Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

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 = .     

Mass transfer with chemical reaction in a laminar falling film

Zusammenfassung Es wird eine analytische Lösung für die Absorption in einem laminaren Rieselfilm mit homogener und heterogener chemischer Reaktion 1. Ordnung vorgestellt, wobei der Stofftransportwiderstand auf der Gasseite liegt. Die Lösung ist eine Funktion von drei dimensionslosen ParameternBi, und, welche die BiotZahl und einen homogenen bzw. heterogenen Reaktionsparameter darstellen. Es wird gezeigt, daß für feste Werte vonBi und die Absorptionsrate (bezogen auf die Breite 1 des Rieselfilms) über eine gewisse Länge (dimensionslos) des Rieselfilms unabhängig von ist, wenn, < 0,6 ist. Die laufende Länge wird von der Stelle aus gemessen, an der die Absorption beginnt. Für b 0,6 nimmt der FlußQ mit zu, erreicht aber einen Sättigungswert bei=10, wonachQ nurmehr sehr langsam anwächst. Jedoch für ein gegebenes und ohne Übergangswiderstand im Film (Bi ) nimmtQ mit für alle 0 zu.

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Mass transfer with chemical reaction in a laminar falling film

An analytical solution is presented for gas absorption in a laminar falling film with first-order homogeneous and heterogeneous chemical reaction and external gas-phase mass transfer resistance. The solution depends on three dimensionless parametersBi, and, wich represent the Biot number, homogeneous and heterogeneous reaction parameters, respectively. It is shown that for fixed values ofBi and, the rate of gas absorption (per unit breadth) over a certain length; (dimensionless) along the falling film measured from the point where surface absorption begins is independent of if < 0.6. For 0.6, this fluxQ increases with but reaches a saturation value at=10 beyond whichQ increases very slowly. But for given and zero gas film resistance (Bi ),Q increases with for all 0.

     

Unsteady forced convection heat/mass transfer from a flat plate

Numerical methods are used to investigate the transient, forced convection heat/mass transfer from a finite flat plate to a steady stream of viscous, incompressible fluid. The temperature/concentration inside the plate is considered uniform. The heat/mass balance equations were solved in elliptic cylindrical coordinates by a finite difference implicit ADI method. These solutions span the parameter ranges 10 Re 400 and 0.1 Pr 10. The computations were focused on the influence of the product (aspect ratio) × (volume heat capacity ratio/Henry number) on the heat/mass transfer rate. The occurrence on the plates surface of heat/mass wake phenomena was also studied.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Torsion of two bonded layers by a rigid disk

The problem considered in this paper describes the torsion of a homogeneous isotropic elastic layer (0zd ₁) of finite thickness d ₁, perfectly bonded to another elastic layer (-d ₂z0) of finite thickness d ₂. The problem is reduced to the solution of a Fredholm integral equation of the second kind. The solutions are given for some particular cases.

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Sommario In questo lavoro si considera il problema della torsione di uno strato elastico omogeneo ed isotropo (0zd ₁) di spessore finito d ₁, perfettamente incollato ad un altro strato elastico (-d ₂z0) di spessore finito d ₂. II problema é ricondotto alla soluzione di una equazione integrale di Freedholm del secondo ordine. Le soluzioni sono ottenute per alcuni casi particolari.

     

Maximum density effects on thermal instability of horizontal laminar boundary layers

Linear stability theory is used to investigate the onset of longitudinal vortices in laminar boundary layers along horizontal semi-infinite flat plates heated or cooled isothermally from below by considering the density inversion effect for water using a cubic temperature-density relationship. The analysis employs non-parallel flow model incorporating the variation of the basic flow and temperature fields with the streamwise coordinate as well as the transverse velocity component in the disturbance equations. Numerical results for the critical Grashof number Gr _L ^* =Gr _X ^* /Re _X< ^{Emphasis>/3/2} are presented for thermal conditions corresponding to –0.5 ₁–2.0 and –0.8 ₂1.2.Nomenclature a wavenumber, 2/ - D operator, d/d - F (f–f)/2 - f dimensionless stream function - g gravitational acceleration - G eigenvalue, Gr _L/Re_L - Gr _L Grashof number based on L - Gr _X Grashof number based on X - L characteristic length, (X/U)^1/2 - M number of divisions in y direction - P pressure - Pr Prandtl number, / - p dimensionless pressure, P/( ² /Re _L) - Re _L, Re_X Reynolds numbers, (U L/)=Re _X< ^1/2 and (U), respectively - T temperature - U, V, W velocity components in X, Y, Z directions - u, v, w dimensionless perturbation velocities, (U, V, W)/U - X, Y, Z rectangular coordinates - x, y, z dimensionless coordinates, (X, Y, Z)/L - thermal diffusivity - coefficient of thermal expansion - ₁, ₂ temperature coefficients for density-temperature relationship - similarity variable, Y/L=y - dimensionless temperature disturbance, /T - dimensionless wavelength of vortex rolls, 2/a - ₁, ₂ thermal parameters defined by equation (12) - kinematic viscosity - density - dimensionless basic temperature, (T _b T )/T - –1 - T temperature difference, (T _w–T ) - * critical value or dimensionless disturbance amplitude - prime, disturbance quantity or differentiation with respect to - b basic flow quantity - max value at a density maximum - w value at wall - free stream condition     

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

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

Diffuse approximation method for solving natural convection in porous media

The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.Nomenclature H height of the cavities - I functional - K permeability - p(M _i ,M) line vector of monomials - p ^T p-transpose - M current point - Nu Nusselt number - Ri inner radius - Ro outer radius - Ra Rayleigh number - x, y cartesian coordinates - u, v velocity components - T temperature - M vector of estimated derivatives - _t thermal diffusivity - coefficient of thermal expansion - practical aperture of the weighting function - scalar field - (M, M _i ) weighting function - streamfunction - kinematic viscosity     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Modell zur Beschreibung des Fließens wandgleitender Substanzen durch Düsen

Zusammenfassung Zur Analyse des Fließens einer direkt an der Düsenwand gleitendenOstwald-deWaele-Flüssigkeit (Potenzgesetz) wird ein Modell entwickelt, das die rheologischen Vorgänge tribologisch, d. h. analog derCoulombschen Reibung fester Körper beschreibt.Es zeigt sich, daß in der Düse zwei Bereiche zu unterscheiden sind: ein Haftbereich in der Nähe des Düseneinlaufs und ein am Düsenaustritt liegender Gleitbereich. Die Länge des Gleitbereichs, der Verlauf des Drucks und der Schubspannung längs der Düse sowie die Änderung des Geschwindigkeitsprofils im Gleitbereich werden ermittelt.Überschreitet die Wandschubspannung einen kritischen Betrag, so entsteht am Düsenende ein labiler Bereich, in dem der Betrag der Wandschubspannung sprunghaft auf einen kleineren Wert sinken kann. Der von verschiedenen Autoren gefundene Sprung in der Fließkurve bestimmter Polymerschmelzen kann damit grundsätzlich erklärt werden.

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Summary Starting from theCoulomb Friction Law for solids, a theoretical model is developed for the pressure flow of a viscous power-law fluid with slip at the wall.It is shown that two flow regions exist in the die: a first region at the upstream part of the die, where the fluid sticks to the wall; and a second region at the downstream part of the die, where the fluid slips at the wall. The length of the slip region, the development of pressure and shear stress along the die as well as the change of the velocity distribution are given for the slip region.For shear stresses above a critical value, an instability region is found at the exit of the die. In this region, a sudden decrease of shear stress can occur. This seems to explain the discontinuity in the flow curve reported by several investigators.

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F Querschnittsfläche der Kapillaren - Volumendurchsatz - K _R Reibkraft - L Düsenlänge - m Stoffwert (Fließexponent) - N Normalkraft - p hydrostatischer Druck - p _L Druck am Düsenende - p ₁ Druck an der Übergangsstelle Haften-Gleiten - p ₀ Druck vor der Düse - p _0H Druck vor der Düse im Falle des Wandhaftens - r Radius - R Düsenradius - v _g Gleitgeschwindigkeit - v _z Strömungsgeschwindigkeit inz-Richtung - z Koordinate in Strömungsrichtung - z ₁ Längskoordinate der Übergangsstelle Haften-Gleiten - Schergeschwindigkeit - Stoffwert - Viskosität - µ Gleitkoeffizient - µ _H Haftkoeffizient - Dichte - dimensionsloser Radiusr/R - _rz Schubspannung in der Flüssigkeit - _rz ^(R) Wandschubspannung in der Flüssigkeit - ₀ Stoffwert - _wg Wandschubspannung im Falle des Gleitens - _wH Haftschubspannung an der Wand Auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologen in Berlin vom 28.–30. April 1975.Mit 10 Abbildungen     

Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature

The purpose of this study is the construction of interpolation formulas for the dependence of Maxwell viscosity, a quantity which is the reciprocal of shear-strain relaxation time , on shear-strain intensity and temperature for several metals: iron, aluminum, copper, and lead. This function was interpolated in various temperature and deformation velocity ranges in accordance with available experimental data for iron (0 10⁷ sec^–1, 200 ° T 1500 °); aluminum (0 10⁷ sec^–1, 300 ° T 900 °); copper (0 10⁵ sec^–1, 300 ° T 1300 °); lead (0 10⁶ sec^–1, 90 ° T 400 °); temperatures in °K.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 114–118, July–August, 1974.     

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     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.

     

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

A class of hypo-elastic non-elastic materials and their thermodynamics

Thermodynamics is developed for a class of thermo-hypo-elastic materials. It is shown that materials of this class obey the laws of thermodynamics, but are not elastic.

Table of Symbols

Latin Letters A _ijkl tensor-valued function of t _ij appearing in hypo-elastic constitutive relation - B _ijkl another tensor-valued function. See equation (4.2) - B the square of - d _ij rate of deformation tensor - d _ij deviator of rate of deformation - f, k functions of pressure, p - g, h functions of the invariant - p pressure - q _i heat flux vector - s _ij stress deviator - _ij co-rotational derivative of stress deviator - t time - t ₁ t ₂ specific values of time - t _ij stress tensor - t _ij ⁰ a specific value of stress - T Temperature - T ₀ a specific value of temperature - u _i velocity - V(t) a material volume as a function of time, t - V ₀ a material volume at a reference configuration - W work (W = work done in a deformation—section 5) Sript Letters Specific internal energy - Specific Helmholtz free energy - G Specific Gibbs function Greek Letters an invariant of the stress deviator—see eq. (2.4) - _ij kroneker delta - (W = work done in a deformation—section 5) - specific entropy - hypo-elastic potential - hypo-elastic potential - mass density - ₀ mass density in a reference configuration - specific volume = 1/ - a function of p - _ijkl a constant tensor—see eq. (2.5) - –G/ - _ij rate of rotation tensor This work is dedicated to Jerald L. Ericksen, without whose influence it would not have been possible     

A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

Shock-induced chemical processes and ‘molecular rocket reaction’ in metallocenes and their cyclodextrin inclusion compounds

Shock-induced yield enhancement has been observed in implantation of recoil atoms into metallocene and its-cyclodextrin (CD) inclusion compounds, as in the case of metal-diketonate compounds previously studied. The enhancement, however, occurs at much lower energy compared with that in metal-diketonates. In acetylruthenocene and benzoylruthenocene--CD inclusion compounds, various aspects of molecular rocket reaction have been discussed.This article was processed by the author using Springer-Verlag T_EX PJour2g macro package version 1.     

Flow Above a Heated Plate in an Ascending Current

An analysis of the results of numerical experiments in which the two-dimensional flow near a plate placed across an ascending fluid current was simulated is presented. The plate temperature was higher than that of the fluid. Fluid flows with a Prandtl number 0.25 Pr 7 were considered on the moderate Reynolds and Richardson number ranges 25 < Re 100 and 0 Ri < 20. Under these conditions, two flow patterns were observable, which differed from each other by the intensity of the transverse oscillations of a system consisting of attached twin vortices and the near wake. For different Prandtl numbers, in the (Re, Ri^1/2) plane the pattern stability boundaries were established, together with the distinctive features of pattern-to-pattern transition. It was found that the vortex arrangement in the wake above the heated plate can differ from that in the von Kàrmàn street in the absence of buoyancy     

The irrotational waves and the convergence of Levi-Civita's series

Summary The paper shows that the existence of irrotational surface waves established by an investigation of Levi-Civita may be extended till the breaking, but that it is valid when is 0,84 p 1 (and when p₁=np with n whole number). No solution is known till to day when is p<0,84 (and when is 1,00<p₁<<1,68).

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Sommario La nota estende la dimostrazione dell'esistenza di onde superficiali irrotazionali, data dal Levi Civita per una ampiezza finita ma sufficientemente piccola dimostrando l'esistenza di queste onde fino al frangimento. Tanto questa dimostrazione che quella originaria di Levi Civita sono valide finchè sia 0,84 p 1 (e quando sia p₁=np con n intero). Nessuna soluzione è nota fino ad oggi quando sia p<0,84 (e quando sia 1 p₁ 1,68).