Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

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Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

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Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

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With 6 figures     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely -invariant measures; (2) given any semiflow , there exists a weak asymptotic pseudotrajectory X of such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for ; and (3) if X is an asymptotic pseudotrajectory for a semiflow , then conditions on that ensure convergence of the empirical measures are derived.     

Nonlinear static behavior of a cantilever subjected to an inclined follower force

The nonlinear static behavior of a linearly elastic cantilever subjected to a nonconservative force of the follower type is formulated and examined. The formulation allows for finite rotations with small strains (the elastica). Exact solutions are found. The investigation is greatly facilitated by means of a phase plane analysis in which the phase plane variables are related to slope angle and bending moment. Some of the interesting and unusual effects occurring in this system are discussed and illustrated with a set of deflection curves for a typical case.Nomenclature x, y coordinates of a point on the deformed elastic axis - slope angle - s arc length - L length of beam (assumed constant) - x _L , y _L , _L values of x, y, at s=L - P applied force - constant angle between P and end tangent - angle between P and the horizontal - EI beam stiffness (assumed constant) - u, dimensionless variables defined by (7) and (8) - c ² load parameter defined by (10) - k, transformation parameters defined by (21) - F(, k), E(, k) elliptic integrals of the first and second kind - argument of elliptic integrals - ₀, ₁ values of at u=0 and u=1 - m, n positive integers - N mode number     

Die Torsionsspannungen in Wellen mit tiefen Längsnuten

Übersicht MitF(x, y) als Spannungsfunktion einer Welle ohne Nut und(, y) als Potentialfunktion des Quelle-Senke-Systems erhält man Spannungsfunktionen(, y) =F(x, y) –(, y) für Wellen mit tiefen Längsnuten. Es wird gezeigt, daß sich damit die Schubspannungen in den Läufern von Schraubenverdichtern ermitteln lassen.

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Shearing stresses in shafts with deep longitudinal grooves

Summary The stress functions(, y) of shafts with deep longitudinal grooves may be represented by(, y) =F(x, y) –(, y) whereF(x, y) is the stress function of a cylindrical shaft without grooves and(, y) denotes the potential function of the source-sink system. It is shown that the shearing stresses in rotors of screw-compressors may be obtained in this way.

     

Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties

A method of analysis is described which yields quasianalytical solutions for one and multidimensional unsteady heat conduction problems with linearly dependent thermal properties, such as thermal conductivity and volumetric specific heat. The method accomodates rather general thermal boundary conditions including arbitrary variations in surface temperature or in surface heat flux or a convective exchange with a fluid having even varying temperature. Once the solution for the identical problem but with constant properties has been developed, its practical realization is rather direct, being facilitated by a reduced number of iterations. The four applied examples given in this work show that a wide variety of nonlinear heat conduction problems can be tackled by this procedure without much difficulty. These simple solutions compare favorably with more laborious results reported in the archival heat transfer literature.

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Berechnung nichtstationärer Wärmeleitvorgänge mit linear temperaturabhängigen Stoffwerten aus der Lösung für konstante Stoffwerte

Zusammenfassung Es werden quasi-analytische Lösungen für ein- und mehrdimensionale nichtstationäre Wärmeleitprobleme mit linear temperaturabhängigen Stoffwerten, wie Wärmeleitfähigkeit und volumetrische Wärmekapazität, mitgeteilt. Die Methode gilt für recht allgemeine Randbedingungen wie beliebige Veränderungen der Oberflächentemperatur, der Wärmestromdichte oder auch konvektiven Wärmeaustausch mit veränderlicher Fluidtemperatur. Ist die Lösung für das identische Problem mit konstanten Stoffwerten bekannt, kann die Methode direkt mit einer begrenzten Zahl von Iterationen angewandt werden. Die vier hier mitgeteilten Beispiele zeigen, daß eine große Zahl nichtlinearer Wärmeleitprobleme auf diese Weise ohne Schwierigkeit angepackt werden können. Die einfachen Lösungen stimmen befriedigend mit komplizierteren Ergebnissen aus der Literatur überein.

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Nomenclature a side of square bar - B _i0 reference Biot number,hR/k₀ - B _i0 ^T transformed Biot number, equation (16) - c geometric parameter, equation (8) - h convective coefficient - k thermal conductivity - k ₀ value ofk atT ₀ - K dimensionless thermal conductivity,k/k ₀ - K _i value ofK at _i - K _i+1 value ofK at _i+1 - m _k slope of theK- line, equation (3) - m _s slope of theS- line, equation (4) - R characteristic length - s volumetric specific heat - s ₀ value of s at T₀ - S dimensionless volumetric specific heat, s/s₀ - S _i value ofS at _i - S _i+1 value of S at _i+1 - t time - T temperature - T ₀ reference temperature - x, y cartesian coordinates - X, Y dimensionless cartesian coordinates,x/a andy/a - thermal diffusivity - _k transformed time, equation (11) - _s transformed time, equation (37) - _k dimensionless time for variable conductivity, equation (8) - _s dimensionless time for variable specific heat, equation (34) - dimensionless temperature,T/T ₀ - dimensionless coordinate,r/R - ₀ value of at T₀ - _i lower value of the interval (_i, _i+1) - _i+1 upper value of the interval (_i, _i+1     

Effect of drop size distribution on the flow behavior of oil-in-water emulsions

The effect of drop size distribution on the viscosity was experimentally examined for oil-in-water emulsions at volume fractions of = 0.5, 0.63 and 0.8. At = 0.5, the hydrodynamic forces during drop collisions govern the viscosity behavior. The viscosity versus shear rate curve is scaled on the root-mean-cube diameter which is related to the number of drops per unit volume. At = 0.8, the resistance to flow arises from the deformation and rearrangement of thin liquid films between drops. The viscosity at a given shear rate is inversely proportional to the volume-surface mean diameter which is related to the total interfacial area per unit volume. However, since the drops come into contact and the liquid film separating adjacent drops is generated without drop deformation at = 0.63, the viscosity curve is not scaled on the mean diameter. The flow behavior near the critical volume fraction strongly depends not only on the mean drop size, but also on the width of the distribution.     

Optimal estimates for the uncoupling of differential equations

Our aim in this note is to give optimal conditions on the spectral gap for the existence of an uncoupling of a differential equation of the form = Cz + H(=) into a system ofuncoupled equations of the form (x, y) = (Ax, By) + (F(x, (x)), G((y),y)), whereC=A×B is a bounded linear operator on a Banach spaceZ=X×Y satisfying a spectral gap condition, andH=(F,G) is a Lipschitz function withH(0) = 0. We also give optimal conditions for the regularity of the manifoldsgraph andgraph , and optimal conditions for the regularity of the leaves of two foliations of the phase space associated to the uncoupling. Sharp estimates for the Lipschitz constant of and and for the Hölder exponent of the uncoupling homeomorphism and its inverse are also given.     

Discretized ordinary differential equations and the Conley index

LetN be a compact isolating neighborhood of an isolated invariant setK with respect to an ODEx=f(x) (C) and(h) x=x + h(x, h) be a consistent one-step-discretization of (C). It is proved in this paper that for someh ₀ > 0 and allh ]0, h₀[, the setN isolates an invariant setK(h) of(h) and the discrete Conley index ofK(h) coincides with the continuous Conley index ofK.     

The theory of a vibrating-rod viscometer

The paper presents a complete theory for a viscometer based upon the principle of a circular-section rod, immersed in a fluid, performing transverse oscillations perpendicular to its axis. The theory is established as a result of a detailed analysis of the fluid flow around the rod and is subject to a number of criteria which subsequently constrain the design of an instrument. Using water as an example it is shown that a practical instrument can be designed so as to enable viscosity measurement with an accuracy of ±0.1%, although it is noted that many earlier instruments failed to satisfy one or more of the newly-established constraints.Nomenclature A, D constants in equation (46) - A _m , B _m , C _m , D _m constants in equations (50) and (51) - A _j , B _j constants in equation (14) - a _j ⁺ , a _j ^– wavenumbers given by equation (15) - C _f drag coefficient defined in equation (53) - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - f(z) initial deformation of rod - f(), F _m () functions of defined in equation (41) - F force in the rod - force per unit length near t=0 - F dimensionless force per unit length near t=0 - g _m amplitude of transient force - G modulus of rigidity - h, h* functions defined by equations (71) and (72) - H functions defined by equation (69) and (70) - I second moment of area - I _0,1, J _0,1, K _0,1 modified Bessel functions - k, k functions defined in equations (2) - L half-length of oscillator - Ma Mach number - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equations (15) and (16) - R radius of rod - R _c radius of container - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - y ₀ initial lateral displacement - y ₁, y ₂ successive maximum lateral displacement - z axial coordinate - dimensionless tension - dimensionless mass of fluid - dimensionless drag of fluid - amplification factor - logarithmic decrement in a fluid - _a , _b logarithmic decrement in fluids a and b - ₀ logarithmic decrement in vacuo - _j logarithmic decrement in mode j in a fluid - spatial resolution of amplitude - _v voltage resolution - r, , , _s, , increments in R, , , _s , , - dimensionless amplitude of oscillation - dimensionless axial coordinate - angular coordinate - _f thermal conductivity of fluid - viscosity of fluid - viscosity of fluid calculated on assumption that * - _a , _b viscosity of fluids a and b - _m constants in equation (10) - dimensionless displacement - _j j the component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - spatial component of defined in equation (11) - _j , _tm jth, mth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - streamfunction - dimensionless frequency (based on ) - angular frequency - ₀ angular frequency in absence of fluid and internal damping - _j angular frequency in mode j in a fluid - _a , _b frequencies in fluids a and b     

Die Analogie zwischen den Verschiebungen und den Spannungsfunktionen in der Biegetheorie der Kreiszylinderschale

Zusammenfassung Für die Kreiszylinderschale wurde eine Biegetheorie aufgestellt, in der die Gleichgewichtsbedingungen (unter Voraussetzung der Symmetrie des Momententensors M _ik ) durch drei Spannungsfunktionen ₁, ₂, ₃ exakt erfüllt sind. Bei der Definition der Deformationsgrößen und der Einführung der Elastizitätsgesetze war die Reißner-Meißnersche Theorie der symmetrisch belasteten Rotationsschale das Vorbild. Die drei Differentialgleichungen für die Verschiebungen _{1 2}, ₃ unterscheiden sich von den drei Differentialgleichungen für die Spannungsfunktionen ₁, ₂, ₃ formal nur im Vorzeichen der Poissonschen Querkontraktionsziffer v. Die beiden Differentialgleichungen achter Ordnung, die man nach Eliminationsprozessen sowohl für ₃ als auch für ₃ erhält, unterscheiden sich nicht mehr voneinander. So trifft man bei der Zylinderschale die Timpe-Wieghardtsche Analogie zwischen Durchbiegung ₃ der Platte und Airyscher Spannungsfunktion ₃ der Scheibe wieder.Es konnte ferner gezeigt werden, daß unsere neue Biegetheorie der bekannten Flüggeschen Theorie an Genauigkeit nicht nachsteht.Es ist wohl nicht zu bezweifeln, daß auch bei Schalen beliebiger Gestalt unsere Analogie vorhanden ist. Sie scheint uns wertvoll als Ordnungsprinzip inmitten der Fülle von Gleichungen, die nun einmal zu einer Schalentheorie gehören.Die Formulierung des Schalenproblems mit Hilfe der drei Spannungsfunktionen ₁, ₂, ₃ wird sich immer dann empfehlen, wenn die Randbelastung vorgegeben ist. Denn dann lassen sich die Randbedingungen in den Spannungsfunktionen übersichtlicher formulieren als in den Verschiebungen. Auch die Gewißheit, daß selbst durch radikales Streichen lästiger Glieder in den Differentialgleichungen der Spannungsfunktionen die Gleichgewichtsbedingungen nicht verletzt werden, mag manchem Rechner angenehm sein.     

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     

A nonequilibrium theory of a slurry

A nonequilibrium theory of a slurry is developed and its practical use is illustrated by a simple stability analysis. Here a slurry is defined as a deformable continuum consisting of a liquid phase containing in suspension a large number of small solid particles which have formed by solidification from the liquid. The liquid is assumed to consist of two components and the solid to contain only one of the two. Consequently, the process of change of phase requires redistribution of material on the scale of the solid particles. This process is assumed to take a finite amount of time, requiring a nonequilibrium macroscopic theory. This theory contains four thermodynamic variables, three to represent the equilibrium state of the binary system and a fourth measuring the departure from thermodynamic equilibrium. The process of microscale diffusion of material is parameterized in the macroscale theory, leading to a Landau-type relaxation term in the equation of evolution of the fourth variable. The theory is simplified to yield a Boussinesq-like set of governing equations. Their practical use is illustrated by analyzing the stability of a simple steady solution of the equations and the effects of a non-zero relaxation time are discussed. A novel instability mechanism involving sedimentation of particles, previously found to occur in the equilibrium case, is found to persist in nonequilibrium, but disappears in the limit of no change of phase.Key to symbols a, b, c thermodynamic coefficients; see (3.36)–(3.38) - sedimentation coefficient; see (5.18) - C _p specific heat; see (3.24) - C _p ^de specific heat of the slurry; see (3.28) and (3.30) - c radius of solid particle (in §4) - D, D diffusive coefficients; see (3.40) and (3.41) - material diffusivity in liquid phase - D ^* modified diffusion coefficient; see (5.15) - d thermodynamic coefficient; see (3.39) - E specific internal energy - f, g, h thermodynamic coefficients; see (3.36)–(3.38) - g acceleration of gravity - reduced gravity; see (5.10) - i total diffusive flux vector of constituent 1 - i diffusive flux vector of constituent 1 in the liquid phase - j diffusive flux vector of solid phase - k thermal conductivity - k entropy flux vector - k _T, k_T thermodiffusion coefficients; see (3.40) and (3.41) - L latent heat of solidification per unit mass; see (3.7) and (3.24) - m wave number - m ^s rate of creation of mass of solid per unit volume through solidification - m ₁ ^s rate of creation of mass of solid constituent 1 per unit volume through solidification - mass rate of freezing per unit area per unit time - N number of solid particles per unit volume - p pressure - p _H hydrostatic component of pressure - p _m mechanical pressure - p ₁ dynamic component of pressure - q heat flux vector - Q _D rate of regeneration of heat through diffusive fluxes - Q _M rate of regeneration of heat through phase-change processes - Q _v rate of regeneration of heat through viscosity - Q vector defined by (3.16) - r heat externally supplied per unit mass (in §3); spherical radial coordinate (in §4) - S specific entropy of slurry - change of specific entropy with mass fraction of constituent 1; also change of chemical potential of liquid phase with temperature barring change of phase - change of chemical potential of liquid phase with temperature in phase equilibrium; see (3.28) and (3.30) - T temperature - t time - t ₀ relaxation time; see (5.30) - u barycentric velocity - u _H horizontal perturbation velocity - V sedimentation speed - w _a upward speed of simple state; see (6.5) and (6.12) - z upward vertical coordinate - upward unit vector - thermal expansion coefficient barring change of phase; see (3.23) - > ^* thermal expansion coefficient in phase equilibrium; see (3.27) and (3.30) - modified thermal expansion coefficient; see (5.1) and (5.4) - isothermal compressibility of slurry barring change of phase; see (3.23) - ^* isothermal compressibility of slurry in phase equilibrium; see (3.27) and (3.30) - dimensionless measure of departure from liquidus equilibrium; see (5.2) - _a deviation from phase equilibrium in simple state; see (6.6) and (6.13) - vertical wave number - volume expansion per unit mass upon melting; see (3.6) - change of chemical potential of liquid phase with pressure; see (3.25) - change of chemical potential of liquid phase with pressure for slurry; see (3.29) and (3.30) - compositional gradient in the static state; see (6.15) - vector defined by (3.35) - constant of integration; see (6.7) and (6.8) - coefficient defined by (6.23) - nonequilibrium expansion coefficient; see (5.1) and (5.4) - thermal diffusivity; =k/C _p - modified thermal diffusivity; see (5.33) - relaxation rate to phase equilibrium; see (2.2) - ₁ relaxation rate to solid-composition equilibrium; see (2.3) - sedimentation coefficient; see (4.29) - horizontal wave number vector - sedimentation coefficient; see (4.30) - ^L , ^s chemical potential of constituent 1 relative to constituent 2 in liquid and solid phase per unit mass; see (2.6) - change of chemical potential of liquid with liquid composition; see (3.8) - coefficient defined by (3.10) - kinematic shear viscosity - total mass fraction of constituent 1 (i.e., solute) - ^L, ^s mass fraction of constituent 1 in liquid and solid phases - density of slurry - _s density of solid phase - - - , growth rate of disturbance - stress tensor - deviatoric stress tensor - dimensionless temperature; see (5,3) - _a constant of integration; see (6.7) - mass fraction of solid phase in slurry - _b vertical gradient of mass fraction of solid; see (6.1) - dimensionless measure of _b; see (6.22) - _c temporal gradient of mass fraction of solid; see (6.1) - specific Gibbs free energy; see (3.13) - _L,_s specific Gibbs free energy of liquid and solid phases; see (2.12) - measure of departure from liquidus equilibrium; see (2.14) - measure of departure from solidus equilibrium; see (2.5) - spherical polar coordinate (in §4); see (4.20); wave angle (in §6); see (6.38)     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

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     

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)     

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f₀ and the diffusion vector fields f₁,...,f_m, and investigate the existence of global random attractors for the associated flows . For this purpose is decomposed into a stationary diffeomorphism given by the stochastic differential equation on the space of smooth flows on R^d driven by m independent stationary Ornstein Uhlenbeck processes z¹,...,z^m and the vector fields f₁,...,f_m, and a flow generated by the nonautonomous ordinary differential equation given by the vector field (_t/x)^–1[f₀(_t)+ _i=1 ¹ f_i(_t)z _t ⁱ ]. In this setting, attractors of are canonically related with attractors of . For , the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f₀ is still a Lyapunov function for , yielding an attractor this way. The criterion is finally tested in various prominent examples.     

Digital phase-stepping holographic interferometry in measuring 2-D density fields

This paper presents a holographic interferometer technique for measuring transparent (2-D or quasi 2-D) density fields. To be able to study the realization of such a field at a certain moment of time, the field is frozen on a holographic plate. During the reconstruction of the density field from the hologram the length of the path traversed by the reconstruction beam is diminished in equal steps by applying a computer controlled voltage to a piezo-electric crystal that translates a mirror. Four phase-stepped interferograms resulting from this pathlength variation are digitized and serve as input to an algorithm for computing the phase surface. The method is illustrated by measuring the basically 2-D density field existing around a heated horizontal cylinder in free convection.List of symbols wavelength - x, y cartesian coordinate system - phase - phase step - K Gladstone-Dale constant - L width of 2-D density field - density - ₀ density at reference conditions - I ₀, I ₁, I ₂, I ₃ recorded interferograms - I _mod modulation intensity - I _bias bias intensity - N numerator determining tan { (x, y)} - D denominator determining tan { (x, y)} - dimensionless temperature - T temperature - T _c temperature of cylinder - T temperature of environment - p pressure - R gas constant     

A distributed electrical analog for transient diffusion-techniques of model fabrication and calibration

Simulation of transient two-dimensional diffusion by means of a distributed electrical analog is discussed. After present techniques of analog model construction and calibration are reviewed, an improved calibration technique is presented and a convenient method of analog fabrication, not previously reported, is described. The proposed new method allows complete access to any point on the analog model during a test. Frequency response and step response measurements indicate that an adequate simulation is provided by this particular type of analog model.

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Zusammenfassung Die Simulierung eines nichtstationären, zweidimensionalen Diffusionsvorganges mittels eines kontinuierlichen, elektro-thermischen Analogapparates wird besprochen. Eine Übersicht der gegenwärtigen Methoden für die Konstruktion und Kalibrierung von elektro-thermischen Analogmodellen wird gegeben. Ein verbessertes Verfahren für die Kalibrierung und eine handliche Fertigungsmethode von Analogmodellen, die noch nicht in der Literatur beschrieben wurden, werden dargestellt. Das vorgeschlagene neue Verfahren gestattet vollständigen Zugang zu jedem Punkt im Analogmodell während des Experiments. Meßbeobachtungen des periodischen Frequenzverhaltens und des nichtstationären Verhaltens zeigen, daß dieses spezielle Analogmodell eine ausreichende Simulierung des Diffusionsvorganges gestattet.

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Nomenclature A ₀ Potential amplitude atx=0 (see Fig. 5) - A _x Potential amplitude at locationx (see Fig. 5) - A _L Potential amplitude atx=L (see Fig. 5) - C Capacitance per unit area - j Frequency - L Characteristic length - R Resistance per square - t Time - x Coordinate - X Dimensionless distance,x/L - y Coordinate - Y Dimensionless distance,y/L - Diffusivity - _x Phase angle at locationx (see Fig. 5) - _L Phase angle atx=L (see Fig. 5) - _d Thickness of dielectric sheet - _r Thickness of resistance sheet - Dielectric constant - Resistivity - Dimensionless time,t/L ² - Potential - ₁ Reference potential - ₂ Reference potential - Dimensionless potential, ( – ₁)/( ₂ – ₁) - Angular frequency, 2f - Dimensionless frequency,L ²/ The investigation was performed while the first author was Visiting Associate Professor at Purdue University during 1967/68.