Solutions for the anisotropic elastic wedge at critical wedge angles

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2 equals , 2 or 2_* where tan 2_* = 2_*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2 = 2_*. A similar situation occurs when the wedge is anisotropic except that 2_* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.     

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2 satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for not equal but very close to 235-3, the classical solution can still be very large as approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2= and 2. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^- and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.     

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     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Dynamic behaviour of the constant temperature anemometer due to thermal inertia of the wire

The time dependent differential equation for the local wire temperature of a constant temperature anemometer is solved by a perturbation method in case of a harmonically changing heat transfer coefficient. The time dependent power supply to the wire follows from the condition of constant mean temperature imposed by the anemometer circuit. The influence of thin supporting wires, or copper-plated wire ends, is evaluated also. Numerical results are given for a number of cases that are of practical interest.Nomenclature c specific heat - D diameter of the wire - D _u diameter of the copper-plated ends of the wire - f D - g I ² r ₀ - I electric current - L length of the wire - P 1/4D ² c - q 1/4D ² - r resistance of the wire per unit length at temperature T' - r ₀ resistance of the wire per unit length at temperature T - T T' – T - T' local temperature of the wire - T ambient temperature - T _w constant mean temperature imposed by the anemometer circuit - T _u local difference between the temperature of the supporting wire and the ambient temperature - t time - x axial coordinate with the origin in the middle of the wire - heat transfer coefficient - temperature coefficient of the resistance - small parameter - time constant = cD ²/4D - _u time constant of the copper-plated ends cD _u ² /4D _u - thermal conductivity of wire material - _u thermal conductivity of the copper-plated wire ends - density - circular frequency     

Quantitative liquid jet instability measurement system using asymmetric magnification and digital image processing

A new technique for measuring the growth of instabilities on the surface of liquid jets flowing into gas is demonstrated. A collimated beam of white light illuminates the jet from behind, forming a shadow image. A pair of cylindrical lenses are arranged to provide different magnifications in the streamwise and cross-stream directions. A number of streamwise diameters and one cross-stream diameter are thus captured with maximum resolution in a single image on a charge-coupled device (CCD) electronic camera. A short-duration spark is used to freeze the jet motion. A mask representing the theoretical edge-response of the imaging system is digitally convolved with the cross-stream gray scale data to obtain sub-pixel resolution of the jet edge profile. The method is demonstrated using the well-known capillary jet instability and a ratio of streamwise to cross-stream magnifications of 40. Well-resolved single images show the development of the instability from small perturbations through the formation of the first drop. The system forms an accurate automated method of measuring the development of liquid jet instabilities. It can readily be applied to practical problems including liquid jet atomization.List of symbols a undisturbed jet radius - k nondimensional wavenumber (= 2a/) - Q gas-to-liquid density ratio - r ₀ mean jet radius, from initial region of image - R Reynolds number (= 2Ua/) - U mean jet velocity - We Weber number - z streamwise coordinate, origin at jet orifice - temporal growth rate - _s measured spatial growth rate - nondimensional temporal growth rate - r absolute value of height of peaks or troughs relative to r ₀ - r ₁ height of first extremum in a particular record - instability wavelength - liquid viscosity - liquid density - surface tension of liquid-gas interface     

On the effects of uniform high suction on the rotationally symmetric flow of a conducting liquid near a stationary disc in the presence of a transverse magnetic field

In the present paper an attempt has been made to find out effects of uniform high suction in the presence of a transverse magnetic field, on the motion near a stationary plate when the fluid at a large distance above it rotates with a constant angular velocity. Series solutions for velocity components, displacement thickness and momentum thickness are obtained in the descending powers of the suction parameter a. The solutions obtained are valid for small values of the non-dimensional magnetic parameter m (= 4 _e ² H ₀ ² /) and large values of a (a2).Nomenclature a suction parameter - E electric field - E _r , E , E _z radial, azimuthal and axial components of electric field - F, G, H reduced radial, azimuthal and axial velocity components - H magnetic field - H _r , H , H _z radial, azimuthal and axial components of magnetic field - H ₀ uniform magnetic field - H* displacement thickness and momentum thickness ratio, */ - h induced magnetic field - h _r , h , h _z radial, azimuthal and axial components of induced magnetic field - J current density - m nondimensional magnetic parameter - p pressure - P reduced pressure - R Reynolds number - U ₀ representative velocity - V velocity - V _r , V , V _z radial, azimuthal and axial velocity components - w ₀ uniform suction through the disc. - density - electrical conductivity - kinematic viscosity - _e magnetic permeability - a parameter, (/)^1/2 z - a parameter, a - * displacement thickness - momentum thickness - angular velocity     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Irrotational outflow of liquid from a stream through a lateral hole of finite depth

An algorithm is constructed for numerical determination of the flow parameters and coefficient of contraction of a jet in the case of irrotational lateral outflow of liquid from a semiinfinite stream through a nozzle of finite depth situated at an arbitrary angle to the mainstream flow. The solution is based on the use of N. E. Zhukovskii's method and the Schwarz-Christoffel formula. The results of calculations for a nozzle situated at an angle = /2 ± , where = /6, are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, January–February, 1977.     

Mass transport in flow of sources

Steady and unsteady local concentration has been determined analytically for two- und three-dimensional sources and is presented for various boundary-concentrations, volumetric flows and diffusion coefficients. The steady cases have been evaluated numerically. In addition an unsteady two-dimensional mass transport has been evaluated.

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Stofftransport in Quellströmungen

Zusammenfassung Es wurden die stationäre und instationäre örtliche Konzentration von einer zwei- und drei-dimensionalen Quellströmung als Funktion verschiedener Randkonzentrationen, verschiedener Stromvolumen und Diffusionskoeffizienten analytisch bestimmt. Die stationären Fälle wurden numerisch ausgewertet. Außerdem wurde ein zwei-dimensionaler instationärer Stofftransport behandelt.

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Nomenclature a inner radius of circle (2-dimensional case), inner radius of sphere (three-dimensional case) - b } >a outer radius of circle (2-dimensional case), outer radius of sphere (three-dimensional case) - c concentration - c ₁,c ₂ given concentration at the boundariesr=a andb resp - c _i initial concentration at the timet=0 - D diffusion coefficient - I _n +1/2 modified spherical Bessel function - J _v ,Y _v Bessel function ofv-th order and first and second kind resp - k =b/a} > 1 diameter ratio - P _n ^o () Legendre polynomials - ¯ r, polar coordinates - r, , spherical coordinates - t time - u velocity in radial direction - V ₀ volumetric flow - ₀ V/4D flow parameter for two-dimensional flow - ₀ V ₀/8 D flow parameter for three-dimensional flow - _mn eigenvalues - _mn te] ² =n ² + ₀ ² ,=cos =r/a roots of determinant (28)     

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.     

Hydrodynamic models as applied to the investigation of magnetic plasma stability

In the present paper magnetohydrodynamic models are employed to investigate the stability of an inhomogeneous magnetic plasma with respect to perturbations in which the electric field may be regarded as a potential field (rot E 0). A hydrodynamic model, actually an extension of the well-known Chew-Goldberg er-Low model [1], is used to investigate motions transverse to a strong magnetic field in a collisionless plasma. The total viscous stress tensor is given; this includes, together with magnetic viscosity, the so-called inertial viscosity.Ordinary two-fluid hydrodynamics is used in the case of strong collisions=. It is shown that the collisional viscosity leads to flute-type instability in the case when, collisions being neglected, the flute mode is stabilized by a finite Larmor radius. A treatment is also given of the case when epithermal high-frequency oscillations (not leading immediately to anomalous diffusion) cause instability in the low-frequency (drift) oscillations in a manner similar to the collisional electron viscosity, leading to anomalous diffusion.Notation f particle distribution function - E electric field component - H₀ magnetic field - density - V particle velocity - e charge - m, M electron and ion mass - _i, _e ion and electron cyclotron frequencies - viscous stress tensor - P pressure - r_i Larmor radius - P pressure tensor - t time - frequency - T temperature - collision frequency - collision time - j current density - _i, _e ion and electron drift frequencies - k_x, k_y, k_z wave-vector components - n₀ particle density - g acceleration due to gravity. The authors are grateful to A. A. Galeev for valuable discussion.     

Elongational flow behavior of viscoelastic liquids: Modelling bubble dynamics with viscoelastic constitutive relations

Summary Earlier parts of this series have described a technique based on the collapse of single bubbles in the fluids for studying the elongational rheology of viscoelastic solutions and melts of moderate viscosities ( ₀ > 10²p) at relatively high strain rates . The present paper describes the modelling of bubble collapse with both rate and integral type constitutive relations using a body coordinate system. Predictions of the stress at the bubble wall as a function of time during collapse from a BKZ model and a modified corotational Maxwell model compared favorably with experimental data for two polymer solutions, 1% polyacrylamide in water/glycerine and 2% hydroxypropyl cellulose in water.

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Zusammenfassung In vorangehenden Veröffentlichungen dieser Reihe wurde eine Methode beschrieben, mit Hilfe derer man aus dem Zerfall von einzelnen Blasen in einer Flüssigkeit auf die Dehn-Rheologie viskoelastischer Lösungen und Schmelzen mittlerer Viskosität ( ₀ > 10² P) bei relativ hohen Dehngeschwindigkeiten schließen kann. Die vorliegende Untersuchung beschreibt Modelle des Blasenzerfalls mit Hilfe von Stoffgleichungen sowohl vom rate- als auch vom Integral-Typ, wobei ein körperfestes Koordinatensystem benutzt wird. Die Voraussagen der Spannung an der Blasenwand als Funktion der Zeit während des Zerfalls bei Verwendung eines BKZ- und eines modifizierten korotatorischen Maxwell-Modells zeigen eine recht gute Übereinstimmung mit experimentellen Werten, die an zwei Polymerlösungen, nämlich einer 1%igen Polyacrylamid-Lösung in einer Wasser-Glycerin-Mischung und einer 2%igen wäßrigen Hydropropylcellulose, erhalten worden sind.

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Nomenclature a material constant - b material constant - g metric tensor, space coordinates - m material constant - n material constant - p pressure - P _G pressure within bubble - P _R pressure outside bubble at the wall - P pressure far away from the bubble - R bubble radius - dR/dt - R ₀ initial bubble radius - t time - u velocity - U potential function - Y R/R ₀ Greek symbols covariant body metric tensor - surface tension - rate of deformation matrix, II -second invariant of - strain rate - ₀ zero shear rate viscosity - _e elongational viscosity - _ef effective viscosity - ¹, ², ³ coordinates in body system - ^{1 1}/R ₀ ³ - body stress tensor - density - space stress tensor - relaxation time - _ef effective relaxation time - bubble pressure function, defined in eq. [19] - vorticity tensor With 11 figures and 1 table     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

An investigation of laminar radial flow between two parallel discs

Summary Experiments have been carried out to test recent theoretical predictions of the pressure distribution for laminar flow between parallel discs, including inertia effects. The experimental investigation covered the condition where the inertia effects were always completely dominant over the central region of the discs in contrast to other recent experimental work on the problem where the central injection diameter was considerably larger. The present experiments subject the theories to a stringent test, due to the dominance of the inertia effects, and it is found that the inertia effects predicted by the various theoretical analyses are significantly smaller than those shown by the experimental results. It is suggested that the theoretical approach requires further development before it will cover the conditions where the central injection diameter is small.Nomenclature r, y, cylindrical co-ordinates - u velocity in r direction - U _m mean velocity in r direction at radius r - density - coefficient of viscosity - Q volume flow per unit time - 2h gap between parallel discs - p static pressure - R r/h - P h ³ p/Q - R _e Q/h     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Viscous heating correction for thermally developing flows in slit die viscometry

In the thermally developing region, d _yy /dx| _y=h varies along the flow direction x, where _yy denotes the component of stress normal to the y-plane; y = ±h at the die walls. A finite element method for two-dimensional Newtonian flow in a parallel slit was used to obtain an equation relating d _yy /dx/ _y=h and the wall shear stress ₀ at the inlet; isothermal slit walls were used for the calculation and the inlet liquid temperature T₀ was assumed to be equal to the wall temperature. For a temperature-viscosity relation /₀ = [1+(T–T₀]^–1, a simple expression [(hd _yy /dx/ _y=h )/ _w0] = 1–[1-F _c(Na)] [M()+P(Pr) ·Q(Gz ^–1)] was found to hold over the practical range of parameters involved, where Na, Gz, and Pr denote the Nahme-Griffith number, Graetz number, and Prandtl number; is a dimensionless variable which depends on Na and Gz. An order-of-magnitude analysis for momentum and energy equations supports the validity of this expression. The function F _c(Na) was obtained from an analytical solution for thermally developed flow; F _c(Na) = 1 for isothermal flow. M(), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate at the inlet can be calculated from the flow rate Q using the isothermal equation.Notation x,y Cartesian coordinates (Fig. 2) - , dimensionless spatial variables [Eq. (16)] - dimensionless variable, : = Gz(x)^–1 - dimensionless variable [Eq. (28)] - t,t ^* time, dimensionless time [Eq. (16)] - , velocity vector, dimensionless velocity vector - _x , velocity in x-direction, dimensionless velocity - _y , velocity in y-direction, dimensionless velocity - V average velocity in x-direction - _yy , ^* normal stress on y-planes, dimensionless normal stress - shear stress on y-planes acting in x-direction - _w , _w ^* value of shear stress stress at the wall, dimensionless wall shear stress - _w0, _w0 ^* wall shear stress at the inlet, dimensionless variable - , ^* rate-of-strain tensor, dimensionless tensor - wall shear rate, wall shear rate at the inlet - Q flow rate - T, T ₀, temperature, temperature at the wall and at the inlet, dimensionless temperature - h, w half the die height, width of the die - l,L the distance between the inlet and the slot region, total die length - T ₂, T ₃, T ₄ pressure transducers in the High Shear Rate Viscometer (HSRV) (Fig. 1) - P, P2, P3 pressure, liquid pressures applied to T ₂ and T ₃ - , ₀, ^* viscosity, viscosity at T = T ₀, dimensionless viscosity - viscosity-temperature coefficient [Eq. (8)] - k thermal conductivity - C _p specific heat at constant pressure - Re Reynolds number - Na Nahme-Griffith number - Gz Graetz number - Pr Prandtl number     

A fourfold generalization of Peaucellier's inversion cell

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

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