Die Versuche zur Stabilität laminarer Strahlen lassen sich gut mit einem Impulsstromparameter korrelieren.

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Experimental and numerical study of laminar, axisymmetrical jets with and without buoyancy

Axisymmetrical, laminar and incompressible jets with and without buoyancy in homogeneous surroundings are investigated experimentally and numerically. The experimental set up is described. Characteristics of jets without buoyancy are presented in a parameterless form. Buoyancy — induced jets are completely determined by three parameters, the Reynolds-Number, the Grashof-Number and the Prandtl-Number. The influence of the characteristic numbers to the numerical solution is discussed. On the other hand this result is controlled by analytical solutions. The experimental results are in good agreement with the predicted values. The experiments for stability of laminar jets are correlated with a parameter of momentum.

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Formelzeichen a Temperaturleitfähigkeit - a₁...a₄ Polynomkoeffizienten - b Breite - c_p spez. Wärmekapazität bei konstantem Druck - D Durchmesser - E kinetische Energie - g Erdbeschleunigung - l Länge - L laminare Lauflänge - m Massenstrom - p Parameter - Q Energiestrom - R Radius - T Temperatur - Geschwindigkeitsvektor - dimensionsloser Geschwindigkeitsvektor - U Parameter - Längenvektor - x_o Korrekturlänge - X Parameter - \ thermischer Ausdehnungsbeiwert - Grenzschichtdicke - dynamische Viskosität - T=T_o-T übertemperatur - =(T-T)/T dimensionslose Temperatur - kinematische Viskosität - dimensionsloser Längenvektor - Dichte - Stromfunktion - Funktion - dimensionslose Stromfunktion Indizes A au\en - I innen - m Werte auf Symmetrieachse - Th Thermoelement - – Mittelwert - o Düsenaustrittsgrö\en - Umgebung - Vektor - * dimensionslose Grö\en Kenngrö\en Re =u_o·R/v Reynoldszahl (Radius) - Reynoldszahl (Durchmesser) - Pr=/a Prandtlzahl - Pe =u_o·R/a Pécletzahl - Grashofzahl     

Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

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     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

A fourfold generalization of Peaucellier's inversion cell

New inversors are proposed which are generalizations of the well-known inversor of Peaucellier. It appears that a kite in the Peaucellier cell is replaceable by an arbitrary 4-bar linkage (abhk) whereas the direction and length of the straight line, produced by the inversor, can be manipulated through the particular choice of the relative polar coordinates of a vertex A of the triangular input link. Formulas are derived for practical inversors with a revolving input link. The ones selected are basically governed by the choice of two transmission angles, ₁ and ₃, by the length L of the acquired line, as well as by its direction represented by the angle /2- comprised between the line L and the frame.

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Sommario Vengono proposti nuovi inversori quali generalizzazioni del ben noto inversore di Peaucellier. Si mostra che un quadrilatero isoscele nella cella di Peaucellier è sostituibile da un'arbitraria connessione a quattro barre, mentre la direzione e la lunghezza della retta, prodotta dall'inversore, possono essere manipolate con la particolare scelta delle relative coordinate polari di un vertice A del collegamento triangolare di input. Vengono derivate formule per inversori funzionali con un collegamento di input rotante. Quelli solezionati sono principalmente governati dalla scelta di due angoli di trasmissione, ₁ e ₃, dalla lunghezza L della linea ottenuta, così come dalla sua direzione rappresentata dall' angolo /2- compreso tra la linea L ed il riferimento.

     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/     

The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

Summary A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.Nomenclature c /, ft/sec - e base of natural logarithms - F(x, y) integration function, equation (38) - (x) mean value of F(x, y) - g local acceleration of gravity, ft/sec² - h width of conduit, ft - k (2m–1)^{2 2} L/h ² c, equation (35) - k* 12L/h ² c, frictional resistance parameter, equation (46) - L length of conduit, ft - m positive integer - n positive integer - p pressure, lb/ft² - p ₀ constant pressure at inlet of conduit, lb/ft² - P pressure plus elevation head, p+gz, equation (4) - mean value of P over the conduit width h - P ₀ p ₀+gz ₀, lbs/ft² - R frictional resistance coefficient for steady state wall shear, lb sec/ft⁴ - s positive integer; also, condensation, equation (6) - t time, sec - t ct/L, dimensionless time - u x component of fluid velocity, ft/sec - u _m mean velocity in conduit, equation (12), ft/sec - u ₀(y) velocity profile in Poiseuille flow, equation (19), ft/sec - transformed velocity - U initial mean velocity in conduit - x distance along conduit, measured from valve (fig. 1), ft - x x/L, dimensionless distance - y distance normal to conduit wall (fig. 1), ft - y y/h, equation (25) - z elevation, measured from arbitrary datum, ft - z ₀ elevation of constant pressure source, ft - isothermal bulk compression modulus, lbs/ft² - _n , equation (37) - _n (2n–1)/2, equation (36) - viscosity, slugs/ft sec - / = kinematic viscosity, ft²/sec - density of fluid, slugs/ft³ - ₀ density of undisturbed fluid, slugs/ft³ - ø angle between conduit and vertical (fig. 1) The research upon which this paper is based was supported by a grant from the National Science Foundation.     

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     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.

     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

The dynamic and steady shear properties of synovial fluid and of the components making up synovial fluid

Steady-shear and dynamic properties of a pooled sample of cattle synovial fluid have been measured using techniques developed for low viscosity fluids. The rheological properties of synovial fluid were found to exhibit typical viscoelastic behaviour and can be described by the Carreau type A rheological model. Typical model parameters for the fluid are given; these may be useful for the analysis of the complex flow problems of joint lubrication.The two major constituents, hyaluronic acid and proteins, have been successfully separated from the pooled sample of synovial fluid. The rheological properties of the hyaluronic acid and the recombined hyaluronic acid-protein solutions of both equal and half the concentration of the constituents found in the original synovial fluid have been measured. These properties, when compared to those of the original synovial fluid, show an undeniable contribution of proteins to the flow behaviour of synovial fluid in joints. The effect of protein was found to be more prominent in hyaluronic acid of half the normal concentration found in synovial fluid, thus providing a possible explanation for the differences in flow behaviour observed between synovial fluid from certain diseased joints compared to normal joint fluid.Nomenclature A Ratio of angular amplitude of torsion head to oscillation input signal - G Storage modulus - G Loss modulus - I Moment of inertia of upper platen — torsion head assembly - K Restoring constant of torsion bar - N ₁ First normal-stress difference - R Platen radius - S ⁽ⁱ⁾ Geometric factor in the dynamic property analysis - t ₁ Characteristic time parameter of the Carreau model - X, Y Carreau model parameters - Z () Reimann Zeta function of - Carreau model parameter - Shear rate - Apparent steady-shear viscosity - ^* Complex dynamic viscosity - Dynamic viscosity - Imaginary part of the complex dynamic viscosity - ₀ Zero-shear viscosity - ₀ Cone angle - Carreau model characteristic time - Density of fluid - Shear stress - Phase difference between torsion head and oscillation input signals - ₀ Zero-shear rate first normal-stress coefficient - Oscillatory frequency     

Development of a full-field planar Mie scattering technique for evaluating swirling mixers

A full-field planar optical diagnostic technique for studying mixing in swirling flows is described. Results were obtained using this technique to provide planar mixing information by seeding a simulated fuel stream with aluminum oxide particles, then inferring concentration from Mie scattering intensity distributions. This facility and measurement technique are unique for several reasons. First, they allow spatial variations in laser sheet energy to be corrected for on a shot-to-shot basis. Second, they allow experiments to be performed for swirlers with practical fuel and oxidizer flow rates, i.e. on the order of 150 g/s (0.33 lbm/s). Finally, they allow full size swirler models to be evaluated, with the entire exit plane imaged simultaneously. Representative results are presented as false color images of the planar mixing fields. These images allow rapid assessment of the mixing process and its changes with variations in operating conditions or swirler geometry.List of Symbols C seed particle concentration, m^–3 - mean component of seed particle concentration, m^–3 - C fluctuating component of seed particle concentration, m^–3 - C ^* time averaged ratio of rms particle concentration fluctuations to average particle concentration, dimensionless - d _p particle diameter, m - I laser energy after passing through the flow, J/m² - mean laser energy, J/m² - I ₀ laser energy before passing through the flow, J/m² - L _v eddy length scale, m - l laser beam path length, m - U _v eddy velocity scale, m/s - V diode voltage reading after passing through the flow, V - mean diode voltage, V - V ₀ diode voltage reading before passing through the flow, V - absorptivity, m² - _rel relative equivalence ratio, dimensionless - fluid viscosity, Ns - _p particle density, kg/m³ - Stokes number= _p / _f , dimensionless - _f flow time scale, s - _p particle response time, s     

Eine umfassende Modelltheorie für Kunststoff-Einschneckenmaschinen

Zusammenfassung In dieser Arbeit wird der Versuch unternommen, eine umfassende Modelltheorie für Kunststoff-Einschneckenmaschinen zu entwickeln. Ausgegangen wird vom analytischen Ähnlichkeitsprinzip.Wesentliche Merkmale der vorgestellten Theorie sind: a) das Energieprinzip, d. h. die dimensionslosen Kennzahlen werden aus Energiebilanzen hergeleitet, b) die strömungstechnische Ähnlichkeit wird durch die Konstanz des Drosselquotienten beschrieben, c) für die schergeschwindigkeitsabhängige Viskosität wird ein Potenzgesetz zugrunde gelegt, d) abgesehen von zwei Ausnahmen wird von einem konstantenL/D-Verhältnis ausgegangen.Es stellte sich heraus, daß man den bisher empirisch ermittelten Exponenten des Gangtiefenverhältnisses theoretisch ermitteln kann und daß dieser Exponent als dimensionslose rheologische Kennzahl deutbar ist. Mit dieser stoffabhängigen Kennzahl, die aufgrund bestimmter Voraussetzungen modifiziert werden kann, und dem dazugehörigen Drehzahlmodellgesetz lassen sich innerhalb bestimmter Grenzen für die übrigen Maschinen- und Verfahrensparameter Modellgesetze herleiten. Hier kann man zwischen einem universellen und zwei speziellen Modellgesetztypen, sowie deren Modifikationen aufgrund unterschiedlicher geometrischer Bedingungen unterscheiden.

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Summary In this paper it is attempted to develop a comprehensive model theory for polymer single-screw machines proceeding from the analytical similarity principle.The essential points of the presented theory are: a) the energy principle, i.e. the dimensionless parameters are derived from energy balances, b) that the flow-characteristical similarity is described using a constant value of the throttle quotient, c) that a power-law approximation is taken as a basis for the shearrate dependent viscosity, d) a procedure using a constantL/D-ratio except for two special cases.It appeared that the until now empirically determined exponents of the channel depth ratios may be determined theoretically and that this exponent is a meaningful dimensionless rheological parameter. With this material-dependent parameter which may be modified on the basis of definite assumptions, and the related speed of rotation model law, model laws may be derived in some range of validity of the remaining machine and process parameters. Here it is distinguished between a universal and two special modellaw types including their modifications owing to different geometrical conditions.

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Symbole A Betriebspunkt - a Drosselquotient - C Konstante - c _p spezifische Wärme - D Schneckendurchmesser - D _z Zylinderdurchmesser - H Heizleistung bzw. Kühlleistung der Ausstoßzone - H Heizleistung bzw. Kühlleistung - h ₁ Gangtiefe der Einzugszone - h=h ₂ Gangtiefe der Ausstoßzone - i(Index) Bezeichnung für die einzelnen Schneckenzonen - L Schneckenlänge - L Länge der Ausstoßzone - L _w wirksame Schneckenlänge, Länge der Druckaufbauzone - Massedurchsatz - M d Drehmoment - n Drehzahl - P am Schneckenschaft abgegebene Leistung - P der Ausstoßzone zugeführte Leistung - p Druck in der Trennebene zwischen Werkzeug bzw. Düse und Maschine - Vom Zylinder pro Flächen- und Zeiteinheit zugeführte Wärme - Volumendurchsatz - Volumendurchsatz infolge Druckströmung - Volumendurchsatz infolge Schleppströmung - v Geschwindigkeit - x frei wählbarer Exponent im Drehzahlmodellgesetz - Wärmeübergangszahl - Schergeschwindigkeit - Viskosität - ₁ Extrudattemperatur - Mittlere Massetemperatur der Ausstoßzone - _z Zylinderwandtemperatur - _h Temperaturdifferenz zwischen Masse und Zylinderwand - ( ₁ – ₀) Gesamtmassetemperaturdifferenz - _L=(₁ – ) Massetemperaturdifferenz der Ausstoßzone - Steigung der Viskositätskurve - Wärmeleitfähigkeit - v Steigung der Fließkurve - Steigung der Massetemperatur — Schergeschwindigkeitskurve bei konstanter Viskosität - Dichte - Schubspannung - Gangsteigungswinkel - dimensionslose rheologische Kennzahl Gekürzte Fassung eines Plenarvortrags, gehalten während des 9. Kunststofftechnischen Kolloquiums des IKV in Aachen vom 8.–10. März 1978.Mit 8 Abbildungen und 5 Tabellen     

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