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.

     

Elastohydrodynamic lubrication in a plane slider bearing

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

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

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

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     

The Most Unstable Profiles of a Plane-Parallel Flow in a Channel

It is well known (after Rayleigh) that a plan-parallel flow in a channel can be unstable only if the basic velocity profile U(z) possesses inflection points. The profile determines (via the Rayleigh equation) the maximal increment c _i of small perturbations and 'eigenvalues' c (see Equation (1)). The increment and the imaginary parts c _i of the eigenvalues c provide a quantitative characterization of the basic stability properties of the flow. Here we find some best possible bounds for these values. The bounds are determined by the following parameters: wave number ; enstrophy of the basic flow; width of the channel L. A similar approach can be applied to models of atmosphere, ocean, plasma etc.     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

Production of temperature fluctuations in grid turbulence: Wiskind's experiment revisited

Measurements have been made in nearly-isotropic grid turbulence on which is superimposed a linearly-varying transverse temperature distribution. The mean-square temperature fluctuations, , increase indefinitely with streamwise distance, in accordance with theoretical predictions, and consistent with an excess of production over dissipation some 50% greater than values recorded in previous experiments. This high level of production has the effect of reducing the ratio,r, of the time scales of the fluctuating velocity and temperature fields. The results have been used to estimate the coefficient,C, in Monin's return-to-isotropy model for the slow part of the pressure terms in the temperature-flux equations. An empirical expression by Shih and Lumley is consistent with the results of earlier experiments in whichr 1.5, C 3.0, but not with the present data where r 0.5, C 1.6. Monin's model is improved when it incorporates both time scales.List of symbols C coefficient in Monin model, Eq. (5) - M grid mesh length - m exponent in power law for temperature variance, x ^m - n turbulence-energy decay exponent,q ² x ^-n - p production rate of - p pressure - q ² - R microscale Reynolds number - r time-scale ratiot/t - T mean temperature - U mean velocity - mean-square velocity fluctuations (turbulent energy components) - turbulent temperature flux - x, y, z spatial coordinates - temperature gradient dT/dy - thermal diffusivity - dissipation rate ofq ²/2 - dissipation rate of - Taylor microscale (₂=5q₂/) - temperature microscale - _v temperature-flux correlation coefficient, /v - dimensionless distance from the grid,x/M     

Fourier Transform in Bounded Domains

The functional analysis, the concept of distributionsu in the sense of Schwartz [7] andtheir extension given by Gelfand and Shilov [5]to ultradistributions u ,enables us to find by the means of the Fourier transform a secondlanguage to characterize physical behaviour. Almost any expressionwith physical meaning can be transformed, even if it isformulated in domains with complicated boundaries and evenif it is not integrable.Numerical procedures in the transformed space can bedeveloped in analogy to those well known in engineeringmechanics like the methods of Finite or BoundaryElements (FEM or BEM). Basis of the approaches presentedhere is the analytical representation of characteristicdistribution of a domain and the theorem of Parseval whichstates the invariance of energy in respect to thetransformation. In addition, the concept of thecharacteristic distribution leads to a very simplederivation of the Green-Gauss formulas fundamental for theBoundary or Finite Elements (e.g. [6]).     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

On properties of solutions of functional differential equations continuously differentiable on (0, +∞)

We establish the saddle-point property of the system of functional differential equations (t) = Ax(t) + Bx((t)) + C ((t)) + f (x(t), x((t))), (0) = 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 302–310, July–September, 2004.     

Transient flow towards a well in an aquifer including the effect of fluid inertia

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.List of symbols b Thickness of the aquifer, m - c ₀ Wave velocity, m/s - k Permeability of the porous medium, m² - n Porosity of the porous medium - p( ,t) Pressure, N/m² - Q Volume flux, m³/s - r Radial coordinate, m - r _w Radius of the well, m - s Transform variable - S Storativity of the aquifer - S _d(r, t) Drawdown, m - t Time, s - T Transmissivity of the aquifer, m²/s - ( ,t) Velocity of the fluid, m/s - Coordinate vector, m - z Vertical coordinate, m - Coefficient of compressibility, m²/N - Coefficient of fluid compressibility, m²/N - Relaxation time, s - (r, t) Hydraulic potential, m - Dynamic viscosity of the fluid, Ns/m² - Dimensionless radius - Density of the fluid, Ns²/m⁴ - (, ) Dimensionless drawdown - Dimensionless time - , x Dummy variables - ₀, ₁ Auxilary functions     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile     

Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

The dynamics of an analytic reversible vector field (X,) is studied in with one real parameter close to 0; X=0 is a fixed point. The differential D_x (0,0) generates an oscillatory dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a slow dynamics which changes from a hyperbolic type—eigenvalues are —to an elliptic type—eigenvalues are —as passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

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)     

Temperature and concentration dependent viscosity and gelation temperature of ABA triblock copolymer solutions

In solutions of ABA-triblock copolymers in a poor solvent for A thermoreversible gelation can occur. A three-dimensional dynamic network may form and, given the polymer and the solvent, its structure will depend on temperature and polymer mass fraction. The zero-shear rate viscosity of solutions of the triblock-copolymer polystyrene-polyisoprene-polystyrene in n-tetradecane was measured as a function of temperature and polymer mass fraction, and analyzed; the polystyrene blocks contained about 100 monomers, the polyisoprene blocks about 2000 monomers. Empirically, in the viscosity at constant mass fraction plotted versus inverse temperature, two contributions could be discerned; one contribution dominating at high and the other one dominating at low temperatures. In a comparison with theory, the contribution dominating at low temperatures was identified with the Lodge transient network viscosity; some questions remain to be answered, however. An earlier proposal for defining the gelation temperature T _gel is specified for the systems considered, and leads to a gelation curve; T _gel as a function of polymer mass fraction.Mathematical symbols {} functional dependence; e.g., f{x} means f is a function of x - p _log logarithm to the base number p; e.g., ¹⁰log is the common logarithm - exp exponential function with base number e - sin trigonometric sine function - lim limit operation - – in integral sign: Cauchy Principal Value of integral, e.g., - derivative to x - partial derivative to x Latin symbols dimensionless constant - b constant with dimension of absolute temperature - constant with dimension of absolute temperature - B dimensionless constant - c mass fraction - dimensionless constant - constant with dimension of absolute temperature - d ^* dimensionless constant - D{0} constant with dimension of absolute temperature - e base number of natural (or Naperian) logarithm - g distribution function of inverse relaxation times - G relaxation strength relaxation function - h distribution function of relaxation times reaction constant enthalpy of a molecule - H Heaviside unit step function - i complex number defined by i ² = –1 - j{0} constant with dimension of viscosity - j index number - k Boltzmann's constant - k _H Huggins' coefficient - m mass of a molecule - n number - N number - p index number - s entropy of a molecule - t time - T absolute temperature Greek symbols as index: type of polymer molecule - as index: type of polymer molecule - shear as index: type of polymer molecule - shear rate - small variation; e.g. T is a small variation in T relative deviation Dirac delta distribution as index: type of polymer molecule - difference; e.g. is a difference in chemical potential - constant with dimension of absolute temperature - (complex) viscosity - constant with dimension of viscosity - [] intrinsic viscosity number - inverse of relaxation time - chemical potential - number pi; circle circumference divided by its diameter - mass per unit volume - relaxation time shear stress - angular frequency     

An asymptotic treatment of the transient development of axisymmetric surface waves

A linearized theory is developed for the derivation of an asymptotic solution of the initial value problem of axisymmetric surface waves in an infinitely deep fluid produced by an arbitrary oscillating pressure distribution. An asymptotic treatment of the problem is presented in detail to obtain the solution for the surface elevation for sufficiently large time. Finally, the main prediction of this analysis for some particular pressure distributions of physical interest is exhibited.Nomenclature R, , Y cylindrical polar coordinates - frequency - g acceleration due to gravity - density of fluid - T time - (R, Y; T) velocity potential - E(R, T) vertical surface elevation - P(R, T) applied surface pressure - r, y nondimensional cylindrical polar coordinates, - p(r, t) nondimensional surface pressure - t nondimensional time, T - (r, y; t) nondimensional velocity potential, - (r, t) nondimensional vertical surface elevation, - (k) Hankel transform of a function p(r) with respect to r - I ₁ transient wave integral - I ₂ steady state wave integral     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …