On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

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     

Laminar flow of some inelastic time-dependent fluids in pipes

The paper presents some modification of the previously published model of thixotropic fluids for the case of fluids with irreversible destruction of the structure. In the second part of the paper the laminar flow of this kind of fluids in pipes is discussed. A simple method of determining the initial value of the structural parameter at the inlet to the pipe is proposed. D pipe diameter, m - k ₁,k ₂ rheological parameter in eq. (5), Nsⁿ/m² - L pipe length, m - m ₁,m ₂ rheological parameter in eq. (5) - n ₁,n ₂ rheological parameter in eq. (6) - p pressure, Pa - v mean linear velocity in the pipe, m/s - rheological parameter in eq. (2), = 1 s - shear rate, s^–1 - structural parameter - ₀ initial value of structural parameter - ₁, ₂ mean value of natural time for breakdown and build-up of the structure, respectively, s - shear stress, Pa - fluid density, kg/m³ - mean value of the friction factor - De modified Deborah number defined by eq. (11) - Re _m generalized Reynolds number defined by eq. (16) - Re _n generalized Reynolds number defined by eq. (10) - Se structural numbers defined eq. (12)     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Use of genetic algorithms for the design of rotors

The problem of finding an optimum shape for rotating discs is a classical one and has received considerable attention; in the present paper an attempt to use genetic algorithms is described.The problem of finding the constant stress profile by using genetic algorithms is tackled, firstly using the well known results of conventional methods. The problem of optimizing the shape of pierced discs is then attempted with results which are affected by strong stress concentrations, owing to simplifying assumptions in the stress analysis implicit in the so-called disc theory. This drawback is exactly the same which limits the usefulness of conventional solutions. In order to overcome this problem, particular formulations of the fitness function aimed to discourage shapes leading to strong stress concentrations are introduced, showing that profiles which lead to very favourable stress patterns when analysed using tri-dimensional methods can be obtained even with an optimization procedure essentially based on the disc theory. A profile of a disc flywheel with the rim and hub obtained using the genetic approach shows the results which can be obtained using this optimization technique.

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Sommario Il problema di ricavare il profilo ottimale per un disco rotante ha ricevuto in passato considerevole attenzione e soluzioni analitiche sono conosciute da circa un secolo. Scopo del presente lavoro è quello di illustrare un tentativo di soluzione basato sugli algoritmi genetici.Per prima cosa è affrontato il problema della definizione del disco di uniforme resistenza, ottenendo un profilo che coincide con la ben nota soluzione descritta in letteratura. Viene poi affrontata l'ottimizzazione di dischi forati, ottenendo profili che portano a forti concentrazioni di tensione, dovute alle ipotesi semplificative usate per il calcolo dello stato di tensione, generalmente note come teoria dei dischi. Per superare questo problema, che limita peraltro l'utilità delle soluzioni classiche, sono state introdotte formulazioni delle funzioni obiettive che scoraggiano profili che portano a forti concentrazioni di tensione. Si mostra cosi come sia possibile ottenere profili che, analizzati con procedure di calcolo tridimensionale, portano a distribuzioni di tensione molto favorevoli, anche operando con procedure di ottimizzazione basate essenzialmente sulla teoria dei dischi. Un volano a disco con mozzo e corona ottenuto mediante l'approccio genetico mostra i risultati che possono essere ottenuti attraverso questa tecnica di ottimizzazione.

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Symbols e kinetic energy - h thickness at radiusr - m mass - r radial coordinate - r _o outer radius - B parameter for constant stress profile - I _p performance index - difference between the maximum and minimum values of - density - stress - _c circumferential stress - _e equivalent stress - _r radial stress - nondimensional radial coordinate (=r/r _o ) - angular velocity     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Take-off threshold velocity of solid particles lying under a turbulent boundary layer

The take-off of free solid particles by wind erosion has been investigated. A preliminary bibliographic study (Foucaut and Stanislas 1995) has enabled an analysis of the Bagnold criterion (1941) to be made and its accordance with the results obtained by White (1982) to be shown, thus leading to a semi-empirical model. This initial study led to a more judicious choice of the primary parameters, allowing a more physical representation of the threshold velocity. In the present study, the criterion validation was carried out in a specific boundary layer wind tunnel, by means of a direct measurement of the threshold velocity. The basic idea was to increase the wind tunnel velocity slowly and linearly and to perform an optical detection of the first take-offs.List of symbols D _p particle diameter - D _Pref reference diameter - _p D _p /D _pref - g gravity - H shape factor = (/) - h ⁺ roughness parameter ( =u D _p /2) - R _f =u D _p / - R _x u _e x/v - û u _e /v - u ⁺ flow velocity ( =u/u ) - u _e external velocity - u _elim external threshold velocity - û _e u _e /u _elim - u friction velocity - u ^* threshold friction velocity - u _tref reference velocity - ^* u ^*/u ^ref - y ⁺ yu / - _p ( _p –)/ - boundary layer thickness - displacement thickness - dynamic viscosity - v kinematic viscosity - momentum thickness - fluid density - _p particle density - standard deviation of particle diameter - shear stress Research carried out at Ecole des Mines de Douai     

Tensiometer reaction related to its filter dimensions

A theoretical model for tensiometers is presented. It is based on a new physical considerations: the tensiometer filter is a quasi-saturated porous medium and the transmission fluid in the cavity is in hydrostatic equilibrium and is incompressible. The evolution equations form a complete system which could be used and coupled in a wide number of situations once filter dimensions and geometry have been correctly defined. The model is applied to tensiometer design and leads to new design recommendations. It predicts the existence of two distinct evolution modes for tensiometers. The time constant of the first varies linearly with the ratio of filter thickness to contact area and that of the second varies according to the square of the filter thickness and is independent on the contract area. The model leads to the formulation of an equation for fine-filter tensiometers. This extends Richards and Neal's equation by taking fine-filter geometry and gravity into account.Nomenclature A area of surfaceS _i - A _n ,B _n coefficients defined in Appendix B - C filter capacity - da boundary integration element - g constant gravity vector field - K permeability - L filter thickness - M _f mass of transmission fluid exchanged for a unit variation of the potential - M _mn components of a matrix defined in Appendix B - n porosity - n outward unit vector to filter rim (boundary) - N number of terms (Appendix B) - p pressure - P _i ,p _e internal, external pressure - q volume flux - r variable defined in Appendix B - r _n coefficients defined in Appendix B - S global sensitivity and gauge sensitivity - S _i ,S _e ,S _r filter rim - S _f saturation of filter - t time - U velocity of the transmission fluid in the cavity - V volume of filter - V _c volume of cavity - V _p volume of parasitic fluid - x positional vector - z spatial coordinate Greek Letters _p compressibility of the parasitic fluid - potential - _e potential outside of tensiometer - _i potential inside of tensiometer - ₀ initial potential - _p potential of parasitic fluid - adimensional parameter defined in (5.8) - conductance - dynamic viscosity - pi - density of transmission fluid - _p density of parasitic fluid - temporal parameter and time constant - adimensional temporal coordinate - adimensional spatial coordinate Symbols gradient operator - a·b scalar product ofa andb - a×b vector product ofa andb - partial derivative of with respect to - partial derivative of with respect to - mean geometrical value of _e(t,x) defined in (4.7) - x V x belongs toV     

Effect of radial and rotational lubricant inertia on the pressure distribution in externally pressurised thrust bearings

Expressions are obtained for the pressure distribution in an externally pressurised thrust bearing for the condition when one bearing surface is rotated. The influence of centripetal acceleration and the combined effect of rotational and radial inertia terms are included in the analysis. Rotation of the bearing causes the lubricant to have a velocity component in an axial direction towards the rotating surface as it spirals radially outwards between the bearing surfaces. This results in an increase in the pumping losses and a decrease in the load capacity of the bearing. A further loss in the performance of the bearing is found when the radial inertia term, in addition to the rotational inertia term is included in the analysis.Nomenclature r, z, cylindrical co-ordinates - V _r, V , V _z velocity components in the r, and z directions respectively - U, X, W representative velocities - coefficient of viscosity - p static pressure at radius r - p mean static pressure at radius r - Q volume flow per unit time - 2h lubricant film thickness - density of the lubricant - r ₂ outside radius of bearing = D/2 - angular velocity of bearing - R dimensionless radius = r/h - P dimensionless pressure = h ³ p/Q - Re channel Reynolds number = Q/h     

Formulation of electro-chemico-osmotic processes in soils

Electrokinetic techniques have been used for various purposes including consolidation of soils, dewatering of sludges, and hazardous waste remediation among others. Estimating the feasibility of employing electro-osmosis in a particular operation depends on the ability to predict the outcome under a variety of conditions. Predictions of this type are frequently facilitated by the use of a mathematical model designed to represent the physical system under consideration in a rigorous fashion. First, a review of fundamental aspects of electro-chemico-osmotic flow in soils is presented. Following a brief outline of previous studies, identification and quantification of the significant processes, and the construction of mathematical representations are given. This is achieved using an approach based on the macroscopic conservation of mass equations and the principle of a continuum, in contrast to an approach based on the irreversible thermodynamics of coupled flows. Special emphasis is given to coupling effects on transport processes. A complete model and associated boundary conditions are then obtained for electrokinetic processes in a compressible porous medium. The proposed model takes into consideration the migration of a contaminant plume in a flow field generated by an applied electric potential.Symbols a _v soil compressibility - A an entity - C _w mass fraction of water component in the water phase - C _s mass fraction of chemical component in the water phase - C ^* capacitance of the porous medium per unit volume of porous volume - D mechanical dispersion coefficient - D _fw ^ps hydrodynamic diffusion tensor for the chemical component in the water phase - D _fw ^pw hydrodynamic dispersion coefficient for the water component in the water phase - D _f( )/Dt material derivative with respect to an observer moving at the water phase velocity V _f - D _s( )/Dt material derivative with respect to moving solids - e void ratio - f a function - F = 0 equation of a moving boundary - g gravitational acceleration - k permeability tensor of the porous medium - k _e coefficient of electro-osmotic permeability - k _ec coefficient of migration potential - k _hc chemico-osmotic coupling coefficient - m _i number of moles of the ith component - m _i0 number of moles of the ith component at a reference level - n porosity - p pore pressure - p _oi pore pressure at a reverence level - q specific discharge of water phase - q _e current density - q _fe ^p0 constant current density applied at a boundary - q ₀ constant flow rate - q _r specific discharge of the water phase relative to the moving solid matrix - R net mass transfer rate of the chemical component in the water phase - t time - u velocity of a moving surface - _i partial molar density of ith component - V _f velocity of the water phase - V _s velocity of the solid (rate of deformation) - x vertical coordinate - coefficient of matrix compressibility - _p compressibility of water phase in motion - total (overburden) stress tensor - effective stress tensor - _h streaming current conductivity - _e electrical conductivity - electrical potential - _f viscosity of the water phase - _hf density of the water phase     

Experimental and analytical aspects of plasma heat transfer

The effect of an applied electrical potential on heat transfer to a tube immersed in a highly ionized flow of atmospheric pressure Argon plasma is experimentally and analytically determined. Bare copper tubes as well as pyrex coated tubes are utilized and the measured heat fluxes to the locally floating copper surfaces are found to be identical to those of pyrex coated surfaces. This effect is due to unimpeded recombination of electrons with ions on the insulating surface. Comparisons of experiment with analysis indicate that ions diffusing through the thermal and concentration boundary layers surrounding the tube recombine only at the wall (i. e., frozen chemistry prevails in the boundary layers).

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Zusammenfassung Diese Arbeit befaßt sich sowohl theoretisch als auch experimentell mit dem Einfluß eines äußeren elektrischen Feldes auf den Wärmeübergang zu einer Sonde (Röhrchen), die sich in einem hochionisierten atmosphärischen Argon-Plasmastrom befindet. Es werden wassergekühlte, blanke Kupferröhrchen als auch solche, die mit einer dünnen Pyrexglasschicht überzogen sind, als Sonden benutzt. Messungen zeigen denselben Wärmestrom für beide Sondentypen solange die blanke Sonde lokal auf dem Potential einer stromlosen Sonde gehalten wird. Dieses Ergebnis wird auf die ungehinderte Rekombination von Elektronen mit Ionen auf der isolierenden Oberfläche zurückgeführt. Vergleiche zwischen Experiment und Theorie zeigen, daß die durch die thermische und Konzentrationsgrenzschicht diffundierenden Ionen nur auf der Sondenoberfläche rekombinieren, d. h. die Grenzschicht ist chemisch eingefroren.

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Nomenclature e charge of an electron - I electrical current - j electrical current density - j_is ion saturation current density of the probe surface - j_e electron current density at the probe surface - k Boltzmann constant - K thermal conductivity of plasma - K_a thermal conductivity of atoms in plasma - L_a Lewis Number (ratio of ambipolar diffusion coefficient to the thermal diffusion of atoms) - m_i mass of ion - m_e mass of electron - n_e number density of electrons - q actual heat flux - q₀ effective heat flux - Q heat flow to the tube - r radial position - T_e temperature of electrons - T_es temperature of electrons at sheath edge - T_w wall temperature - U work function of surface - V electrical potential of wall - V_f wall floating potential - V_m potential of wall for minimum heat flux - V_i ionization potential - V_i, v_e ion and electron average thermal velocities, respectively - w_i ion drift velocity - X off axis position - degree of ionization - Q heat flow at potential V minus the heat flow at floating potential - q the Abel inversion of Q - _e electron mean free path - _De Debye length - ₀ permittivity of free space Support of this work by the National Science Foundation under Grant GK 15924 is gratefully acknowledged.     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

Relation between viscoelasticity and shear-thinning behaviour in liquids

Summary The shear-thinning behaviour of a liquid is represented in terms of a relaxation time, defined by the ratio ₀/G₀ of initial viscous and elastic constants. The relationship provides a very simple basis for the evaluation of andG ₀ from viscosity/shear data. Results are compared with relaxation times and moduli from primary normal-stress measurement, from stress relaxation and from direct measurement of recoverable shear strain. Good agreement is found but there is experimental evidence the recoverable shear strain _e is related to normal stressN ₁ and shear stress by _e = N₁/3, which does not agree with the theoretical prediction of eitherWeissenberg orLodge.

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Zusammenfassung Das Scherentzähungsverhalten einer Flüssigkeit wird mittels einer Relaxationszeit beschrieben, die durch das Verhältnis der Anfangswerte von Viskosität und Elastizitätsmodul ₀/G₀ definiert ist. Diese Beziehung eröffnet eine einfache Methode zur Bestimmung von undG ₀ aus Scherviskositätsmessungen. Die damit erhaltenen Ergebnisse werden mit Relaxationszeiten und Moduln verglichen, die durch Messung der ersten Normalspannungsdifferenz, der Spannungsrelaxation und der Scherdehnungsrückstellung (recoverable shear strain) gewonnen worden sind. Es wird eine gute Übereinstimmung gefunden, zugleich aber wird der experimentelle Nachweis geführt, daß die Scherdehnungsrückstellung _e mit der ersten NormalspannungsdifferenzN ₁ und der Schubspannung durch die Beziehung _e = N₁/3 verknüpft ist, was sowohl zu der theoretischen Voraussage vonWeissenberg als auch zu derjenigen vonLodge im Widerspruch steht.

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With 10 figures and 1 table     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

The behaviour of a mass-spring system provided with a discontinuous dynamic vibration absorber

Summary The influence of a specific discontinuous dynamic vibration absorber on the motion of a vibrating system is investigated. Attention is paid to the effectiveness in the case of free vibrations and of vibrations due to sinusoidal excitations, where structural damping is also taken into account. For certain configurations numerical results are given.Notation M mass of the vibrating system - m mass of the ball - c spring constant - k structural damping coefficient - g acceleration of gravity - F ₀ amplitude of external force - frequency of external force - 2 distance over which the ball may move - x displacement of centre of tube - y displacement of the ball x and y are measured positive downward from the equilibrium position of the system when no gravitational force would act     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Die maximale Wärmestromdichte beim Behältersieden an einem waagerechten Draht

Zusammenfassung In der vorliegenden Arbeit wurde die maximale Wärmestromdichte für Kältemittel R13, R114, und R115 durch Messungen an einem waagerecht eingespannten Platindraht (d=0,1 mm) bestimmt. Die Messungen erstreckten sich in einem großen Druckbereich (p*=p/p _k=0,005 bis 0,96). Die Meßergebnisse zeigen, daß die relative Druckabhängigkeit vonq _max aus eigenen Messungen am Draht mit der für Rohre recht gut übereinstimmt. Auch die absoluten Werte vonq _max am Draht lassen sich mit einer für Rohre aufgestellten Beziehung gut wiedergeben.

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The maximum heat flux in pool boiling on a horizontal wire

In the present study the maximum heat flux of refrigerants R13, R114 and R115 in pool boiling was obtained experimentally on a horizontal platinum wire (d=0.1 mm). The measurements are performed in a wide pressure range (p*=p/p _c=0.05 to 0.96). The experimental results show that the relative pressure dependence of the maximum heat flux obtained on the wire is the same as that on tubes. Also, the absolute values ofq _max for the wire can be well represented by a relation established for tubes.

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Formelzeichen d Drahtdurchmesser - g örtliche Fallbeschleunigung - h _v Verdampfungsenthalpie - K ₁ Konstante - p Druck - p* normierter Druck (p/p _k) - q Wärmestromdichte - q _max maximale Wärmestromdichte - T thermodynamische Temperatur - Wärmeübergangskoeffizient - Differenz - Celsius-Temperatur - Flüssigkeitsdichte im Sättigungszustand - Dampfdichte im Sättigungszustand - Oberflächenspannung Indices D Draht - F Flüssigkeit - k kritischer Zustand Herrn Prof. Dr.-Ing. E. Hahne zum 60. Geburtstag gewidmet