The Kepler problem spinor regularization and quantization on the manifolds S2

Summary In a previous research we have shown that the KS-transformation, developed by Kustaanheimo and Stiefel for the regularization of the Kepler problem, may be interpreted as the correspondence which associates to each null 4-vector of the space of Minkowski a one-index spinor, defined up to a phase factor, and we have obtained a new form of the KS-transformation. In the present research we show that this formulation allows a straight derivation of the Hopf fibering of the sphere S³ (characterized by unit spinors) having the base space given by the section (sphere S²) of the light cone, and we show that the KS-transformation allows the quantization of the symplectic manifold S² in the sense of Souriau. The sphere S³ turns out to be a contact quantized manifold. The bilinear relation characteristic of the KS-theory and the column vectors of the KS-matrix are intimately related to the contact structure.

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Sommario In un precedente lavoro si è mostrato che la trasformazione KS, introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero, è riconducibile alla ben nota corrispondenza fra vettori del cono isotropo dello spazio di Minkowski e spinori semplici, definiti a meno della fase, e si è pervenuti ad una nuova formulazione della KS. Nel presente lavoro si mostra come da tale formulazione scaturisca in modo naturale la fibrazione di Hopf della sfera S³ (caratterizzata dagli spinori unitari) avente quale base una sezione (sfera S²) del cono isotropo e si mette in luce come la trasformazione KS consenta di effettuare la quantizzazione della varietà simplettica S² nel senso di Souriau e di ottenere la sfera S³ quale varietà quantica di contatto. La relazione bilineare caratteristica della teoria KS ed i vettori colonna della matrice KS risultano intimamente legati alla struttura di contatto.

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Presented at the VI Congresso Nazionale dell'Associazione Italiana di Meccanica Teorica ed Applicata (AIMETA), Genova, October 1982. Work performed under the auspices of G.N.F.M. of the C.N.R. (Consiglio Nazionale delle Ricerche).     

Columnar vortices in isotropic turbulence

The structure of the intense vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the rangeRe =35–170, and is found to be organized in coherent, cylindrical or ribbon-like, vortices (worms). At the Reynolds numbers studied, they are responsible for much of the extreme intermittent tails observed in the statistics of the velocity gradients, but their importance seems to decrease at higherRe . Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow, while their circulation increases monotonically withRe . An explanation is offered for this latter scaling, based in the assumed presence of axial inertial waves along their cores, excited by a random background strain of the order of the root mean square vorticity. This explanation is consistent with the presence of comparable amounts of stretching and compression along the vortex cores.

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Sommario La struttura di regioni ad intensa vorticità in campi di flusso turbolento omogenei, isotropi ed in equilibrio, simulati numericamente, viene studiata per quattro differenti numeri di Reynolds nell'intervalloRe =35÷170, e si trova che tali regioni si organizzano in vortici coerenti, cilindrici o a forma di nastro (vermi). Con rifermento ai numeri di Reynolds studiati, si vede che tali vortici sono responsabili per gran parte delle code estreme ed intermittenti, osservate nelle statistiche dei gradienti di velocità, ma la loro importanza sembra decrescere a più altiRe . I loro raggi scalano con la microscala di Kolmogorov e le loro lunghezze con la scala integrale del flusso, mentre la loro circolazione cresce monotonicamente conRe . Per quest'ultimo riscalamento viene offerta una spiegazione basata sull'assunzione della presenza di onde inerziali assiali lungo i loro nuclei, eccitate da una deformazione di fondo casuale dell'ordine della radice quadrata della velocità media. Questa spiegazione è consistente con la presenza di incrementi paragonabili di allungamenti e compressioni lungo i nuclei dei vortici.

     

“Modified” phase equilibria

Summary The Authors examine first and second order phase equilibria for classical bodies in the sense of Toupin and Truesdell[9], by assuming a partial equality between the intensive thermal substates of the two phases.In the case of first order equilibria, a system of partial differential equations for the equilibrium manifold is proposed. This system, which reduces to the generalized Clapeyron equations on the assumption of complete equality of the intensive variables[13], is shown to be always integrable.In the case of second order equilibria, a similar modification of the generalized Ehrenfest equations[14] is proved. A straight-forward derivation of the integrability conditions for the system of partial differential equations is given.

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Sommario Gli Autori esaminano gli equilibri di fase del primo e del secondo ordine per corpi classici nel senso di Toupin and Truesdell[9], nella ipotesi di eguaglianza parziale dei sottostati termici intensivi delle due fasi.Nel caso di equilibri del primo ordine gli Autori deducono un sistema di equazioni alle derivate parziali per la varietà di equilibrio, estendendo le equazioni di Clapeyron generalizzate relative agli equilibri di fase con completa eguaglianza delle variabili intensive[13]. Vengono ricavate e discusse le condizioni di integrabilità del sistema.Per gli equilibri del secondo ordine, gli Autori pervengono ad un sistema di equazioni che estende le equazioni di Ehrenfest generalizzate, perchè rimuovono l'ipotesi di totale eguaglianza delle variabili intensive[14]. Anche di questo sistema vengono ricavate le condizioni di integrabilità.

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Work supported by the Gruppo Nazionale per la Fisica Matematica, C.N.R.     

Properties of parallel — surfaces coordinate systems

Summary The paper is devoted to the study of the basic properties of the set of parallel-surfaces coordinate (p.s.c.) system previously introduced by the author for the most appropriate formulation of the differential form of the balance equations in fluid dynamic problems, where a «direction field» plays a fundamental role. This field is usually defined by the unit normal to a given surface S either known or unknown a-priori.To properly exploit the invariant properties enjoyed by both holonomic and anholonomic sets of basic vectors are introduced in E ₃ , the latter ones resting on the coordinate vectors of the variety V ₂ (0)=S. The corresponding tensor algebra and tensor calculus are based on an appropriate affine connection of E ₃ reducing to the metric connections for holonomic basis vectors. They are developed and suitably commented upon in the paper.

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Sommario Il presente lavoro deriva le proprietà fondamentali dell'insieme di sistemi di coordinate in cui una delle famiglie di superfici coordinate è la famiglia di superfici parallele ad una data superficie S, nota o meno a-priori. Tali sistemi di coordinate sono stati precedentemente introdotti dall'autore per la formulazione più appropriata delle equazioni differenziali del bilancio per problemi fluidodinamici in cui un «campo di direzioni», definito dal versore normale ad S, gioca un ruolo fondamentale.Per sfruttare nel modo più adeguato le proprietà di invarianza godute da, vengono introdotti in E ₃ sistemi di vettori base sia olonomici che anolonomici, questi ultimi basati sui vettori coordinati della varietà V ₂ (0)=S. L'algebra tensoriale corrispondente ed il calcolo tensoriale basato su una opportuna connessione affine di E ₃ riducentesi a quella metrica per sistemi olonomici, vengono sviluppati e commentati evidenziando i vantaggi offerti dall'insieme.

     

Base-pressure pulsations behind a cylinder and disk in a subsonic stream flow

Base-pressure fluctuations behind a long cylinder (l/d 5–10) and the disk (l/ d 0.0) is investigated experimentally in this paper. The spectral and correlation characteristics of the base-pressure fluctuations behind axisymmetric bodies at a Mach number M 1.0 are generalized on the basis of the data obtained and the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–183, January–February, 1977.     

On Szebehely's problem extended to holonomic systems With a given integral of motion

Summary We consider a family of curves in the n-dimensional configuration space S_n of a holonomic system with n degrees of freedom. We obtain first-order partial differential equations for the potential function U of forces under which any trajectory belonging to the given family of curves can be described by the representative point of the system. We write the potential function U supposing that, in addition to the energy integral, a first integral of motion linear in the lagrangian velocities is assigned. Next we obtain the compatibility conditions between the energy constant E, the parameter which appears in the first integral, and the n–1 geometric constants c₁, c₂,, c_n–1 which characterize the family of trajectories. Finally we discuss two simple examples.

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Sommario Assegnata una famiglia di curve nello spazio S_n delle configurazioni di un sistema olonomo conservativo ad n gradi di libertà, si determinano le equazioni differenziali alle quali deve soddisfare il potenziale delle forze applicate affinchè il punto rappresentativo del sistema possa descrivere una qualunque delle traiettorie appartenenti alla data famiglia. Si scrive poil' espressione del potenziale supponendo che sia assegnato, oltre l'integrale dell'energia, un integrale primo del moto, lineare nelle velocità lagrangiane. Infine si ricavano le condizioni di compatibilità fra la costante dell'energia E, il parametro che interviene nell'integrale primo e le n-1 costanti geometriche c₁, c₂, c₂,, c_n–1 che caratterizzano la famiglia di curve. Si conclude con due semplici esempi che illustrano quanto esposto.

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Work performed under the auspices of G.N.F.M.-C.N.R., Italy.     

Zur Verknüpfung von Stoffübergang und physikalischen bzw. chemischen Wandvorgängen in technischen Reaktoren

In technical reactors like catalytic honeycombreactors with a reaction at the wall and with an unknown field of concentrationc _A (r,z) the diffusive flux _A,D.w is replaced by the transferflux _A,. The transferflux _A, i.e. the Sherwood-number only depends on processes, which effect the diffusive flux, i.e. the gradient of concentration c _A /r¦ _w . For vaporisation with a constant concentrationc _A,w at the wall or for a heterogeneous reaction with a variable concentrationc _A,w the fluidstream would be such a process. In any case Sherwood-numberSh depends only on Bodenstein-numberBo and is — even for a heterogeneous reaction — no function of Damköhler-number. Only in case of a homogeneous reaction in the fluid phase there is a influence on the gradient of concentration and it followsSh (Bo, Da _I).According toSh _neu (Bo, Da _II) the gradient of concentration c _A /r in _A,D is replaced in _A, by the mean concentration and not, as usual, by the difference of concentration . Both concentrations depends on the Damköhler-numberDa _II. The difference of concentrations shows either no or only little dependence ofDa _II (this illustrates the quality of representation of _A,D,w by _A, ). If is defined by , than _neu depends onDa _II Sh _neu =Sh _neu (Bo, Da _II). According to this definition of _neu no new facts will arise. The common theoretical or experimental values of orSh are applicable to every process with heterogeneous reactions. In analogous cases the following explanations are also valid for heat-transfer at the wall, if a heterogeneous reaction takes place.Zusammenfassung In technischen Reaktoren, wie z. B. einem katalytischen Wabenrohrreaktor, in denen sich das Konzentrationsfeldc _A (r, z) nicht genau ermitteln läßt, ersetzt die Übergangsstromdichte _A, vereinfachend die Diffusionsstromdichte _A,D,w an die Wand. Die Übergangsstromdichte _A, bzw. dieSh-Zahl ist nur von den Vorgängen abhängig, die die Diffusionsstromdichte an die Wand, _A,D,w , d. h. den Konzentrationsgradienten c _A /r¦ _w beeinflussen. Bei Verdampfung mit konstanter Wandkonzentrationc _A,w oder bei einer heterogenen Reaktion an der Wand mit veränderlichemc _A,w ist ein solcher Vorgang z. B. die Strömung, d. h. in beiden Fällen istSh =Sh (Bo) und hängt auch bei einer heterogenen Reaktion nicht von der Damköhler-II-Zahl ab. Nur wenn in der strömenden Phase (zusätzlich) eine homogene Reaktion vorliegt [12], hat diese einen Einfluß auf den Gradienten und es giltSh =Sh (Bo, Da _I).Die AbhängigkeitSh _neu (Bo, Da _II) entsteht definitorisch dadurch, daß der in _A,D auftretende Gradient c _A /r in _A, z. B. durch eine mittlere Konzentration ( _neu), statt wie üblich durch eine Konzentrationsdifferenz , erfaßt wird. Beide Konzentrationen sind von der DamköhlerzahlDa _II abhängig, ihre Differenz aber nicht bzw. wenig (worin sich die Güte der Abbildbarkeit von _A,D,w durch _A, verdeutlicht). Läßt man also in der Definitionsgleichung für die Stoffübergangszahl die veränderliche Wandkonzentrationc _A,w weg, dann entsteht eine entsprechende starke Abhängigkeit der Größe _neu (bzw.Sh _neu) von derDa-II-Zahl:Sh _neu=Sh _neu (Bo, Da _II). Neue Sachverhalte werden mit solchen Definitionen von _neu nicht begründet. Die üblichen theoretisch oder experimentell ermittelten - oderSh-Werte können beim Auftreten von Stoffwandlungsvorgängen an der Wand für reaktionstechnische Berechnungen verwendet werden. Die folgenden Ausführungen gelten bei gegebener Analogie auch für den Wärmeübergang mit physikalischen oder chemischen Wandvorgängen, wie Verdampfung oder chemischer Wandreaktion.     

Effect of nonlinear constitutive equation on vibrations of beams

Summary This paper is concerned with the analytical investigation of static and dynamic nonlinear behaviors of beams with different boundary conditions. While geometric type of nonlinearities on beams have been investigated extensively, material type nonlinearities have received very little attention. Therefore, material nonlinearities of the Ramberg-Osgood type are considered in this analysis. The use of Self-Generating functions for nonlinear beam problems is demonstrated for this type of nonlinearity. Transverse shear and rotatory inertia effects have been included in the formulation to study moderately thick beams. For all the cases investigated here nonlinear frequency ratios are calculated at various amplitudes of vibration and geometric parameters of beams. Numerical results indicate that the Ramberg-Osgood type nonlinearity produces softening-type responses. The study is limited to materials which are nonlinearly elastic and the effect of geometric nonlinearity is not considered in this paper.

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Sommario Questo lavoro riguarda lo studio dei comportamenti non lineari statici e dinamici di travi con diverse condizioni a contorno. Mentre le non linearità di tipo geometrico sono state studiate estesamente, quelle di tipo non lineare hanno ricevuto un'attenzione molto ridotta. Perciò in questa ultima analisi si considerano non linearità materiali del tipo di Ramberg-Osgood. Si dimostra uso delle funzioni autogeneratrici nei problemi non lineari per le travi con questo tipo di non linearità. Nella formulazione dello studio di travi moderatamente spesse si sono inclusi effetti di taglio e inerzia rotatoria. Per tutti i casi qui studiati si calcolano i rapporti di frequenza non lineare per varie ampiezze di vibrazione e parametri geometrici delle travi. I risultati numerici indicano che la non linearità del tipo di Ramberg-Osgood produce risposte del tipo ammorbidimento. Lo studio si limita a materiali con nonlinearità elastica e non si considera nel lavoro l'effetto della non linearitá geometrica.

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List of symbols T _s Transverse shear - k ₁ Shear connection factor - A, B, m Material constants - I Moment of inertia - G Shear modulus - Mass per unit length - w Lateral displacement - h Beam thickness - b Bredth of beam - t Time - R _i Rotatory inertia - a Area of cross-section of beam - q(x) Lateral load on beam - _x Stress - _x Strain - Length of beam - x Beam coordinate - - r Radius of gyration - w Nondimensional maximum deflectionw _max/r - q ₀ ^* Nondimensional load, (q ₀ ³/Al - Thickness parameter,h/ - (T _sK₁/Ga) - ₀ Linear frequency - Nonlinear frequency - w _max w measured at the point of maximum deflection.     

Similarity Solutions for Thawing Processes with a Heat Flux Condition at the Fixed Boundary

Similarity solutions for a mathematical model for thawing in a saturated semi-infinite porous medium is considered when change of phase induces a density jump and a heat flux condition of the type is imposed on the fixed face x=0. Different cases depending on physical parameters are analysed and the explicit solution is obtained if and only if an inequality for the thermal coefficient q ₀ is verified. An improvement for the existence of a similarity solution for the same free boundary problem with a constant temperature on the fixed face x=0 is also obtained. Sommario. Vengono considerate soluzioni di similarità per un modello matematico di disgelo di un mezzo poroso saturo semi-infinito allorquando il cambiamento di fase induce un salto di densità ed una condizione di flusso di calore del tipo viene imposta sulla faccia fissa x=0. Si analizzano differenti casi dipendenti da parametri fisici e la soluzione esplicita viene ottenuta se e solo se risulta verificata una diseguaglianzo per il coefficiente termico q ₀. Si ottiene altresi un miglioramento della condizione di esistenza di una soluzione di similarità per lo stesso problema al contorno libero con temperatura costante sulla faccia fissa x=0.     

Experimental study of mutual interference between two spheres placed on a plane boundary

This paper describes an experimental study of the mutual interference between two spheres placed on a plane boundary. The experiment was carried out in an N. P. L. type wind-tunnel having a working section of 500×500×2000 mm³ in size at a Reynolds number of 4.74×10⁴. The surface-pressure distributions of two spheres were measured for the various relative positions of two spheres and the drag, side-force, and lift coefficients were determined from surface-pressure distributions. The separation of the flow and the formation of vortices were observed by the method of visualization. The distributions of velocities, and turbulent intensities of the flow past two spheres were measured. The experimental results for two spheres were compared with those of a single sphere.List of symbols C _D drag coefficient - C _L lift coefficient - C _p surface-pressure coefficient of sphere=(P-P )(qU ² ) - C _s coefficients of side force - D diameter of sphere [mm] - P static pressure [Pa] - P static pressure in free stream [Pa] - Re Reynolds number= DU/v - S spacing between the centers of two adjoining spheres in plane view [mm] - U time-mean velocity in X-direction [m/s] - [m/s] free stream velocity [m/s] - u, v, w X, Y and Z-components of velocity fluctuation [m/s] - X, Y, Z coordinate axes with origin at the bottom center of test sphere, X, Y, Z axis being taken in the streamwise, lateral and vertical directions respectively [mm] (Fig. 1) - latitude angle [°] - longitude angle [°] - angle between the line connected with the centers of two spheres and wind direction [°] (Fig. 2) - kinematic viscosity of air [m²/s] - density of air [N/m³] This paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, Sept. 22–24, 1986     

First order phase equilibrium for classical bodies

Summary In this note the Author gives a general and unified treatment of first order phase equilibria for classical bodies like those considered by Truesdell and Toupin in[3].The Author reaches a system of partial differential equations (generalized Clapeyron equations) the conditions of whose solution are shown always to be satisfied.In particular, the Author derives the equations governing the polarized phase equilibrium for a fluid.Besides the equations ruling the phase equilibrium for a two phase n-component fluid mixture are given and the equivalence with the statical Gibbs-Duhem relation is shown.

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Sommario In questa nota l'Autore presenta una trattazione generale ed unificata degli equilibri di fase del primo ordine per corpi classici come quelli definiti da Truesdell e Toupin in[3].L'Autore perviene ad un sistema di equazioni alle derivate parziali (equazioni di Clapeyron generalizzate) del quale si dimostra la integrabilità.In particolare, si deducono le equazioni che governano gli equilibri di fase polarizzati.Inoltre si ottengono le equazioni che regolano l'equilibrio di fase per una miscela fluida a n componenti; in questo caso si dimostra l'equivalenza delle equazioni con la relazione statica di Gibbs-Duhem.

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This work was supported by the Gruppo Nazionale per la Fisica Matematica of C.N.R.     

An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows

Summary A method developed by the author for the systematic study of the thermodynamic and dynamic properties of the gas behind a shock wave is reported.The method is applicable to supersonic flow regimes for which the excitation, dissociation and ionization effects invalidate the usually adopted hypothesis of ideal gas.An Ideal Equivalent Gas, having the ratio of the specific heats _s dependent on Mach number and altitude of flight is postulated.On the basis of the mass, momentum and energy conservation equations, valid through the shock wave, the relations defining the thermodynamic and dynamic state of the gas behind the shock wave are derived. These relations establish an extension of the classic relations valid for the ideal gas and reduce to them identically for _s=.The dependence of the ratio of specific heats _s of the Ideal Equivalent Gas on Mach number and altitude has been established, over a wide range, on the basis of the real gas solutions derived by Huber.

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Sommario Nella presente nota viene esposto un metodo sviluppato dall'autore per lo studio sistematico dello stato termodinamico e dinamico del gas a valle di un'onda d'urto in regime supersonico, allorchè cioè gli effetti dell'eccitazione dei gradi di libertà vibrazionali delle molecole e della loro dissociazione e successiva ionizzazione invalidano l'ipotesi di gas ideale generalmente adottata.Viene definito un gas ideale equivalente avente rapporto dei calori specifici _s funzione del numero di Mach e della quota di volo ed in base alle equazioni di conservazione della massa, della quantità di moto e dell'energia, valide attraverso all'onda d'urto, vengono derivate delle relazioni definenti lo stato termodinamico e dinamico del gas a valle dell'onda d'urto. Tali relazioni costituiscono una estensione delle classiche relazioni dell'urto valide per il gas ideale alle quali si riducono per _s=.La dipendenza del rapporto dei calori specifici _s del gas ideale equivalente, dal numero di Mach e dalla quota è stata stabilita sulla base delle soluzioni ottenute da Huber per il gas reale.

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New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow

The shape evolution of a two-dimensional bubble, bounded by a simple closed curve, which is initially placed within a potential viscous flow, is analysed. It is assumed that the influence of gravity and inertia forces is negligible, so the quasi-steady approximation can be applied. Reformulating the problem for Stokes equations with relevant boundary conditions at the free surface in terms of the bianalytic stress-stream function, and using the time-dependent conformal mappingz(,t) of a unit disk onto an unbounded flow domain sought, an infinite system of ordinary differential equations for the Laurent coefficients ofz(,t) is derived. A class of exact solutions is found for the case when the principal part of the complex velocity of the dominant flow at infinity is a polynomial, and the problem of formation of a pointed bubble is discussed.

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Sommario E analizzata l'evoluzione di una bolla bi-dimensionale, limitata da una curva chiusa semplice, inizialmente posta in un flusso potenziale viscoso. Si assume che l'influenza della gravità e delle forze inerziali sia trascurabile, cosicchè si può applicare l'approssimazione quasi-stazionaria. Riformulando il problema per le equazioni di Stokes con le opportune condizionial contorno sulla superficie libera in termini della funzione stress-stream bianalitica, e usando la tecnica delle trasformazioni conformi dipendenti dal tempoz(,t) di un disco unitario su un dominio di flusso non limitato incognito, viene derivato un sistema infinito di equazioni differenziali ordinarie per i coefficienti di Laurent diz(,t). Viene trovata una classe di soluzioni esatte per il caso in cui la parte principale della velocità complessa del flusso dominante all'infinito è una polinomiale, ed è discusso il problema della formazione di una bolla lenticolare.

     

A note on the limit loads of non-standard materials

Summary The paper discusses the collapse theorems for materials which do not exhibit normality of the strain rate vector to the yield surface in the superposed stress and strain rate spaces. The consequences of the lack of normality on the designs for minimum weight are considered.

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Sommario Si presentano i teoremi fondamentali dell'analisi limite con riferimento a materiali per i quali i vettorï velocità di deformazione plastica non sono normali alla superficie rappresentativa, nello spazio delle tensioni, della condizione di plasticità. Se ne considerano quindi le conseguenze sui progetti di minimo peso.

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Study supported by the C.N.R. (Plasticity Group).

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We use here the term non-standard materials as currently used in the French literature. English-speaking Authors usually prefer the equivalent expression materials with non-associated flow law.     

The displacements solution for steady-state longitudinal vibrations of segment-wise inhomogeneous elastic rods

Summary The Laplace Integral Transform is used to construct the displacements, solution for steady-state vibrations of an inhomogeneous elastic rod made up of n, (n2) homogeneous elastic segments of the same cross section. The method consists in determining the displacements function of the inhomogeneous rod for zero initial conditions and for displacements type superimposed boundary conditions, with the residues calculated for the poles of the Laplace images of the ends boundary conditions. The correctitude of the constructed solution is then proved and several examples are given to illustrate the method (harmonic end excitations are included).

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Sommario Si usa la trasformata integrale di Laplace per costruire gli spostamenti soluzione del problema delle vibrazioni stazionarie di una barra elastica non omogenea fatta di n, (n2) segmenti elastici omogenei di uguale sezione. Il metodo consiste nel determinare la funzione per gli spostamenti della barra non omogenea in corrispondenza di condizioni iniziali nulle e per sovrapposte condizioni al contorno del tipo spostamento, coi residui calcolati per i poli delle immagini di Laplace delle condizioni al contorno agli estremi. Si dimostra poi la correttezza della soluzione costruita e si danno diversi esempi per illustrare il metodo (comprese le armoniche e le eccitazioni).

     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, 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 -{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 large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for 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, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

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 mirror relations and nonlinear viscoelasticity of polymer melts

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS ₁₂ () = – a ( – 1) (a = 0 for 1,a = 1 for 1) for the mirroring of the viscosities,S ₁₂ () = – a (–k ²/) (a = 0 for k, a = 1 for k) for the mirroring of the first normal-stress coefficients, and for the third relation. Here denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.