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.

     

Dynamic shakedown by modal analysis

Summary Dynamic shakedown of discrete elastic-perfectly plastic structures under a specified load history is studied using the dynamic characteristics of the structure provided by modal analysis. Several statical and kinematical theorems are presented, including lower and upper bound theorems for the minimum adaptation time of the structure. In the formulation of the kinematical theorems a crucial role is played by the appropriate definition of admissible plastic strain cycle.

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Sommario Si studia il problema dell'adattamento dinamico (shakedown) di una struttura discreta elasto-perfetta-mente plastica e soggetta ad una storia di carichi prestabilita, facendo uso a tale scopo delle caratteristiche dinamiche della struttura fornite dalla analisi modale. Vengono presentati svariati teoremi, sia di tipo statico che cinematico, tra cui taluni teoremi di delimitazione superiore ed inferiore del tempo minimo di adattamento. Nella formulazione dei teoremi cinematici ha un ruolo cruciale la corretta definizione di ciclo deformativo ammissibile.

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This paper is part of a research project sponsored by the National (Italian) Research Council, C.N.R., Group of Structural Engineering, and by the National Department of Education (M.P.I.).     

A Stefan-type free boundary problem in a thermal switch

Summary We study a mathematical model of thermal switch in a glass under the Joule heating. We prove the existence of solutions for the free boundary problem related to the model. This problem is a Stefan-type problem with a source depending on the free boundary and the latent heat depending on the history of the free boundary.

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Sommario Si studia un modello matematico che descrive un fenomeno di cambiamento di fase in un vetro attraversato da una corrente elettrica. In particolare si dimostra l'esistenza di soluzioni per il problema a frontiera libera connesso con il modello. Questo problema è del tipo di Stafan, con sorgente funzione della frontiera libera e con calore latente che dipende dalla storia della frontiera.

     

Variational principles and differential inclusions for unilateral constraints in analytical mechanics

Summary The present paper deals with variational principles in terms of hemivariational inequalities and with differential inclusions for unilateral constraints in analytical mechanics. It is discussed how it is possible to describe one-sided constraints in a very general manner by the help of Clarke's generalized gradient.

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Sommario Questo scritto prende in esame i principi variazionali nella forma delle hermivariational inequalities e le inclusioni differenziali per vincoli unilaterali nella meccanica analitica. Si deduce come sia pissobile descrivere vincoli unilaterali, in modo generale facendo uso di gradienti generalizzati di Clarke.

     

A simplified flood routing model: variable prameter muskingum (VPM)

Summary Flood routing methods are numerical methods for estimating the movement of a flood wave along a channel reach, on the basis of the knowledge of the discharge hydrograph at the upstream end and of the hydraulic characteristics of the reach and, usually, in the hypothesis that no perturbation is coming from downstream (free boundary condition). The flood routing method wich is proposed is similar to the Muskingum one, but with variable and hydraulic parameters; it is able to estimate water levels too; is effective even if kinetic terms are not completely negligible; take advantage of the insignificance of the downstream condition and make it possible to obtain results starting upstream and proceeding downstream; for simplicity's sake, take advantage of the fact that the discharge loop of normal flood waves is quite small. Obtained results are much better that those obtainable from constant parameters methods and indeed, if the flood loop is less that 10%, very similar to those obtainable from more complex and time consuming models.

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Sommario I metodi di flood routing studiano la propagazione di un'onda di piena lungo un tratto di un corso d'acqua, assegnato l'andamento temporale della portata nella sezione di monte e le caratteristiche dell'alveo, e usualmente nell'ipotesi di assenza di perturbazioni provenienti da valle (condizione di valle passiva). Viene qui proposto un procedimento di flood routing, formalmente simile ad un Muskingum ma con i parametri variabili e calcolati per via idraulica; idoneo a stimare anche i livelli idrici; valido anche se i termini cinetici non sono del tutto trascurabili; che sfrutta l'irrilevanza della condizione di valle procedendo a cascata da monte a valle; che sfrutta, a vantaggio della semplicità, il fatto che per le normali onde di piena dei corsi d'acqua il cappio di portata è di dimensioni modeste. I risultati ottenuti sono molto migliori di quelli ottenibili con metodi a parametri costanti e, almeno per i casi in cui il cappio relativo è inferiore al 10%, paragonabili a quelli ottenuti con metodi molto più complessi ed onerosi.

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List of symbols x, t channel distance, starting upstream; time - z water surface height above datum - Q volumetric rate of discharge - P(x, z) steady rating curve - q=Q–P flood loop - g acceleration of gravity - A, B cross section wetted area and free surface width - I, S water surface slope and friction slope - c kinematic wave velocity - F Froude number - L,L ₀,L ₁,L ₂,L ₃ characteristic lengths of the channel - T=L/c characteristic time of the channel - D diffusion - p, l time and space steps - K, X Muskingum parameters - C ₁,C ₂,C ₃,C ₄ Muskingum coefficients - f _x=f/x,f _t=f/t etc. for the partial derivatives Paper presented at the First Italian Meeting of Computational Mechanics held in Milan, June 24–26, 1986.     

On mean rotations in finite deformations

Summary Firstly, the classical interpretation of the mean rotation tensor of Cauchy and Novozhilov is revised. Indeed, two distinctive and defective features of this interpretation reflect a severe restriction on the class of admissible deformations. Secondly, an alternative measure of mean rotation is introduced and its explicit aspect for pure rotations, pure strains and additively pure rotations is determined.

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Sommario In questa nota viene riesaminata l'interpretazione classica del tensore di rotazione media di Cauchy e Novozhilov. Preliminarmente, si mostra come due tratti distintivi di questa interpretazione ne limitino l'ampiezza e riflettano una severa restrizione della classe delle deformazioni ammissibili. Successivamente, si perviene ad una misura alternativa di rotazione media, il cui calcolo esplicito è condotto nei casi di rotazioni pure, deformazioni pure e rotazioni additivamente pure.

     

The Italian contribution to the mathematical theory of elasticity

Summary The contributions of Italian mathematical physicists to the mathematical theory of elasticity are reviewed, starting from the second half of the last century. The work of Piola, Betti, Beltrami, Volterra, Somigliana, Signorini and many other eminent scholars is briefly outlined, showing how Italian mathematics achieved a leading position, comparable to that unanimously accorded Italy in other areas of mathematics, in this field where theory is so close to application.

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Sommario Vengono passati in rassegna i contributi che i fisico-matematici italiani, a partire, circa, dall'inizio della seconda metà del secolo scorso, hanno dato alla teoria matematica dell'elasticità. L'opera di Piola, Betti, Beltrami, Volterra, Somigliana, Signorini e di tanti altri eminenti studiosi viene brevemente riassunta e posto in luce come la matematica italiana, anche in questo campo, cosi vicino alle applicazioni, abbia raggiunto posizioni di primato, comparabili a quelle che, unanimamente, sono state riconosciute all'Italia in altri settori della matematica.

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Invited paper.This article was taken from a lecture given by the author at the University of Palermo on December 16, 1977 and published in the Rendiconti del Circolo Matematico di Palermo (vol. XXVIII, sr. II). The editors of this publication have kindly given Meccanica permission to publish this abbreviated version of the lecture in English.     

Some problems concerning nonstationary currents in dense plasma systems confined by walls

Nonstationary currents are examined in a dense magnetized plasma with 1, in which energy release and heat loss by thermal conduction and radiation are possible. Solutions are found in two limiting cases: ¦f¦ ¦ div (T)¦ and ¦f¦ ¦ div(T)¦ (f is the radiation intensity, is the coefficient of heat conduction, and T is the temperature). In the first case a solution was obtained of some problems of the cooling and heating of a plasma illustrated in part by the evolution in time of the temperature profile in the boundary layer. In the second case an isomorphic solution was found for an arbitrary dependence of the coefficient of heat conduction on the temperature, pressure, and magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–8, January–February, 1972.The author is grateful to G. I. Budker for formulating the problem.     

Thermal stresses in plane elasticity: Numerical solutions based only on harmonic functions

Summary A theoretical treatment is outlined allowing solution of thermal stress problems in plane elasticity by using only numerical methods suited to solving — in 2 D — the Laplace equation. Only one type of element matrix (supposing for the sake of simplicity F.E.M.s are used) and only one mesh would thus be required, both for the determination of the thermal field and of the displacement/stress field. The numerical solutions required in the plane domain of interest entail, consequently, only one variable per node in place of two. Even if numerous unit solutions are required in order to impose arbitrary boundary conditions, this reduction of nodal variables allows to spend less computation time in solving linear systems, at least for problems of a certain extent.

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Sommario Si delinea una tecnica che permette di risolvere i problemi di coazioni termiche in elasticità piana facendo uso solo di metodi numerici atti a risolvere l'equazione di Laplace. Un solo tipo di matrice degli elementi (nel caso si usi una formulazione a E. F.) e una sola reticolazione sono pertanto richiesti tanto per il problema termico come per quello elastico. Ne consegue altresì il vantaggio che le soluzioni numeriche richieste nel dominio di interesse cornportano una sola variabile per nodo, anzichè due; anche tenuto conto che sono richieste numerose soluzioni unitarie per poter imporre condizioni al contorno comunque definite, ne deriva per problemi di una certa ampiezza una riduzione del tempo di calcolo speso nella soluzione di sistemi.

     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)     

Derivation of a non-linear model equation for wave propagation in bubbly liquids

Summary Though the use of an asymptotic approach, it is deduced a fourth order partial differential equation governing the evolution of nonlinear (sound) wave propagation in bubbly liquids. Remarkably it involves also higher order nonlinear terms. Some results concerning the steady-state solution as well as the dispersion relation are obtained.

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Sommario Mediante l'usa di uno schema asintotico viene dedotta una equazione differenziale alle derivate parziali del quarto ordine, atta a descrivere l'evoluzione di un'onda non lineare propagantesi con la velocità del suono in un liquido con bolle. Tale equazione in particolare presenta delle non linearità anche nette derivate di ordine più elevato. Infine vengono discussi alcuni risultati numerici connessi con la ricerca di soluzioni del tipo onda stazionaria e con la relazione di dispersione.

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This work was supported by M.P.I. through Fondi per la ricerca scientifica 40% e 60%.     

The Kepler problem spinor regularization and pre-quantization of the negative-energy manifold

Summary We extend the results of our two foregoing papers to the whole phase space of the Kepler problem, by: i) developing a natural spinor extension of the KS-regulari-zation of the Kepler problem, due to Kustaanheimo and Stiefel, and ii) applying the results to the pre-quantization (in the sense of Souriau) of the negative-energy manifold, the energy levels satisfying a well-known condition. We show that the quantized manifold is diffeomorphic to the symplectic productS ₃×S ₃ of two 3-spheres of the spinor phase space, factored through a suitable equívalence relation.

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Sommario Estendendo i risultati ottenuti in due prece denti lavori all'intero spazio delle fasi del problema di Keplero, si perviene nella presente nota ad una naturale estensione spinorialedella trasformazione KS (introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero) la quale consente di prequantizzare, nel senso di Souriau, la varietà delle orbite kepleriane isoenergetiche, con energia negativa spddisfacente ad una ben nota condizione. Si mostra che la varietà quantica associata è diffeomorfa al prodotto simplettico S ₃×S ₃,opportunamente quozientato, di due 3-sfere dello spazio spinoriale delle fasi.

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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.).     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - 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 - v traditional superficial volume averaged 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 - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

On a path-independent integral approach to fast tearing of elastomers

Summary The feasibility of a path-independent integral approach to fracture mechanics of elastomers and the objectivity of the pertinent crack stability criterion are discussed with reference to tearing with negligible mechanical dissipations.

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Sommario In questo articolo si discutono le possibilità offerte dall'impiego dell'integrale J e del relativo criterio di stabilità della cricca nello studio della frattura veloce di materiali elastomerici.

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List of symbols and abbreviations F Deformation gradient. - Feled Finite Element for Large Elastic Deformations. - T Strain energy release per unit crack growth (termedG in fracture mechanics of metals). - T _l First Piola-Kirchhoff stress tensor. - u Displacement vector. - W Stored energy function per unit of volume in the undeformed configuration. - x Spatial co-ordinates. - X Material co-ordinates.     

Thermal and diffusive relaxation of an evaporating drop with internal heat liberation

The problem of nonsteady-state evaporation or growth of a radiating drop with uniformly distributed internal heat sources is considered. The Reynolds R=ua/v 1 and Peclet P_D= ua/D 1 numbers are assumed to be small (a is the radius of the drop, u the velocity of its relative motion, andv, D, the coefficients of viscosity, diffusion and thermal diffusivity of the vapor-gas medium). This enables the convective transfer of vapor and heat to be neglected, and the concentration and temperature fields to be regarded as spherically symmetric [1]. In view of the fact that the density of saturated vapor is less than the density of liquid the convective flow caused by the change in radius of the drop is not taken into account [2]. It has already been shown [3,4], that for _r (, _r are the coefficients of molecular and radiative thermal conductivity) there exists a bounded region ryo (1/) /_r ( is the absorption coefficient for radiation in the gas), in which the effect of radiation on the temperature relaxation of the vapor-gas medium is negligible. If the conditiona (1/) /_r is satisfied, then the temperature at the outer boundary of this region will be practically the same as the temperature at infinity T=T. This means that terms in the energy equation connected with energy transferred by radiation can be neglected. It is assumed that the free path of molecules in the gas is less than the radius of the drop, and so concentration and temperature discontinuities close to the surface of the drop can be neglected [2].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 78–87, January–February, 1970.The authors are grateful to V. G. Levich for discussing the results of the paper.     

Heat transfer at a wedge in a hypersonic low-density gas stream

Results are presented of an approximate analysis of steady heat transfer on a sharp thin (sin < 1) strongly cooled (t_w 1) wedge, washed by a hypersonic (M 1) gas stream at zero angle of attack under almost free-molecule conditions. Dimensionless parameters on which the heat transfer depends have been established; approximate formulas for estimation purposes are given.The present analysis is not a rigorous quantitative theory and its results should be regarded only as estimates; its conclusions may be useful for experimental planning and for generalization of test data. The method used here is analogous to that applied in [1] in analyzing flow over a flat plate.