On the evolutions of the discontinuities of orders ⩾ m in solutions for quasi-linear systems of PDEs of order m,or for the linearizations of these systems

Summary One considers a system L[u]=0 of PDEs, quasi-linear (according to [1]) and of order m, which possesses a bicharacteristic line , as it happens in the hyperbolic case. For v=0, , –m (>0) let u(^v) be a discontinuity wave of order m+v that solves the system above and whose discontinuity hypersurface includes . The corresponding transport equations along are considered. Furthermore some interesting cases are pointed out, in which these equations turn out to be mutually equivalent in a suitable sense. Some theorems are stated to compare the transport equations for the discontinuities of the above kinds, that are connected with the systems d^hL[u]/dt^h=0 (h=0, , –m) and/or the linearization of the system L[u]=0 around any regular solution of it.

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Sommario Si considera un sistema L[u]=0 di equazioni alle derivate parziali, quasi lineare (secondo [1]) e di ordine m, il quale sia dotato di qualche bicaratteristica , come accade nel caso iperbolico. Per v=0, , –m(>0) sia u(^v) un'onda di discontinuità di ordine m+v risolvente il detto sistema e avente ipersuperficie di discontinuità contenente Si considerano le relative equazioni di trasporto lungo e si determinano casi interessanti in cui queste equazioni sono mutuamente equivalenti in senso opportuno. Si stabiliscono teoremi di confronto per il trasporto delle discontinuità del tipo suddetto, relative ai sistemi d^hL[u]/dt^h=0 (h=0, , –m) e/o alla linearizazione del sistema L[u]=0 attorno a qualche sua soluzione regolare.

     

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.

     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

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.     

Saint-Venant's principle in linear two-dimensional elasticity for non-striplike domains

Let be a simply connected domain in the x ₁-x ₂ plane which lies within the strip 0<x ₂, is a simple closed piecewise smooth curve. Let l= [(x ₁, x ₂): (x ₁, x ₂) and x ₁>0], _l = [(x ₁ x ₂): (x ₁ ,x ₂) and x ₁>1>0].Suppose that a two-dimensional homogeneous isotropic elastic body occupies , that a self-equilibrated stress loading is applied to - l, and that l is stress-free. Knowles [2] and Flavin [6] showed that the elastic energy in _l decays exponentially with respect to l with an exponential decay constant of the form k/b, where k is a universal constant. It is shown here that a decay constant of the form c/ may be obtained where c is a universal constant and is a characteristic dimension of , which is more appropriate than b for general non-striplike domains. In addition, an appropriate decay theorem is obtained for coil-like domains.     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

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).

     

Berechnung des dynamischen Verhaltens von instationären Temperaturfeldern in zylindrischen Bauteilen

Zusammenfassung Im ersten Teil dieser Untersuchung wird zur Betrachtung des dynamischen Verhaltens instationärer Temperaturfelder in den Wandungen zylindrischer Rohre ein mathematisches Modell erstellt und mit Hilfe der Laplace-Transformation ausgewertet. Im einzelnen werden dabei die Übertragungsfunktionen der Rohrwandtemperaturen hergeleitet und für den Fall der Abweichung vom stationären Zustand unter dem Einfluß äußerer Störungen explizit dargestellt.Im zweiten Teil der Untersuchung wird das sich daraus ergebende dynamische Verhalten der Wandtemperatur fluiddurchströmter Rohre für einige Beispiele in Form von Ortskurven dargestellt.

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Computation of the dynamic behaviour of unsteady-state temperature fields in cylindrical structures

In the first part of this paper a mathematical model is developed allowing the investigation of unsteady-state temperature fields in the walls of cylindrical pipes. Evaluation is done by means of Laplace-transformation. In particular the transfer function of the pipe wall temperature is derived, explicitly shown for the case of deviations from steady-state influenced by external disturbances. In the second part of this paper the resulting dynamical behaviour of the wall temperature of heat pipes containing a fluid is shown by means of Nyquist plots for several examples.

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Formelzeichen a Temperaturleitzahl m²/sec - A, B, A^*, B^*, , ¯B Integrationskonstanten °C - ber, bei, ker, kei Kelvin-Funktionen - ber₁, bei₁, ker₁, kei₁ kelvin-Funklionerion - Bi Biot-Zahl - c spezifische Wärme kJ/kg K - F Übertragungsfunktion - i –1 (imaginäre Einheit) - I₀, K₀, I₁, K₁ modifizierte Bessel-Funktionen - N Nenner (Gl. (39)) - r Rohrradius m - R normierter Abstand von der Innenwand % - s (komplexe) Laplace-Variable 1/sec - t Zeitvariable sec - T Zeitkonstante sec - u Integrationsvariable (Gl. (15)) - Y₀₀, Y₁₀, Y₁₁ Hilfsfunktionen (Gl. (35)-(37)) - Wärmeübergangszahl kW/K m² - kleine Änderung - Laplace-Operator 1/m² - Umgebungstemperatur °C - Rohrwandtemperatur °C - Wärmeleitfähigkeit kW/K m - Dichte kg/m³ - (komplexe) Kennvariable (Gl. (11)) - Frequenz 1/sec - Variable (Gl. (45)) Indizes a Rohraußenwand - FDS Frischdampfsammelrohr - F Fluid - H Heizgas - i Rohrinnenwand - m Mittel - VD Verdampferrohr - W Rohrwand - 0 zum Zeitpunkt t=t₀ - -(Überstreichung) stationärer Zustand Herrn Prof. Dr.-Ing. R. Quack zum 65. Geburtstag gewidmet.     

Eine Methode zur Bestimmung des Transportkoeffizienten bei nichtlinearen Transportvorgängen

Zusammenfassung Es wird gezeigt, daß man nichtlineare Transportkoeffizienten aus Potentialprofilmessungen auch ohne die Lösung der nichtlinearen Transportgleichung bestimmen kann. Die Methode wird auf zwei Ziegeltone angewendet, deren Feuchtigkeitsdiffusionskoeffizienten recht komplizierte Funktionen des Feuchtegehaltes sind. Die Ergebnisse werden mit den Ergebnissen aus einer numerischen Lösung für einen empirischen Ansatz nach Fujita verglichen.

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A method for evaluation of conductive properties in nonlinear transport phenomena

A prediction procedure for the conductive properties based upon the measured data on potential field has been developed. The method is particularly related to the nonlinear phenomena when the analytical solution is not available. The approach proposed is applied to the determination of the moisture diffusivity for two kinds of brick clay that posses a pronounced dependance of properties on moisture content. The results are used for Fujita's numerical solution which is compared with measured moisture field. The comparison has been performed and an agreement was found.

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Bezeichnungen u [kg_w/kg_s] Feuchtigkeit bezogen auf trockene Substanz - w [kg_w/kg_s] nach Gl. (10) - V allgemeines Potential - _s, _so [kg_s/m³] Dichte des trockenen Skeletts - D, D₀ [m²/s] isotherme Feuchteleitkoeffizient; Diffusionskoeffizient - x [m] Koordinate längs Zylinder - , ₀ [s] Zeit - J_w [kg_w /m²s] Feuchtestrom - , Koeffizienten der allgemeinen Leitungsgleichung (4) - A, B, Konstanten der Gl. (6) - , Konstanten der Gl. (7) auch der Gl. (12) - d [m] Zylinderdurchmesser - PE Petrovaradin - SK Sremski Karlovci     

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.     

Beitrag zur Berechnung des Wärmeübergangs bei Kondensation in Anwesenheit von Inertgas

Zusammenfassung Wärme- und Stoffaustausch erfolgen bei der Abkühlung eines Dampf-Gas-Gemisches unter anderen Voraussetzungen wie bei der Kondensation reinen Dampfes. Während bei kleinen Dampfgehalten die Analogie zwischen Wärme-und Stoffübergang als gültig betrachtet werden kann, müssen bei größeren Dampfanteilen Korrekturfaktoren verwendet werden; diese können für bestimmte Bereiche vereinfacht werden. Von großem Einfluß ist die Wahl der Beziehung für die Diffusionsgrenzschichtdicke entsprechend den Strömungsbedingungen. Zur näherungsweisen Ermittlung der Kühlfläche wird ein vereinfachtes Rechenverfahren beschrieben.

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When cooling a vapor-gas-mixture heat and mass transfer take place under other conditions than in case of pure vapor condensation. With a small vapor content the analogy between heat and mass transfer can be considered valid; with a great vapor content, however, corrective factors are necessary. Within certain ranges these factors may be simplified. The choice of the equation for the diffusion boundary layer corresponding to the stream conditions is of great importance. A simplified calculation method for an approximate determination of the cooling area is reported.

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Bezeichnungen d Rohrdurchmesser - D Diffusionszahl - F Fläche - k Wärmedurchgangswiderstand - p Druck - Wärmestromdichte - Wärmestrom - r Verdampfungswärme - R Gaskonstante - t, T Temperatur - w Strömungsgeschwindigkeit - x Länge - C =p _g/p Gasgehalt - relative Molekülmasse (Molekulargewicht) - Wärmeübergangskoeffizient - Stoffübergangskoeffizient - Grenzschichtdicke - Wärmeleitkoeffizient Indices a Austritt - d Dampf - D Diffusion - e Eintritt - g Gas - gem Gemisch - k Konvektion - K Kondensatfilm - m mittel - W Kühlwasser Vorgetragen auf der Arbeitssitzung des Ausschusses Wärme-und Stoffübertragung der VTG am 26. 3. 1968 in Baden-Baden.     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.     

Stabilitätsrechnung der Rohrströmung einer inkompressiblen viskoelastischen Flüssigkeit zweiter Ordnung

Zusammenfassung Für die Durchsatzströmung im Rohr wird mit Hilfe der klassischen hydrodynamischen Stabilitätstheorie gezeigt, daß die inkompressible Flüssigkeit zweiter Ordnungs = –pI + 2(d + 2t ₁ d ² –t ₀ d) stabil ist gegenüber kleinen rotationssymmetrischen Störungen.

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Summary For Poiseuille pipe flow it is shown by means of the classical theory of hydrodynamic stability, that the incompressible second-order fluids = –pI + 2(d + 2t ₁ d ² –t ₀ d) is stable with respect to small disturbances of rotational symmetry.

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Nomenklatur a _n Koeffizienten der Reihenentwicklung - c = /k komplexe Wellengeschwindigkeit - d Deformationsgeschwindigkeitstensor - D, D dimensionsloser Deformationsgeschwindigkeitstensor (Grund- und Störtensor) - e _i kovariante Basis - g Vektor der Erdbeschleunigung - I Einheitstensor - k Wellenzahl - M, O, S, Q, T Funktion vonk, Re, ₀ - p, P, p Gesamt-, Grund-, Stördruck - r, (r, , z) dimensionsloser Ortsvektor (Zylinderkoordinaten) - R Rohrradius - Re =U _M R/ Reynoldszahl - s(s ^*=s*pI) Spannungstensor (Isotroper Anteil des ) - t ₀,t ₁ Stoffzeiten, Parameter der Flüssigkeit zweiter Ordnung - t Zeit - u, U, u Vektor der Gesamt-, Grund-, Störgeschwindigkeit - U _M Maximale Grundgeschwindigkeit - v, V, v Vektor der dimensionslosen Gesamt-, Grund-, Störgeschwindigkeit - w Rotationsgeschwindigkeitstensor - W, W Rotationsgeschwindigkeitstensor, dimensionslos (Grund-, Störtensor) - x (x _r ,x ,x _z ) Ortsvektor (Zylinderkoordinaten) - Viskosität - ₀, ₁ dimensionslose Stoffzeiten - dimensionsloser Druck - Dichte - dimensionslose Zeit - Stromfunktion, dimensionslos - komplexe Frequenz, dimensionslos - = e ⁱ /x _i Nablaoperator (e ⁱ kontravariante Basis) - _* Nablaoperator, dimensionslos - R, I Real-, Imaginärteil Mit 4 Abbildungen     

Symmetries and Global Solvability of the Isothermal Gas Dynamics Equations

We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the -law equation p() with =1. Our results complete those obtained earlier for >1. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density.One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed on the line =1. This is in contrast to the common approach (when >1) which prescribes initial data on the vacuum line =0. The entropies we construct here are weak entropies, i.e., they vanish when the density vanishes.Another feature of our proof lies in the reduction theorem, which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.Acknowledgement P.G.L. and V.S. were supported by a grant from INTAS (01-868). The support and hospitality of the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, where part of this research was performed during the Semester Program Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics (January to July 2003) is also gratefully acknowledged. P.G.L. was also supported by the Centre National de la Recherche Scientifique (CNRS).     

A distributed electrical analog for transient diffusion-techniques of model fabrication and calibration

Simulation of transient two-dimensional diffusion by means of a distributed electrical analog is discussed. After present techniques of analog model construction and calibration are reviewed, an improved calibration technique is presented and a convenient method of analog fabrication, not previously reported, is described. The proposed new method allows complete access to any point on the analog model during a test. Frequency response and step response measurements indicate that an adequate simulation is provided by this particular type of analog model.

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Zusammenfassung Die Simulierung eines nichtstationären, zweidimensionalen Diffusionsvorganges mittels eines kontinuierlichen, elektro-thermischen Analogapparates wird besprochen. Eine Übersicht der gegenwärtigen Methoden für die Konstruktion und Kalibrierung von elektro-thermischen Analogmodellen wird gegeben. Ein verbessertes Verfahren für die Kalibrierung und eine handliche Fertigungsmethode von Analogmodellen, die noch nicht in der Literatur beschrieben wurden, werden dargestellt. Das vorgeschlagene neue Verfahren gestattet vollständigen Zugang zu jedem Punkt im Analogmodell während des Experiments. Meßbeobachtungen des periodischen Frequenzverhaltens und des nichtstationären Verhaltens zeigen, daß dieses spezielle Analogmodell eine ausreichende Simulierung des Diffusionsvorganges gestattet.

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Nomenclature A ₀ Potential amplitude atx=0 (see Fig. 5) - A _x Potential amplitude at locationx (see Fig. 5) - A _L Potential amplitude atx=L (see Fig. 5) - C Capacitance per unit area - j Frequency - L Characteristic length - R Resistance per square - t Time - x Coordinate - X Dimensionless distance,x/L - y Coordinate - Y Dimensionless distance,y/L - Diffusivity - _x Phase angle at locationx (see Fig. 5) - _L Phase angle atx=L (see Fig. 5) - _d Thickness of dielectric sheet - _r Thickness of resistance sheet - Dielectric constant - Resistivity - Dimensionless time,t/L ² - Potential - ₁ Reference potential - ₂ Reference potential - Dimensionless potential, ( – ₁)/( ₂ – ₁) - Angular frequency, 2f - Dimensionless frequency,L ²/ The investigation was performed while the first author was Visiting Associate Professor at Purdue University during 1967/68.     

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 cross sectional area difference method — a new technique for determination of particle concentration by laser doppler anemometry

A new technique for the determination of particle concentration from the signals of a laser Doppler anemometer (LDA) is described. It is based on a statistical relation between the number of Doppler periods, or the amplitude of the Doppler signals, and the particle concentration. The technique allows the mass flux of the dispersed phase of a two-phase flow to be obtained from the data set of a conventional one-dimensional (ID) LDA. The technique has been called the cross sectional area difference method. Simulations and first experimental results are presented and discussed.List of symbols a, b, c half-axes of measurement control volume (mcv) - a ₁, b ₁, c ₁ half-axes of detection volume - c _L velocity of light - d _m beam waist diameter - d _p particle diameter - d _pc diameter of the calibration particle - d _pmin minimum detectable particle diameter - e elementary charge - h Planck's constant - i number of particle size classes - k wavenumber - m visibility - m refractive index - n(d _p ) particle concentration - n(d _pi ) concentration of ith particle class - n vector of n(d _pi ) - q exponent of size dependence of G(d _p ) - v _x x-velocity component - x fringe spacing - y ₀, z ₀ coordinates of particle trajectory and cross sectional area - A cross sectional area of mcv - A matrix of A ₁ - a ₁ cross sectional area of detection volume - A ₁ difference of neighbouring cross sectional areas - C _A normalisation constant for linear graduation of amplitude - C _N normalisation constant for Doppler periods - C _scat non-size-dependent factor of G(d _p ) - C _x normalisation constant for nonlinear graduation of amplitude - F() power spectral density - G(d _p ) integral scattering function - H number of accumulated counts - H _max maximum number of accumulated counts - I amplitude of Doppler signal - I _max I for a particle passing through the origin of the mcv - I _s trigger level - K logarithmic amplitude ratio - K _max logarithmic amplitude ratio for I _s - K _x degree of linear class width of amplitude - K _A degree of nonlinear class width of amplitude - N number of Doppler periods - N _m number of Doppler periods required by signal validation - N _max N for a particle passing through the origin of the mcv - N ₀ fringe number inside mcv along x-axis - P _L laser power - S ₀ particle arrival rate - S ₁ trigger rate - S ₁ contribution to trigger rate coming from A ₁ - S ₁ vector of S _1i - S _1i contribution to trigger rate coming from ith class of distribution - _Q quantum efficiency - wavelength of laser light - off-axis angle - elevation angle - angular frequency - beam intersection angle - phase difference