Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

Nonlinear rheology of a concentrated spherical silica suspension:

Time-dependent nonlinear flow behavior was investigated for a model hard-sphere suspension, a 50 wt% suspension of spherical silica particles (radius = 40 nm; effective volume fraction = 0.53) in a 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The suspension had two stress components, the Brownian stress ^B and the hydrodynamic stress ^H After start-up of flow at various shear rates , the viscosity growth function ₊ (t, ) was measured with time t until it reached the steady state. The viscosity decay function _– (t, ) was measured after cessation of flow from the steady as well as transient states. At low where the steady state viscosity ( ) exhibited the shear-thinning, the _– (t, ) and ⁺ (t, ) data were quantitatively described with a BKZ constitutive equation utilizing data for nonlinear relaxation moduli G (t, ). This result enabled us to attribute the thinning behavior to the decrease of the Brownian contribution ^B = ^B / (considered in the BKZ equation through damping of G (t, )). On the other hand, at high where ( ) exhibited the thickening, the BKZ prediction largely deviated from the ₊ (t, ) and ₊ (t, ) data, the latter obtained after cessation of steady flow. This result suggested that the thickening was due to an enhancement of the hydrodynamic contribution ^H = ^H / (not considered in the BKZ equation). However, when the flow was stopped at the transient state and only a small strain (<0.2) was applied, ^H was hardly enhanced and the _– (t, ) data agreed with the BKZ prediction. Correspondingly, the onset of thickening of ₊ (t, ) was characterized with a -insensitive strain ( 0.2). On the basis of these results, the enhancement of ^H (thickening mechanism) was related to dynamic clustering of the particles that took place only when the strain applied through the fast flow was larger than a characteristic strain necessary for close approach/collision of the particles.     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

Multiaxial elongation of polyisobutylene with various and changing strain rate ratios

The multiaxial elongational rheometer equipped with rotary clamps is modified such that in addition to simple, equibiaxial and multiaxial elongations also tests with new modes of elongation can be performed. As an example, polyisobutylene is elongated with a ratio of the principal strain rates of and magnitudes of the maximum strain rate , 0.04 and 0.08 s^–1. As a test result, the first elongational viscosityµ ₁ (t) is obtained which follows closely the linear viscoelastic shear viscosity . In contrast, the second elongational viscosityµ ₂ (t) remains below . By means of a further modification of the rheometer, the test modes can be varied during the deformation period. This allows one to investigate the influence of a well-defined rheological pre-history on the following rheological behaviour. As an example a variation ofm = 0.5 2 was performed. The measured normal-stress differences superpose from the single steps of deformation similar to the linear viscoelastic prediction.Dedicated to Prof. F. R. Schwarzl on the occasion of his 60th birthday     

Grundzüge einer Stabilitätstheorie für elastische Systeme unter nichtkonservativer Belastung

Zusammenfassung Um den Einfluß von nichtkonservativen Kräften zu erfassen, muß die Stabilitätstheorie nichtkonservativer elastischer Systeme auf dem Prinzip der virtuellen Verrückung aufgebaut werden, das von genügender Allgemeinheit ist. Als mathematische Methode steht nach Klärung von Konvergenzfragen das Galerkinsche Verfahren zur Verfügung, das dem ursprünglichen Problem der Stabilität elastischer Systeme das Problem der Stabilität der Bewegung eines mechanischen Systems von unendlich vielen Freiheitsgraden als gleichwertig zuordnet und die Algebraisierung des Stabilitätsproblems erlaubt. Damit wird die Ljapunowsche Theorie anwendbar. Als Wichtigstes ist die Berechtigung der Linearisierung zu untersuchen, wozu der Einfluß der Dämpfung festzustellen ist. Es ergibt sich, daß bei Berücksichtigung von Dämpfung die Stabilität stets asymptotisch ist, so daß die Linesarisierung gerechtfertigt wird. Darüber hinaus zeigt es sich, daß die Struktur der Matrizen und von entscheidender Bedeutung für die Zuständigkeit eines der in Frage kommenden Stabilitätskriterien ist: Ist symmetrisch und eine beliebige positive Diagonalmatrix bzw. schiefsymmetrisch, oder symmetrisierbar bzw. unsymmetrisch und noch gleichzeitig (die Dämpfungskoeffizienten alle gleich), so wirkt in diesen Fällen die Dämpfung günstig, und die Stabilitätskriterien des ungedämpften Vorganges, das statische bzw. das kinetische, bleiben daher hinreichend. Man darf dann auch ohne Dämpfung rechnen. Ist dagegen unsymmetrisch und eine positive Diagonalmatrix mit verschiedenen Elementen, so darf in solch einem Fall nicht ohne Dämpfung gerechnet werden, weil gegenüber dem ungedämpften Vorgang ein Sprung in den Stabilitätsbedingungen auftritt, durch welchen das Ergebnis der Rechnung ohne Dämpfung in Frage gestellt wird.     

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, and , the size of the HRR dominant zone is much smaller than forV and , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( ) and over most of the region where measurements were obtained ( ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Rheology of concentrated disperse systems III. General features of the proposed non-newtonian model. Comparison with experimental data

Summary A non-newtonian viscosity equation where is the volume concentration and is an intrinsic viscosity, function of a relative shear rate and being structural parameters, has been proposed in a previous paper (1). From empirical grounds, the valuep = 1/2 holds for a large class of systems, like suspensions of rodand disc-shaped particles. In the high shear rate limit, aCasson law-type is recovered and discussed, especially the concentration dependence of the yield stress. However, the latter disappears in the low shear limit, and must be considered as a pseudo-yield stress. Good agreement is found in this low shear limit with some theoretical results ofBueche for polymers. More generally, the viscosity equation displays pseudo-plastic behaviour and fitting it on experimental data allows the determination of the structural parameters. Some examples (especially Red Blood Cell suspensions and Blood) are studied and support the model. Nevertheless, for spherical particle suspensions, the best fitting is reached forp = 1. Accurate values of particle diameters can be deduced from the structural parameter , in this case.

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Zusammenfassung In einer vorangegangenen Arbeit (1) wurde eine Viskositätsgleichung für eine nicht-newtonsche Flüssigkeit von der Form vorgeschlagen, worin die Volumenkonzentration und eine Grenzviskosität bedeutet; die letztere stellt eine Funktion der relativen Schergeschwindigkeit dar, die Konstantenk ₀,k und bezeichnen Strukturparameter. Empirisch wird gefunden, daß für eine große Klasse von Systemen, wie z. B. stäbchen- und scheibchenförmigen Teilchen,p = 1/2 gilt. In der Grenze hoher Schergeschwindigkeiten wird ein Verlauf gemäß einer Casson-Gleichung gefunden und diskutiert, insbesondere bezüglich der Konzentrationsabhängigkeit der Fließspannung. Allerdings verschwindet diese in der Grenze niedriger Schergeschwindigkeiten und muß daher als Pseudo-Fließspannung betrachtet werden. In diesem Grenzfall wird eine gute Übereinstimmung mit theoretischen Voraussagen vonBueche an Polymeren gefunden. Ganz allgemein beschreibt die obige Viskositätsgleichung ein pseudoplastisches Verhalten, und ihre Anpassung an experimentelle Werte erlaubt die Bestimmung der Strukturparameter. Einige Beispiele, insbesondere Suspensionen von roten Blutkörperchen und Blut, werden untersucht und bestätigen das Modell. Allerdings erhält man bei Suspensionen kugelförmiger Teilchen die beste Anpassung fürp = 1. In diesem Fall kann man mit Hilfe des Strukturparameters genaue Werte der Teilchendurchmesser bestimmen.

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With 5 figures and 3 tables     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators

An elliptic Lindstedt--Poincaré (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with are studied in detail.     

Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems

The concept of an interval stochastic matrix is introduced. We prove a combinatorial theorem which describes the network flow associated with an interval matrix. The semi-invariant vectors of are characterized in terms of eigenvectors with unit eigenvalue of stochastic matrices . These results are then applied to the approximation and machine computation of invariant measures of dynamical systems.Funded under Australian Research Council Grant A 8913 2609.     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

Dehnungsverhalten von Polyäthylen-Schmelzen

Zusammenfassung In einem Dehnungsrheometer werden Spannungs-Dehnungs-Diagramme von Polyäthylen-Schmelzen bei 150 °C und bei konstanter Dehnungsgeschwindigkeit gemessen ( zwischen 0,001 und 1 sec^–1). Weiterhin wird der reversible (elastische) Dehnungsanteil bestimmt. Messungen mit einem Dehnungstester für Kunststoff-Schmelzen ergänzen die Ausführungen.Die Ergebnisse zeigen deutlich, daß bei Dehnung mit zunehmender Verformungsgeschwindigkeit die Dehnungsviskosität nicht abnimmt, im Gegensatz zu dem bekannten strukturviskosen Verhalten bei Scherung.Bei Dehnungen bis zu=1 kann das Verhalten unabhängig von beschrieben werden, wenn als viskoelastische Materialfunktion die Dehnungs-Spannviskosität betrachtet wird. In diesem Bereich von gilt dabei die BeziehungT(t)=3 _s (t) mit _s (t) als zeitabhängige Scherviskosität im linear-viskoelastischen Bereich.Bei größeren Dehnungen und nicht zu kleinen Dehnungsgeschwindigkeiten zeigt verzweigtes Polyäthylen eine zusätzliche starke Spannungszunahme. In dem Bereich dieser zusätzlichen Verfestigung ist das Verhalten im wesentlichen eine Funktion der Dehnung und fast unabhängig von . Die zusätzliche Verfestigung scheint eine Folge der Verzweigungsstruktur des verzweigten Polyäthylens zu sein, da bei Linear-PE ein derartiger Verlauf des Spannungs-Dehnungs-Diagramms nicht beobachtet wird.Die Betrachtung des reversiblen Dehnungsanteils _R zeigt bei der ausführlich untersuchten Schmelze I (verzweigtes PE) drei verschiedene Bereiche: Unterhalb einer Grenzdehnungsgeschwindigkeit ist _R =0, unterhalb einer Versuchszeitt ^** ist _R =. Im dazwischenliegenden Bereich treten elastische und viskose Dehnungsanteile auf,= _R + _V , wobei für niedrige gilt, daß _R lg . Die Grenze wird der Frequenz der thermisch aktivierten Platzwechsel zugeordnet,t ^** erscheint als Zeit, innerhalb der die Verhakungen wie fixierte Vernetzungen wirken.In dem Anhang wird der Einfluß der Grenzflächenspannung zwischen PE-Schmelze und Silikonöl auf die Ergebnisse der Dehnungsversuche diskutiert.

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Summary Stress-strain relations for different PE melts are measured at 150 °C in an extensional rheometer under the condition of a constant extensional strain rate ( between 0,001 and 1 sec^–1). Further, the recoverable (elastic) portion _R of the total strain ( in Hencky's measure) is determined and additional measurements with a tensile tester for polymer melts are described.The results show clearly that in extension there is no decrease of the tensile viscosity with increasing deformation rate, in contrast to the well-known pseudoplastic behaviour in shear. Within total strains<1 the tensile behaviour can be described independently from by means of a viscoelastic material function called stressing viscosity . In this range of the relation _T (t)=3 _s (t) holds, where _s (t) is the stressing viscosity in shear in the linear viscoelastic range. For larger tensile strains and not too small branched PB shows a remarkable increase in stress. This work-hardening behaviour is mainly a function of and almost independent from . This additional hardening seems to be due to the branches in branched PE, because linear PE does not show this effect.The discussion of the recoverable tensile strain _R gives three regions of tensile rate: Below a critical there is _R =0. At times shorter thant ^** the equation _R = is valid. Within these limits both elastic and viscous portions of the total strain= _R + _V exist. may correlate with the frequency of the thermally activated position changes of the molecular segments.t ^** is assumed to be the time for the entanglements to act as fixed cross-links.In the appendix the influence of the interface tension between PE melt and silicone oil on the results of the tensile experiments is discussed.

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Vorgetragen auf der Deutschen Rheologen-Tagung, Berlin, 11.-13. Mai 1970.

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An der Weiterentwicklung des Dehnungsrheometers, an der Durchführung und Auswertung der Messungen waren die HerrenB. Kienle, F. Landmesser, M, Reuther undF. Scherr beteiligt. Herr Dr.F.Ramsteiner und HerrH. Schroeck haben sich um die Herstellung der Stränge aus Linear-PE bemüht. Herr Dr.W. Ball besorgte die GPC-Messungen und Herr Dr.P. Simak die Ultrarot-Untersuchung. Den vorgenannten Herren sei für ihre Hilfe beim Zustandekommen dieser Arbeit gedankt. Herrn Dr.H. Baur danke ich für wertvolle Diskussionen.     

Effects of couple-stresses on the dispersion of a soluble matter in a pipe flow of blood

Summary An analysis of the effects of couple-stresses on the effective Taylor diffusion coefficient has been carried out with the help of two non-dimensional parameters based on the concentration of suspensions and , a constant associated with the couple-stresses. It is observed that the concentration distribution increases with increasing or The effective Taylor diffusion coefficient falls rapidly with increasing when is negative.

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Zusammenfassung Der Einfluß der Momentenspannungen auf den effektiven Taylorschen Diffusionskoeffizienten wird untersucht. Dabei treten zwei dimensionslose Parameter and auf: Der erste bezieht sich auf die Suspensionskonzentration, der zweite kennzeichnet die Momentenspannungen. Man findet, daß die Verteilungsgeschwindigkeit mit wachsendem oder zunimmt. Dagegen fällt der Taylorsche Diffusionskoeffizient bei wachsendem stark ab, wenn negativ ist.

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a Tube radius - C Concentration - C _i Body moment vector - C ₀ Concentration at the axis of the tube - C _m Mean concentration - D Molecular diffusion coefficient - d _ij Symmetric part of velocity gradient - F Function of and characterising effective Taylor diffusion coefficient - f _i Body force vector - H A function of and - K ₂ Integration constant - K ^* Effective Taylor diffusion coefficient - k Radius of gyration of a unit cuboid with its sides normal to the spatial axes - I _n Modified Bessel's function ofnth order - L Length of the tube over which the concentration is spread - M Function ofH and - M _ij Couple stress tensor - P Function of - p Fluid pressure - Q Volume rate of the transport of the solute across a section of the tube - r Radial distance from the axis of the tube - T _ij Stress tensor - t Time coordinate - T _ij ^A Antisymmetric part of the stress tensor - u Relative fluid velocity - Average velocity - v _i Velocity vector - Fluid velocity at any point of the tube - v ₀ ⁿ Velocity of Newtonian fluid at the axis of the tube - _i Vorticity vector - x Axial coordinate - x ₁ Relative axial coordinate - z Non-Dimensional radial coordinate - Density - _ij Symmetric part of the stress tensor - µ Viscosity of the fluid - µ _ij Deviatoric part ofM _ij - , Constants associated with couple-stress With 3 figures     

Der selbsterregte unsymmetrische Kreisel

Zusammenfassung Wir können alle bisherigen Erkenntnisse über die Selbsterregung des unsymmetrischen Kreisels dahin zusammenfassen, daß — abgesehen von den trivialen Fällen der Drehungen um jede der drei Hauptachsen und abgesehen von den permanenten Drehungen um beliebige Achsen — zu jeder achsenfesten zeitunabhängigen Selbsterregung achsenbewegliche Drehungen gehören, wie wir solche in Ziff. 2 bis 7 untersucht haben; ferner daß es, wieder abgesehen von jenen trivialen Fällen, nur den einzigen Typ der achsenfesten zeitabhängigen Selbsterregung von Ziff. 9 gibt, welche eine achsenfeste zeitabhängige Drehung zu beeinflussen, wenn auch weder zu erzeugen noch zu vernichten vermag; endlich daß der Momentvektor der Selbsterregung, welcher eine beliebige achsenfeste Drehung erzeugen oder vernichten oder in beliebiger Weise beeinflussen soll, im allgemeinen (also abgesehen von den trivialen Fällen und von dem Fall in Ziff. 9) zeitabhängig in einer körperfesten Ebene wandern muß. Die achsenbeweglichen Drehungen, welche von achsenfester zeitunabhängiger Selbsterregung erzeugt oder unterhalten werden, sind je nach der Lage (und Größe) des Momentes der Selbsterregung periodisch hinsichtlich der Bewegung des Drehvektors im Kreisel oder asymptotisch mit unbeschränkt wachsendem Drehvektor oder asymptotisch mit einem Drehvektor , der sich einem permanenten Drehvektor (von endlichem Betrag) annähert (oder auch sich von ihm entfernt).     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Dependence of the yield stresses of certain materials on loading rate

The dependence of the dynamic yield stresses of certain metals and alloys on loading rate has been investigated at temperatures of +20 and –196°C. In the experiments the loading rate remained constant up to the appearance of plastic strains, i.e., , where . The yield stresses of the materials and the time to the appearance of plastic strains were determined from the oscillograms representing the strain and load as functions of time.     

Evaluation of the performance characteristics of a thermal transient anemometer

The Goertler instability of a hypersonic boundary layer and its influence on the wall heat transfer are experimentally analyzed. Measurements, made in a wind tunnel by means of a computerized infrared (IR) imaging system, refer to the flow over two-dimensional concave walls. Wall temperature maps (that are interpreted as surface flow visualizations) and spanwise heat transfer fluctuations are presented. Measured vortices wavelengths are correlated to non-dimensional parameters and compared with numerical predictions from the literature.List of symbols c _p Specific heat coefficient at constant pressure of the free stream - F Input (true) image - F ₀ Fourier number - Restored image - G Recorded (degraded) image - G Goertler number based on the boundary layer thickness, as defined by Eq. (3) - H System transfer function - M Mach number - Pr Prandtl number - p ₀ Stagnation pressure - Exchanged net heat flux - Convective heat flux - Radiative heat flux - r Recovery factor - Re _m Unit Reynolds number - Re _x Local Reynolds number based on the distance from the leading edge - Re Local Reynolds number based on the boundary layer thickness - Curvature radius - St Stanton number, as defined by Eq. (7) - T _aw Adiabatic wall temperature - T _w Wall temperature - T ₀ Stagnation temperature - t Time - V Free stream velocity - x Streamwise spatial coordinate - y Normal-to-wall spatial coordinate - z Spanwise spatial coordinate - Thermal diffusivity coefficient - Disturbance wavenumber - Non dimensional wavenumber - Boundary layer thickness - Goertler number based on the vortices wavelength - Vortices wavelength - Free stream density - Disturbance total amplification, as defined by Eq. (3) - Disturbance (spatial) growth rate - Non-dimensional growth rate - Perturbation amplitude of a generic quantity - Perturbation amount     

The theory of dispersion of chemically active solutes in a rectilinear flow field: The vector problem

In this paper and its sequel we seek an understanding of diffusion and reaction processes in multicomponent systems under convective conditions. To do this, we construct dispersion approximations to c, the solution of the vector convective diffusion equation, and to its transverse average, . Dispersion coefficients are not the direct route to dispersion approximations in the vector problem. Yet constant long time dispersion coefficients exist and imply a rearrangement of the Hermite expansion of . We deduce the rearrangement and introduce useful approximations thereto.     

The unsteady aerodynamics of a first stage stator vane row

The fundamental unsteady aerodynamics on a vane row of an axial flow research compressor stage are experimentally investigated, demonstrating the effects of airfoil camber and steady loading. In particular, the rotor wake generated unsteady surface pressure distributions on the first stage vane row are quantified over a range of operating conditions. These cambered airfoil unsteady data are correlated with predictions from a flat plate cascade inviscid flow model. At the design point, the unsteady pressure difference coefficient data exhibit good correlation with the nonseparated predictions, with the aerodynamic phase lag data exhibiting fair trendwise correlation. The quantitative phase lag differences are associated with the camber of the airfoil. An aft suction surface flow separation region is indicated by the steady state surface static pressure data as the aerodynamic loading is increased. This separation affects the increased incidence angle unsteady pressure data.List of symbols b airfoil semi-chord - C airfoil chord - C _p dynamic pressure coefficient, - _p static pressure coefficient, - i incidence angle - k reduced frequency, - N number of rotor revolutions - p dynamic pressure difference - static pressure difference, - S stator vane circumferential spacing - U _t rotor blade tip speed - u longitudinal perturbation velocity - V absolute velocity - V _axial absolute axial velocity - v transverse perturbation velocity - x _sep location of separation point - inlet angle - inlet air density - blade passing angular frequency