Propagation of a spherical wave in nonlinearly compressible and elastoplastic materials

The problem of spherical wave propagation in soil under the action of an intense uniformly decreasing load ₀(t) applied to the boundary of a cavity with radius r₀ is considered. Soil with a high stress level is modeled either by ideally nonlinearly compressible or elastoplastic material, taking account of linear irreversible unloading for the material. In contrast to [1–7], in order to describe material movement use is made of strain theory [8] with determining functions = (), _i=_i(_i), where , _i, , _i are the first and second invariants of strain and stress tensors. During material loading these functions are presented in the form of polynomials ()=(_i+₂¦¦), _ii)=(_i-_2i)_i, in which constant coefficients _i, _i=1, 2) are determined by experiment, taking account of the triaxial stressed state of soil. Solution of the problem is constructed by an analytically reversible method, with prescribed shape for the shock-wave (SW) surface in the form of a second-degree polynomial relating to time t and a numerical method of characteristics for a prescribed arbitrarily decreasing load _i(t). On the basis of the analytical equations obtained, calculations are carried out for material parameters (including loading profile) in a computer and stresses and mass velocity of plastic and elastoplastic materials are compared.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–100, July–August, 1986.The authors express their sincere thanks to Kh. A. Rakhmatulin for discussing the results of this work.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Stress analysis of the cylinder subjected to arbitrarily distributed axisymmetrical loads

Summary Stress analysis has been carried out for a finite cylinder subjected to arbitrarily distributed axisymmetrical surface loads. Direct stress _x in the axial direction is assumed to be of the form _x = ₀+r ₁ +r ₂ where ₀ to ₂ are functions of x. Using the equations of equilibrium and compatibility the other direct stresses and the shearing stress are expressed by ₁ and ₂. Fundamental equations governing ₁ and ₂ are introduced using the variational principle of complementary energy. From the results of the present analysis it is evident that the boundary conditions can be satisfied completely even for the case where the external forces are specified in complicated form, and that more accurate solutions can easily be obtained by introducing additional terms in _x.

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Spannungsanalyse für den Zylinder unter axialsymmetrischer Last in beliebiger Verteilung

Übersicht Für einen endlichen Zylinder unter axialsymmetrischer Oberflächenlast in beliebiger Verteilung werden die Spannungen ermittelt. Die Normalspannung in Axialrichtung wird in der Form _x = ₀+r ₁ +r ₂ angesetzt mit ₀, ₁, ₂ als Funktionen von x. Mit Hilfe der Gleichgewichtsund Verträglichkeitsbedingungen werden die anderen Normalspannungen und die Schubspannung durch ₁ und ₂ ausgedrückt. Über das Variationsprinzip für die Komplementärenergie werden die grundlegenden Gleichungen für ₁ und ₂ eingeführt. Die Ergebnisse zeigen, daß die Randbedingungen selbst für komplizierte Belastungsarten vollständig erfüllbar sind und mit zusätzlichen Termen in _x mühelos noch genauere Lösungen bestimmt werden können.

     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

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     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Formulation of Gyarmati's principle for heat conduction equation

Gyarmati's principle is formulated in various pictures for the heat conduction phenomenon in solid. Since the heat current density and the internal energy function can be given in three different pictures for heat conduction phenomena, we get the nine forms of the principle from which the heat conduction equation can be derived. This formulation has been shown using the generalized picture. In the subsequent section the principle is formulated in proper picture from which three proper pictures namely Fourier, entropy and energy follow.

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Formulierung des Prinzips von Gyarmati für Wärmeleitprobleme

Zusammenfassung Das Prinzip von Gyarmati wird in verschiedenen Arten für Wärmeleitphänomene formuliert. Da die Wärmestromdichte und die innere Energie in drei verschiedenen Arten für Wärmeleitphänomene angegeben werden können, erhalten wir die neun Formen des Prinzips, von denen die Wärmeleitgleichung abgeleitet werden kann. In dieser Formulierung wird ein verallgemeinertes -Bild verwendet. Im folgenden Teil wird das Prinzip in einem geeigneten -Bild formuliert, von dem drei geeignete Bilder folgen, nämlich das Fourier-, das Entropie- und das Energie-Bild.

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Nomenclature rate of entropy production - dissipation potential function of thermodynamic forces only - dissipation potential function of fluxes only - v volume of the system - x _i thermodynamic forces - J _i thermodynamic currents - f number of irreversible processes taking place in the system - L_iK phenomenological coefficients representing conductivity of the material - R_iK phenomenological coefficients representing resistances - density of the material - a specific value of the extensive transport quantity - _i state parameters, the gradients of which give rise to the thermodynamic forces - _i source density of a_i - s specific entropy - T absolute temperature - J _q heat current density vector - heat conductivity coefficient - L_qq phenomenological coefficient corresponding to heat conductivity coefficient - x _q thermal dissipative force - _q entropy production due to heat transfer - u specific internal energy - L phenomenological coefficient in picture - c_V specific heat at constant volume     

Boundary layer stability on a rotating disk with corotation of the surrounding fluid

The stability of the steady self-similar flow in the boundary layer on a rotating disk of infinite radius with corotation of the surrounding fluid is analyzed by the normal mode method. The spectral problem for infinitesimal three-dimensional disturbances is solved by a collocation method with expansion of the amplitude functions in Chebyshev polynomials. It is established that for all values of the parameter 0, equal to the ratio of the angular velocities of the fluid and the disk, the lower critical Reynolds number is determined byA-type, waves, whose development is governed by the parallel instability mechanism typical of an Ekman layer. TheB-type instability, associated with the presence of an inflection point on the velocity profile, disappears when 4. The neutral surfaces are calculated for Karman flow (=0) and Bödewadt flow (). It is found that in Karman flowA-type waves may grow at values of the Reynolds number several times smaller than the critical Reynolds number for spiral vortices. The results of the analysis are compared with the available experimental data.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 69–77, September–October, 1992.     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

Zur Variationsformulierung nichtlinearer Randwertprobleme

Übersicht Es werden verschiedene Bedingungen aufgestellt, die es erlauben, die durch die beiden (Systeme von) nichtlinearen DifferentialgleichungenA (u, ) = q, B (u, ) = und Randbedingungen zusammen mit den nichtlinearen algebraischen Relationenq = C(u, ), = D(u, ) beschriebene Aufgabe durch äquivalente Variationsprobleme zu ersetzen. Dabei zeigt sich ein enger Zusammenhang mit den in der Festkörpermechanik wohlbekannten Prinzipien der virtuellen Verschiebungen und der virtuellen Kräfte. Die auf systematischem Weg konstruierten Variationsfunktionale enthalten viele in der Physik bekannte Funktionale als Sonderfälle, insbesondere jene, die in der Elastomechanik nach Green, Castigliano, Hellinger, Reißner, Hu und Washizu benannt werden.

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Summary In this paper there are established various conditions which allow a variational formulation of the problem described by the two (systems of) nonlinear differential equationsA(u, ) = q, B(u, ) = and boundary conditions together with the nonlinear algebraic relationsq = C(u, ), = D(u, ). Besides a close relationship is revealed to the principles of virtual displacements and virtual forces which are wellknown in solid mechanics. The systematically constructed variational functional contain many functionals in physics as special cases, mainly those of Green, Castigliano, Hellinger, Reißner, Hu and Washizu in elastomechanics.

     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

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 _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points 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 - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

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Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

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Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche     

The moving, plane heat source in an elastic medium

Summary This note presents an exact solution for the stress and displacement field in an unbounded and transversely constrained elastic medium resulting from the motion of a plane heat source travelling through the medium at constant speed in the direction normal to the source plane.Nomenclature mass density - diffusivity - thermal conductivity - Q heat emitted by plane heat source per unit time per unit area - speed of propagation of plane heat source - shear modulus - Poisson's ratio - T temperature - _x, _y, _z normal stress components - u _x, u_y, u_z displacement components - c speed of irrotational waves - t time - x, y, z Cartesian coordinates - =x–vt moving coordinate     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

Über eine Anwendung der Hillschen Minimalbedingung in der Plastizitätstheorie

Zusammenfassung Auf dem gezeigten Weg wurden die Spannungen _r , , _z berechnet, wobei an Stelle der Veränderlichen r und die dimensionlosen Größen x _i = r _i /, x=r/ und x _a = r/ in die Rechnung eingeführt wurden. Die Funktion (r, ) wurde dann für den Bereich 0,45x_i1,0, 1x_a 2 tabuliert. Hierbei zeigte sich, daß der Rechenaufwand bei der Durchrechnung eines Einzelbeispiels nach der Charakteristikenmethode wesentlich geringer ist. Bei der Anlage von Zahlentafeln zur Berechnung von Spannungen für beliebige Durchmesserverhältnisse ergab sich, daß der aufgezeigte Wege zu geringerem Rechenaufwand führt. Für das Beispiel r _i /r _a=1/2 wurden die Rohraufweitungen bestimmt und diese Werte noch durch praktische Versuche nachgeprüft. Hierbei ergab sich, daß die theoretisch bestimmten Rohraufweitungen in dem Streubereich der gemessenen Rohraufweitungen lagen, wobei Messungen an drei Rohren aus demselben Material und demselben Rohrverhältnis durchgeführt wurden. Insbesondere stimmten die theoretischen Rohraufweitungen auch mit den gemessenen Rohraufweitungen überein, wenn das Rohr entlastet wurde und die Restdeformationen bestimmt wurden.Daraus kann geschlossen werden, daß durch die berücksichtigte lineare Verfestigung die tatsächlichen Verhältnisse außerordentlich gut erfaßt werden.Der sogenannte Platzdruek eines Rohres kann auf rein rechnerischem Weg nicht erfaßt werden, da für =r_a die geometrische Gestalt des Rohres instabil wird. Bei den Versuchen zeigt sich, wenn der Innendruck über p _i ( =r **** _a ) gesteigert wird, daß das Rohr schon bei geringen Überschreitungen aufzubauchen beginnt.Meinem Lehrer Herrn Prof. Dr. Dr. R. Grammel zum 65. Geburtstag gewidmet.     

Periodic axisymmetrical laminar flow of second-order liquids in cylindrical pipes of circular cross section

Summary After a few basic remarks, equations of unsteady slow of second-order incompressible fluids are given for the axifymmetrical case. The solution is worked out to give the expression of velocity and discharge, when the hydraulic slope represents any periodic function of time; the average energy dissipated in a period is also calculated. Particularly, when the hydraulic slope reduces to the expression a sin t+b cos t only, the dependence of pressure on r is also given, and it is recognized and emphasized that some components of the stress tensor oscillate with a frequency which is twice the frequency of motion.

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Sommario Dopo brevi richiami, si scrivono le equazioni del moto vario per un fluido incomprimibile del secondo ordine, nel caso di simmetria assiale. Viene trovata la soluzione, esprimendo la velocità e la portata, nel caso in cui la pendenza motrice sia una qualsiasi funzione periodica del tempo; si calcola anche l'energia dissipata mediamente in un periodo. Nel caso particolare in cui la pendenza motrice si riduca all'espressione a sin t+b cos t, si dà la dipendenza della pressione da r, riconoscendo e mettendo in evidenza che alcune componenti del tensore degli sforzi oscillano con frequenza doppia rispetto a quella del moto.

     

Flow of viscous and viscoelastic fluids through and around porous bodies

The slow flow of a viscous fluid through and around porous spheres is considered. The numerical simulation uses a special mixture of computational techniques: quadratic approximation and expansion in power series. The resulting calculations predict the evolution of the main features of the flow if the boundary conditions are varying, particularly if the tangential velocity is neglected or if a viscous filtration velocity is assumed at the sphere surface. The cases of full and hollow spheres with uniform and non uniform permeabilities are considered, the external impermeable walls of the flow being concentric spheres or cylinders. Some influence of viscoelastic properties of the fluid is also given.Nomenclature AA _n , A_n, B_n, b_n, C_n, c_n, D_n constants of integration - C _n (t) Gegenbauer functions with degree n and order –1/2 - e shell thickness - K, K* permeability - P _n (t) Legendre functions - Q _v volumetric rate of flow - p, p ₀, p _e pressure, far away pressure, average pressure - R* sphere radius - r, spherical coordinates - Re Reynolds' number (see equation 37) - s, t sinus and cosinus - V ₀ ^* uniform velocity - v velocity component - We Weissenberg's number (see equation (37)) - permeability coefficient - thickness coefficient - structural coefficient - diameter ratio sphere-cylinder - * dynamic viscosity of the fluid - stream functions - normal stress ( _rr ) - tangential stress ( ) - ₀ ^* relaxation time of the fluid     

Cavity nucleation, growth and stress field in elastic-plastic medium

A finite element method is used to study the cavity nucleation and growth in an elastic-plastic medium. The critical cohesion strength, _c at IPM (interface between second phase particle and matrix) is employed as the criterion of cavity formation. Three different values of _c are taken to examine their influence on the overall mechanical behaviour and process of cavity formation.