Weighted L∞ bounds and uniqueness for the Boltzmann BGK model

We prove rather general L bounds for hydrodynamical fields in terms of weighted L norms of the kinetic density. We use these estimates to prove L bounds and uniqueness for the solution of the BGK Equation, which is a simple relaxation model introduced by Bhatnagar, Gross & Krook to mimic Boltzmann flows.     

On patches of uniform vorticity in a plane of irrotational flow

We derive an evolution equation for the motions of patches of vorticity (vortex). Steady state solutions of this equation that include those of Kirchhoff and Moore & Saffman are established. The m-fold symmetric, m3, hypotrochoid is an exact steady solution of this equation when rotation and strain are present. When strain is absent but rotation is present, the m-fold symmetric, m2, hypotrochoid is a perturbation solution with a dispersion relation extending that of Lamb. The case of m=2 is exact and is the Kirchhoff elliptical vortex.     

Die Stabilität der Strömung in einem gekrümmten Kanal

Zusammenfassung Die Wirbelinstabilität einer laminaren Strömung durch einen gekrümmten, unendlich hohen Kanal, dessen Breite klein ist gegenüber den Krümmungsradien der Kanalwände, wird durch die beiden Differentialgleichungen (2) beschrieben. Das System (2) stellt ein Eigenwertproblem dar, bei dem vor allem die kleinsten Eigenwerte S ₁ in Abhängigkeit von dem Parameter mit den zugehörigen Eigenfunktionen interessieren; dabei beziehen wir uns bereits auf (3), wo neutrale Störungen (=) betrachtet werden, und S ₁() ist bis auf einen Zahlenfaktor die kritische Kurve des Parameters 2 Re² d/R ₁ in Abhängigkeit von der Dicke /2 der angesetzten Wirbel. Mit Hilfe Greenscher Funktionen werden die Differentialgleichungen (3) in die Integralgleichungen (4) verwandelt, die es zunächst erlauben, untere Schranken für S ₁() anzugeben (Fig. 2). Das Iterationsverfahren nach [9] liefert S ₁() mit hinreichender Genauigkeit nach wenigen Iterationsschritten (Fig. 3). Ebenso ergeben sich damit die Eigenfunktionen (Fig. 4), also die Wirbelkomponenten, so daß in Fig. 5 das Aussehen der neutralen Wirbel angegeben werden kann. Der kritische Wert — aus dem kleinsten unter den Eigenwerten. S ₁() — ist 2Re² d/R ₁=5740, während Dean [2] 1928 den ebenfalls richtigen Wert 5832±2,5% gefunden hatte. Die zugehörige kritische Wirbeldicke ist /2=0,79 d. Die Ergebnisse von Yih und Sangster [3] erweisen sich als falsch. Die Arbeit verfolgte zwei Ziele: Eine Neubehandlung der Differentialgleichungen (5) mit exakt begründeten Methoden, um eine Beurteilung der Ergebnisse von Dean und von Yih und Sangster möglich zu machen. Dann die Angabe der Eigenfunktionen, die es gestatten, das Aussehen der entstehenden Wirbel zu beschreiben.Diese Arbeit wurde durch das Wirtschaftsministerium des Landes Baden-Württemberg gefördert. Vorgelegt von H. Görtler     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

Die umkehrung der stabilitätssätze von Lagrange-Dirichlet und Routh

Summary The Lagrange-Dirichlet theorem states that the equilibrium position of a discrete, conservative mechanical system is stable if the potential energy U(q) assumes a minimum in this position. Although everything seems to indicate that the equilibrium is always unstable in case of a maximum of the potential energy, this has yet to be proven. In all existing instability theorems the function U(q) has to satisfy additional requirements which are very restrictive.In this paper instability is proved in the case of a maximum of U(q)C ², without further restrictions. The instability follows directly from the existence of certain types of motions which are not found as solutions of differential equations, but as the solutions of a variational problem. Existence theorems are given for the variational problem, based on a result found by Carathéodory.In similar way an inversion of Routh's theorem on the stability of steady motions in systems with cyclic coordinates is also given. The result obtained here is not as general as the inversion of the Lagrange-Dirichlet theorem because the equations of motion are of a more complex type.

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Vorgelegt von C. Truesdell

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Von der Fakultät für Mathematik der Universität Karlsruhe (TH) angenommene Habilitationsschrift.     

Uniqueness of positive solutions of Δu−u+up=0 in Rn

We establish the uniqueness of the positive, radially symmetric solution to the differential equation u–u+u^p=0 (with p>1) in a bounded or unbounded annular region in R ⁿ for all n1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

An elementary proof of the polar factorization of vector-valued functions

We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ^d satisfying a nondegeneracy condition admit the polar factorization (*) u(x)=(s(x)), where is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory; whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (*).     

Mass and rated characteristics of planetary gear reduction units

A sample of planetary gear reduction units for industrial appliances is analysed from a statistical point of view. A correlation between the variables mass/rated torque M/T and rated torque T of the above machines is pointed out, which is independent of the number of stages z. Parallel correlations for single values of z are moreover pointed out between the variables P _t/T ^2/3 and T, where P _t is the thermal capacity, with natural cooling, of the above machines. The results are compared with those found for a sample of ordinary gear units, chosen and analysed with the same criteria. The well-known advantages of planetary over ordinary gear units, as far as mass is concerned, are confirmed but a corresponding disadvantage is pointed out as far as thermal capacity is concerned.

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Sommario Si analizza da un punto di vista statistico un campione di riduttori epicicloidali per applicazioni industriali. Viene evidenziata l'esistenza di una correlazione tra le variabili massa/coppia nominale M/T e coppia nominale T delle macchine in questione, indipendente dal numero z di stadi delle medesime. Correlazioni parallele per singoli valori di z sono egualmente evidenziate tra le variabili P _t/T ^2/3 e T, dove P _t indica la potenza limite termica, con refrigerazione naturale, delle macchine studiate. I risultati dell'analisi sono confrontati con quelli relativi ad un campione di riduttori ordinari ad ingranaggi cilindrici, scelto ed analizzato con gli stessi criteri. I noti vantaggi dei riduttori epicicloidali su quelli ordinari, per quanto riguarda la massa, vengono confermati ma nello stesso tempo vengono evidenziati corrispondenti svantaggi per quanto riguarda la potenza termica limite.

     

An exactly solvable Lorentz gas

We introduce and analyse a new and special case of the Lorentz gas or the Wind-Tree model of Ehrenfest. This model (which has a number-theoretic character) is shown to exhibit normal diffusion, the diffusion coefficient C() being obtained in closed form as a function of the density . The function C() turns out to be an entire analytic function of , in spite of the model's non-classical high density behaviour. A collision expansion which is appropriate for high densities is also given.     

Experimentelle Bestimmung des Staupunktswärmeüberganges in Argon im Stoßwellenrohr

Zusammenfassung Es wird die Theorie der elektrisch isolierten Kalorimetersonde mit endlicher Schichtdicke entwickelt. Anhand des Ergebnisses lassen sich die möglichen Fehler der Sonde leicht abschätzen. Außerdem wird der Einfluß der Temperaturverteilung auf den Widerstand diskutiert. Die experimentellen Ergebnisse des Wärmeüberganges im Staupunkt einer Halbkugel in Argon und Luft werden befriedigend mit der Theorie vonFay undRiddell erklärt. Bisherige experimentelle Ergebnisse lagen entweder zu hoch oder zu niedrig.

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Experimental determination of the stagnation point heat transfer in argon in a shock tube

The theory of the electrically insulated calorimeter gauge with finite thickness has been developed. From the theoretical results the possible errors of the gauge can be estimated. The influence of the temperature distribution on the electrical resistance is given. The experimental results of the heat transfer to the stagnation point of a spherical cylinder in argon and air agree reasonably with the theory ofFay andRiddell. Earlier experimental results have been found too high or too low.

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Bezeichnungen (Verwendete Einheiten in Klammern) a=/c· Temperaturleitfähigkeit (cm²/s) - c spezifische Wärme (Ws/g grd) - d, L, B Längen (cm) - e=c Wärmeeindringzahl (Ws^1/2/cm² grd) - F Pläche (cm²) - I Strom (A) - M _s Stoß-Machzahl - q Wärmestromdichte (W/cm²) - R Widerstand () - t Zeit (s) - T Temperatur (°K, grd) - U Spannung (V) - x, y, z Raumkoordinaten (cm) - Temperaturkoeffizient des Widerstandes (grd^–1) - Wärmeleitfähigkeit (W/cm grd) - Dichte (g/cm³)     

On the cross flow through in-line tube banks with regard to the effect of surface roughness

The behaviour of the flow through an in-line tube bank is studied at variable roughness conditions of the surface. The total pressure loss, the local static pressure and the skin friction distribution were measured in the range of Reynolds numbers 4·10⁴<Re<10⁷. Opposing to the staggered arrangement the drag coefficient of an in-line bundle diminishes in the transcritical flow range with increasing roughness parameters. As at the same time heat transfer is improved the in-line bundle is of great practical interest. By means of the local flow parameters the interaction of the point of impact, separation point of the boundary layer and wake width can be pointed out.

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Zusammenfassung Die Strömung durch ein fluchtendes Rohrbündel glatter und rauher Oberfläche wird untersucht. Der Gesamtdruckverlust sowie die örtliche Druck- und Wandschubspannungsverteilung wurden im Bereich der ReynoldsZahlen 4·10⁴<Re<10⁷ gemessen. Im Gegensatz zum versetzten Wärmetauscher verringert sich bei der fluchtenden Anordnung im transkritischen Strömungsbereich der Widerstandsbeiwert mit zunehmender Oberflächenrauhigkeit. Da gleichzeitig eine Verbesserung des Wärmeüberganges beobachtet wird, kommt dem fluchtenden Bündel große praktische Bedeutung zu. Anhand der experimentell gewonnenen örtlichen Strömungsparameter wird die Wechselwirkung von Auftreffpunkt der Strömung, Grenzschichtablösung und Nachlauf breite verdeutlicht.

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Nomenclature D m tube diameter - k _s m sand grain roughness - L m tube length - p N/m₂ pressure - s ₁ m longitudinal pitch - s _tm transversal pitch - U _c m/s velocity in the smallest cross section between the tubes - z number of rows Greek symbols pressure drop coefficient (index t=total, index f=friction) - kg/sm fluid viscosity - kg/m³ fluid density - ₀ N/m² wall shear stresses - grd angle of circumference Characteristics Reynolds number - pressure drop coefficient The investigations have been carried out in the Institut für Reaktorbauelemente der Kernforschungsanlage Jülich GmbH The author wishes to thank Dr.von der Decken, new director of the institute, who supported the already started work in its final stages with great interest. He also expresses his gratitude to his co-workers H.Gillessen, F.Hoffmanns, H.Reger, W.Schmidt and G.Türk for their activity and engagement to bring these investigations to a successful end.     

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem

Conclusions We have investigated solutions of equation (3) when ² is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and ² = _i ² we have exactly two nontrivial solutions which bifurcate to the right of _i ² ; these solutions are shown to exist in an interval ( _i ² , _i ² + ₀). The method of Section 3 may then be used to extend these two solutions to the right of _i ² + ₀ providing that ²= _i ² + ₀ is not an eigenvalue of the linear operator (13) evaluated at = ± ₁. Either a solution can be uniquely extended, or there exists a value of ²where the bifurcation method must be applied again³.While the method used here gives the exact number of solutions bifurcating from _i ² , other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels [4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval _i ² < _i ^+1/2 remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner [7] and Coffman [8].     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity

The aim of this article is to study the quasistatic evolution of a three-dimensional elastic-perfectly plastic solid which satisfies the Prandtl-Reuss law. The evolution of the field of stresses -which solves a time dependent variational inequality — and that of the field of displacements u, have been described in previous works [15], [26], [35], [36], [37] but it was not shown there that and u satisfy indeed the Prandtl-Reuss constitutive law. In this article we find and u in a class of functions which are sufficiently regular for the Prandtl-Reuss law to make sense and we prove that and u satisfy the constitutive law. This result is attained by considering the elastic-perfectly plastic model as the limit of a family of elastic-visco-plastic models like those of Norton and Hoff. The Norton-Hoff type models which we introduce depend on a viscosity parameter > 0; we study the perturbed models (i.e. > 0 fixed) and then we pass to the limit 0.Dedicated to James Serrin on the occasion of his 60^th Birthday     

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.     

Existenz- und Eindeutigkeitsbeweis für ein freies randwertproblem für die stationären Navier-Stokesschen Bewegungsgleichungen

Ohne Zusammenfassung Vorgelegt von S. Antman Dem Andenken an Herrn Professor Dr. S. I. Gheorghia     

Quelques relations fondamentales entre solutions de l'équation biconfluente de heun

Résumé On établit trois relations fondamentales existant entre certaines solutions de l'équation biconfluente de l'équation de Heun. Ces trois relations constituent une généralisation de trois relations bien connues dans le cas des fonctions confluentes de Kummer; l'une apparait dans [11] et les deux autres peuvent être attribuées à Gordon [5] et à Meixner [9].On caractérise certaines constantes fondamentales non élémentaires à l'aide d'une intégrale notée J (, , , ) qui généralise la fonction eulérienne gamma.En particularisant certains paramètres on obtient de façon naturelle des résultats connus.     

Stability transitions for periodic orbits in hamiltonian systems

The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h ₀. First considered is the case of a critical point (with eigenvalues ±, ±i, and >0) of the Hamiltonian at energy h ₀ with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h ₀ on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h ₀, on which the orbits are hyperbolic, involve the coexistence problem for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±±i), and >0, investigated by Henrard and Devaney.     

Eine Fluoreszenzmethode zur Untersuchung des Transportmechanismus bei der Gasabsorption im Rieselfilm

Zusammenfassung Gewissep h-Indikatoren zeigen bei einem bestimmtenp h-Wert eine stufenförmige Änderung der Fluoreszenzfähigkeit. Diese Eigenschaft wurde für die Untersuchung der Absorption eines sauren oder basischen Gases in einem Rieselfilm ausgenutzt; das Gas geht eine schnelle Neutralisationsreaktion mit der basischen bzw. sauren Rieselflüssigkeit ein, die den Indikator enthält. Der Rieselfilm wird durch ultraviolette Strahlung zur Fluoreszenz angeregt; aus der gemessenen Lichtemission wird der Abstand zwischen der Phasengrenze und der Neutralfläche im Film ermittelt. Es ist möglich, aus den Meßergebnissen die Konzentrationsverteilung in der Flüssigkeit und daraus die Abhängigkeit des effektiven Diffusionskoeffizienten vom Abstand zur Phasengrenze zu berechnen. Letztere bestätigt das verallgemeinerte Diffusionsmodell des Stofftransportes. Blitzaufnahmen zeigen, daß die Absorption nicht gleichmäßig über die Fläche verteilt ist, sondern es bilden sich flecken- oder streifenförmige Muster.

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The fluorescence emission of certainp h-indicators changes according to a step function at a particularp h value. This property has been used to study the absorption of an acidic or alkaline gas into a falling film; the gas undergoes an instantaneous neutralisation reaction with a dilute solution of an alkali or acid containing the indicator. The falling film is exposed to ultraviolet radiation; from the measured intensity of fluorescence the distance between the phase boundary and the neutralisation plane within the film is evaluated. From these measurements one can calculate the concentration distribution in the liquid and consequently the variation of effective diffusivity with distance from the interface. The latter supports the generalised diffusion model of mass transfer. It is evident from flash photographs that absorption is not uniform across the surface but shows characteristic patterns of spots or stripes.

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Bezeichnungen A Konzentration der absorbierten Komponente im Rieselfilm - A₀ Eintrittskonzentration der zu absorbierenden Komponente in der Gasphase - A_0z KonzentrationA₀, erforderlich für Fluoreszenzbeginn bei Koordinate z - A* formale Konzentration=A+(B ₀–B)D _B/D _A - B Konzentration des Reaktionspartners im Rieselfilm - B ₀ B am Zulauf - c Konzentration - c _i Indikatorkonzentration - D molekularer Diffusionskoeffizient - D _eff effektiver Diffusionskoeffizient - d Dicke des Rieselfilms (zeitlicher Mittelwert) - g Erdbeschleunigung - I Fluoreszenzintensität (fotoelektrische Anzeige) - I _k Eigenfluoreszenz der Küvette - I _n Nullwert der Fluoreszenz=dc _i i ₀+I _k - i spezifische Fluoreszenzintensität des Indikators beid=1 mm undc _i=1 n - i ₀ i imp h-Bereich schwacher Fluoreszenzfähigkeit - i ₁ i imp h-Bereich starker Fluoreszenzfähigkeit - k beliebiger Faktor - k Geschwindigkeitskonstante für Reaktion 1. Ordnung - k Geschwindigkeitskonstante für Reaktion 2. Ordnung - q absorbierte Mengenstromdichte - t Zeit - w(x) Strömungsgeschwindigkeit in der Flüssigkeit - w _s Strömungsgeschwindigkeit der Flüssigkeitsoberfläche - mittlere Strömungsgeschwindigkeit der Flüssigkeit - x Abstand von der Flüssigkeitsoberfläche - x _N Dicke der stark fluoreszierenden Schicht - z Abstand vom Beginn des Rieselfilms - _g Stoffübergangskoeffizient auf der Gasseite - kinematische Viskosität der Flüssigkeit - Grenzflächenspannung - Re _fl Reynoldszahl der Flüssigkeit= - Re _g Reynoldszahl der Gasströmung Über Teile der vorliegenden Arbeit wurde auf dem Jahrestreffen der Verfahrensingenieure im Oktober 1966 in Hamburg vorgetragen. Eine Zusammenfassung wurde veröffentlicht [1].