Waves and stability

The connection between stability of equilibrium and propagation of small disturbances through a general elastic medium has been stated, since 1903, by J. Hadamard. However a doubtful element in the original proof of the theorem was pointed out by P. Duhem. A more elaborated proof, which seems to be satisfactory even in a more general hypothesis, has been later proposed by the author of this report. That proof is reconsidered in this article which deals, in a general context, with the relations between wave propagation and linear stability. A new elegant proof of Hadamard's theorem, recently proposed by G. Fichera, is briefly reported at the end of the article.

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Résumé Les liens entre la stabilité de l'équilibre et la propagation des ondes dans un milieu élastique ont été codifiés dès 1903, par J. Hadamard. La démonstration originaire du théorème présente toutefois un point douteux, mis en évidence par P. Duhem. Une démonstration plus élaborée, qui parait mathématiquement corrècte même en des conditions physiques plus générales, a été proposée plus tard par l'auteur de cet exposé. Cette démonstration est reprise dans l'article suivant, qui envisage dans un contexte général les relations entre la propagation des ondes et la stabilité linéaire. Une nouvelle et élégante démonstration du théorème de Hadamard, proposée récemment par G. Fichera, est résumée à la fin de l'article.

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Lecture delivered at the Italy-USA Seminar on Existence and Stability in Elasticity, Udine, June 26–30, 1971.     

Vergleichende Untersuchungen der Strömungsprofile und des Fließverhaltens Nicht-Newtonscher Flüssigkeiten im Rotationsviskosimeter

Übersicht Es wird die Verträglichkeit der aus drei verschiedenen Meßverfahren resultierenden Aussagen über das Fließverhalten Nicht-Newtonscher Flüssigkeiten im Couette-Spalt für die beiden Fälle untersucht, daß das Material entweder dem Ostwaldschen oder dem Prandtlschen Fließgesetz gehorcht. Für die benutzten Verfahren: 1. Ausmessung des Strömungsprofils, 2. Entspannungsmessung, und 3. Messung der Drehmomente bei konstanten Drehzahlen des Rührers, werden zweckmäßige, graphische Auswerteverfahren entwickelt, die bereits durch einen Formvergleich zwischen Meßkurven und Musterkurven eine Entscheidung zugunsten des einen oder anderen Fließgesetzes ermöglichen, wenn die Streuung der Meßdaten es zuläßt. Am Beispiel einer Tonsuspension wird gezeigt, daß eine Entscheidung über die Gültigkeit eines der beiden Ansätze jedoch auch bei größeren Streuungen der Versuchswerte gefällt werden kann, wenn man die aus den drei Verfahren resultierenden Ergebnisse auf ihre Verträglichkeit prüft.

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Summary The compatibility of the results of three different testing methods concerning the flow behaviour of Non-Newtonian liquids within the Couette-gap is examined for both cases, whether the material follows either Ostwald's or Prandtl's law of flowing. For the employed procedures: 1. Measuring of the flow profile, 2. Measuring of the torsion release, 3. Measuring of the torque of constant rotational speed of the stirrer, suitable graphical methods of valuation are developed, to decide in comparison with the form between measuring curves and master curves in favour of one flowing law or the other, provided that the scattering of test results permits it. At the example of a clay suspension it is shown that a decision concerning the validity of one of the two formulae can be given, if the results are being judged with respect to their compatibility.

     

On the orders of the best approximations of integrals of functions by integrals of rank σ

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ƒ is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^ϕ. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 528–559, October–December, 2007.     

Flow evolution of a turbulent submerged two-dimensional rectangular free jet of air. Average Particle Image Velocimetry (PIV) visualizations and measurements

The paper presents average flow visualizations and measurements, obtained with the Particle Image Velocimetry (PIV) technique, of a submerged rectangular free jet of air in the range of Reynolds numbers from Re = 35,300 to Re = 2200, where the Reynolds number is defined according to the hydraulic diameter of a rectangular slot of height H. According to the literature, just after the exit of the jet there is a zone of flow, called zone of flow establishment, containing the region of mixing fluid, at the border with the stagnant fluid, and the potential core, where velocity on the centerline maintains a value almost equal to the exit one. After this zone is present the zone of established flow or fully developed region. The goal of the paper is to show, with average PIV visualizations and measurements, that, before the zone of flow establishment is present a region of flow, never mentioned by the literature and called undisturbed region of flow, with a length, L_U, which decreases with the increase of the Reynolds number. The main characteristics of the undisturbed region is the fact that the velocity profile maintains almost equal to the exit one, and can also be identified by a constant height of the average PIV visualizations, with length, L_CH, or by a constant turbulence on the centerline, with length L_CT. The average PIV velocity and turbulence measurements are compared to those performed with the Hot Film Anemometry (HFA) technique. The average PIV visualizations show that the region of constant height has a length L_CH which increases from L_CH = H at Re = 35,300 to L_CH = 4–5H at Re = 2200. The PIV measurements on the centerline of the jet show that turbulence remains constant at the level of the exit for a length, L_CT, which increases from L_CT = H at Re = 35,300 to L_CT = 4–5H at Re = 2200. The PIV measurements show that velocity remains constant at the exit level for a length, L_U, which increases from L_U = H at Re = 35,300 to L_U = 6H at Re = 2200 and is called undisturbed region of flow. In turbulent flow the length L_U is almost equal to the lengths of the regions of constant height, L_CH, and constant turbulence, L_CT. In laminar flow, Re = 2200, the length of the undisturbed region of flow, L_U, is greater than the lengths of the regions of constant height and turbulence, L_CT = L_CH = 4–5H. The average PIV and HFA velocity measurements confirm that the length of potential core, L_P, increases from L_P = 4–5H at Re = 35,300 to L_P = 7–8H at Re = 2200, and are compared to the previous experimental and theoretical results of the literature in the zone of mixing fluid and in the fully developed region with a good agreement.     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

Influence of single rectangular groove on the flow past a circular cylinder

In the present study, flow control mechanism of single groove on a circular cylinder surface is presented experimentally using Particle image velocimetry (PIV). A square shaped groove is patterned longitudinally on the surface of the cylinder with a diameter of 50 mm. The flow characteristics are studied as a function of angular position of the groove from the forward stagnation point of the cylinder within 0° ≤ θ ≤ 150°. In the current work, instantaneous and time-averaged flow data such as vorticity, ω streamline, Ψ streamwise, u/U_o and transverse, v/U_o velocity components, turbulent kinetic energy, TKE and RMS of streamwise, u_rms and transverse, v_rms velocity components are utilized in order to present the results of quantitative analyses. Furthermore, Strouhal numbers are calculated using Karman vortex shedding frequency, f_k obtained from single point spectral analysis. It is concluded that a critical angular position of the groove, θ = 80° is observed. The flow separation is controlled within 0° ≤ θ < 80°. At θ = 80°, the flow separation starts to occur in the upstream direction. The instability within the shear layer is also induced on grooved side of the cylinder with frequencies different than Karman vortex shedding frequency, f_k.     

Application of the Galerkin-Kantorovich-Dorodnitsyn method of integral relations to the solution of steady laminar boundary layers

Summary The Galerkin-Kantorovich-Dorodnitsyn N-parameter method of integral relations belongs in the same general category as those of Ritz and Galerkin. The method is employed to solve the steady, incompressible, two-dimensional flow of a laminar boundary layer. General Nth-order solutions are obtained for regions of accelerated flow and, with a modification to account for a possible separation point, for regions of retarded flow. The convergence and accuracy of the method are evaluated by comparing the solutions with a large number of similar and nonsimilar flow problems. The method permits a separation point to be approached as closely as desired without encountering numerical difficulties or requiring special continuation techniques.

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Übersicht Die N-Parameter-Methode der Integralbeziehungen von Galerkin, Kantorovich und Dorodnitsyn gehört zur gleichen Klasse wie die Verfahren von Ritz und Galerkin. Diese Methode wird benützt, um Lösungen für die stationäre, inkompressible, zwei-dimensionale Strömung einer laminaren Grenzschicht zu finden. Allgemeine Lösungen N. Ordnung werden für Gebiete mit beschleunigter Strömung erhalten und, mit gewissen Abänderungen zur Berücksichtigung eines möglichen Ablösepunktes, auch für Gebiete einer verzögerten Strömung. Die Konvergenz und die Genauigkeit der Methode werden durch Vergleich der Lösungen mit zahlreichen ähnlichen und nicht-ähnlichen Strömungsproblemen untersucht. Die Methode erlaubt eine beliebig weitgehende Annäherung an den Ablösepunkt, ohne daß numerische Schwierigkeiten auftreten oder besondere Grenzwertbetrachtungen erforderlich wären.

     

The influence of side group structure on the glass transition temperature of isomeric vinyl ester polymers

Summary The development of vinyl esters of C₉–C₁₁ carboxylic acids for internal flexibilization of paint latex polymers has led us to study the effect of structural variations. Flexibility being closely connected with the glass-transition temperature,T _g , we have investigated the transition behaviour of a large number of homopolymers and copolymers by dynamic-mechanical methods.The polymers studied were model compounds of uniform structure and a wide variety of copolymers. Of the latter, some were polymers of distillation fractions of the technical vinyl ester mixture, others copolymers of this mixture with a well-defined monomer, e.g. vinyl acetate.As we found a linear dependence ofT _g on composition, we have, in the calculations, treated the technical vinyl ester mixture as a single monomer. Inspection of polymerized fractions clearly revealed the effect of the structure of isomeric side groups, the higherT _g values being found for the more compact structures. The fact that the differences inT _g between polymers made from the various fractions are fairly small implies that their type of branching has led to moderately compact side groups. This has been derived fromT _g determinations on model polymers of highly branched vinyl esters of C₉ carboxylic acids of known structure.

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Zusammenfassung Die Entwicklung von Vinylestern auf der Grundlage von C₉–C₁₁ Carboxylsäuren für die innere Flexibilisierung von Farblatexpolymeren hat uns veranlaßt, den Effekt struktureller Unterschiede zu studieren. Da die Flexibilität eng mit der GlastemperaturT _g zusammenhängt, haben wir das Übergangsverhalten einer großen Zahl Homopolymere und Copolymere mit Hilfe von dynamisch-mechanischen Verfahren untersucht.Die verwendeten Polymere umfaßten sowohl Modellverbindungen einheitlicher Struktur als auch verschiedenartige Copolymeren. Einige der letzteren bestanden aus Polymeren von Destillationsfraktionen des handelsüblichen Vinylestergemisches, andere aus Copolymeren dieses Gemisches mit einem gutdefinierten Monomer, z. B. Vinylacetat.Da wir eine lineare Abhängigkeit der Glastemperatur von der Zusammensetzung feststellten, wurde das handelsübliche Vinylestergemisch bei unseren Berechnungen als ein einfaches Monomer betrachtet. Prüfung der polymerisierten Fraktionen erwies deutlich den Einfluß der Struktur isomerer Seitengruppen, wobei sich die höherenT _g -Werte bei den kompakteren Strukturen ergaben. Aus der Tatsache, daß die auf den verschiedenen Fraktionen basierenden Polymeren einen ziemlich kleinen Unterschied inT _g aufweisen, läßt sich schließen, daß die Verzweigung bei ihnen zu mäßig kompakten Seitengruppen geführt hat. Dieser Befund beruht auf Messungen der Glastemperatur an Modellpolymeren stark verzweigter Vinylester aus C₉ Carboxylsäuren bekannter Struktur.

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Paper presented at the Symposium at Leende of De Nederlandse Rheologische Vereniging on 21st and 22nd May 1970.     

On the first boundary value problem for the classical theory of elasticity in a three-dimensional domain with a singular boundary

An existence and uniqueness theorem is proved for the first boundary value problem of classical elasticity, relating to a broad class of three-dimensional domains whose boundaries may have edges and vertexes.The qualitative properties, up to the boundary, of the solution are investigated.

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Résumé On démontre un théorème d'existence et unicité pour le premier problème de valeurs au contour de l'élasticité classique, concernant une large classe de domaines de l'éspace à trois dimensions, avec des frontières présentant des arètes et des sommets.On étudie les propriétés qualitatives, jusque'au contour, de la solution.

     

Anwendung des Singularitätenverfahrens der Oberflächenbelegung auf rotationssymmetrische Körper

Übersicht Zur Berechnung der inkompressiblen Potentialströmung um beliebig dicke, angestellte Rotationskörper wird das Verfahren der Oberflächenbelegung mit Singularitäten angewendet. Für mehrere Körper mit verschiedenen Konturen und Dickenverhältnissen werden Geschwindigkeits-, Druck- und Auftriebsverteilungen numerisch ermittelt. Einige Vergleiche der so gewonnenen Ergebnisse mit Näherungslösungen, die man bei einer Singularitätenbelegung auf der Rotationsachse erhält, zeigen, inwieweit die Anwendungsmöglichkeit der letzten Methode begrenzt ist.

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Summary For the calculation of the incompressible potential flow about bodies of revolution of arbitrary thickness ratio at angle of incidence, the method of surface singularities is used. The distributions of velocity, pressure, and lift are evaluated for bodies with different shapes and various thickness ratios. Comparisons of the results with those obtained by arranging singularities on the axis show the range of validity of the approximate solutions.

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Gekürzte Fassung der von der Fakultät für Maschinenwesen und Elektrotechnik der TH München genehmigten Dissertation (1. Berichterstatter: Prof. Dr.-Ing. E. Truckenbrodt, 2. Berichterstatter: Prof. Dr. rer. nat. H. St. Stefaniak). Der Verfasser dankt Herrn Prof. Dr.-Ing. E. Truckenbrodt für die Anregung zu dieser Arbeit und für die stetige Förderung und wertvollen Hinweise, die er ihr im weiteren Verlauf zuteil werden ließ.     

Stability of a water tower

Summary The stability of a water tower, consisting of a spherical shell rigidly attached to the top of a vertical column, is analyzed. At first, it is shown on a simple conceptual model that problems of this type are imperfection insensitive; thus, the first bifurcation load from the straight state of equilibrium could be considered the buckling load. This is followed by the analysis of the water tower problem taking into consideration the flexibility of the column, the own weight of the structure, the weight of the liquid, and the rotational stiffness of the soil base. Although the differential equation of the resulting eigenvalue problem has a variable coefficient, the problem is solved exactly in closed form. To simplify the utilization of the obtained results, they are presented graphically for a range of water tower and soil stiffness parameters. Model tests were conducted that confirmed the analytical findings. However, nearP _cr , at the smallest disturbance the model underwent large oscillatory motions. This finding suggests that, for water tower design, only a fraction ofP _cr should be used.

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Stabilität eines Wasserturms

Übersicht Untersucht wird die Stabilität eines Wasserturms aus einer senkrechten Säule mit kugelförmigem Behälter an der Spitze. Zunächst wird anhand einer einfachen Modellabstraktion nachgewiesen, daß Probleme dieser Art unempfindlich sind gegen Imperfektionen; deshalb kann die erste aus dem geraden Gleichgewichtszustand führende Verzweigungslast als Knicklast angesehen werden. Es schließt sich die Untersuchung des Wasserturms mit Nachgiebigkeit und Eigengewicht der Säule, mit Gewicht von Behälter und Inhalt sowie Drehsteifigkeit des Fundaments an. Die Lösung des Eigenwertproblems, dessen Differentialgleichung einen variablen Koeffizienten besitzt, läßt sich geschlossen angeben. Zur bequemen Anwendung werden die Ergebnisse für weite Parameterberciche von Turm und Fundament graphisch dargestellt. Durchgeführte Modellversuche unterstützen die analytischen Ergebnisse. Allerdings vollführte das Modell in der Nähe der kritischen LastP _cr selbst bei kleinsten Störungen große Schwingbewegungen. Deshalb sollte bei einer Auslegung nur ein Bruchteil vonP _cr zugrunde gelegt werden.

     

Burnout at very high heat fluxes

The paper provides a brief overview of the present state-of-the-art in the field of the critical heat flux (CHF) in subcooled flow conditions, with particular regard to high liquid velocity and highly subcooled conditions. These thermofluid-dynamic conditions to reach very high values of the CHF (up to 80 MW/m²) as requested by fusion technology requirements. After reporting the general features of the problems, some details are given about the main parameters affecting CHF in subcooled flow boiling (subcooling, pressure, duct diameter and length, fluid velocity, duct orientation and flow direction, heat flux distribution), and the respective knowledge and possible enhancement techniques for CHF (extended surfaces, hypervapotron, surface roughness, swirl flow, helically coiled wires) are reviewed. As fundamental aspects of highly subcooled flow boiling are not known and, on the other hand, such knowledge is necessary for any possible enhancement technique evaluation and for a theoretical modelling of the phenomenon, recent experiments carried out over the last five years on CHF in subcooled flow boiling are reviewed together with the main correlations and models currently used for data prediction, and suggestions for particular and general research guidelines are given.Der Aufsatz gibt einen kurzen Überblick über den gegenwärtigen Stand der Technik im Bereich der kritischen Wärmestromdichte bei unterkühltem Strömungssieden, mit besonderem Schwerpunkt auf hohe Flüssigkeitsgeschwindigkeiten und hohen Unterkühlungsbedingungen. Diese thermo- und fluiddynamischen Bedingungen ermöglichen es, sehr hohe Werte der kritischen Wärmestromdichte (bis zu 80 MW/m²) zu erreichen, wie sie in den Anforderungen der Fusions-Technik verlangt werden. Nach einem Abschnitt über allgemeine Gesichtspunkte des Problems werden ein paar Einzelheiten der Hauptparameter bezüglich der kritischen Wärmestromdichte bei unterkühltem Strömungssieden (Unterkühlung, Druck, Rohrleitungsdurchmesser und Länge, Fluidgeschwindigkeit, Rohr-Orientierung und Strömungsrichtung, Wärmestromverteilung) genannt. Es wird ein Überblick gegeben über den jeweiligen Wissensstand und mögliche Verbesserungstechniken für die kritische Wärmestromdichte (vergrößerte Oberflächen, Hypervapotron, Oberflächenrauheit, Drallströmung, spiralförmig gewickelte Drähte). Da grundlegende Aspekte von extrem unterkühlten Strömungssieden unbekannt sind, eine Kenntnis dieser Aspekte andererseits für jede mögliche Entwicklung von Verbesserungstechniken sowie für eine theoretische Modellbildung über das Phänomen notwendig ist, wird eine Übersicht über die neuesten Experimente gegeben, die in den letzten fünf Jahren über die kritische Wärmestromdichte bei unterkühltem Strömungssieden durchgeführt wurden. Gleichzeitig werden die wichtigsten Korrelations-Gleichungen und Modelle besprochen, die heutzutage für Daten-Vorausberechnungen verwendet werden sowie Vorschläge für besondere und allgemeine Forschungsrichtlinien unterbreitet.Dedicated to Prof. Dr.-Ing. U. Grigull's 80th birthday     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

Rheological properties of thin gelatin films

Summary The stability of oil-in-water emulsions depends ultimately on the physical nature of the oil-water interface. If a hydrocolloid or protein is used as an emulsifying agent an interfacial film is formed at the oil-water interface and this may be liquid or viscoelastic solid in properties. The film is formed by the two processes of diffusion from the bulk to the interface and the formation of secondary intermolecular cross-links.Gelatine solutions of up to 2% w/v were used to form an interface with light liquid paraffin over a restricted range of p _h . For the 2% w/v gelatin solutions, the p _h range was confined to the acid region p _h 2.5–3.2 owing to the onset of gelation of the whole aqueous phase, at room temperature, if the p _h was increased numerically above this range. At lower concentrations of gelatin, e. g., 0.5% w/v, the useful p _h range could be extended to p _h 4.The interfacial rheological properties were examined using the surface rheometer described in a previous paper. Preliminary results showed that, in the case of the 1% w/v gelatin solution at p _h 3, the interfacial film was liquid up to the end of the first half-hour after formation. Solid properties then developed. After 4 hs it was possible to carry out a creep experiment on the film which exhibited a longest retardation time of several minutes. Further creep and recovery experiments conducted up to 200 h after the formation of the film clearly showed that the growth of solid properties of the film was continuing.The gelatin interfacial films were considerably more compliant than similar films of acacia Senegal, over a comparative period of time, and at the same bulk concentration. This difference in behaviour can be attributed to differences in the basic molecular structure of the two polymers.

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Zusammenfassung Die Stabilität von Öl-in-Wasser-Emulsionen hängt letztlich von der physikalischen Eigenschaft der Öl/Wasser-Grenzfläche ab. Wenn ein Hydrokolloid oder ein Protein als Emulgator verwendet wird dann bildet sich in der Öl/Wasser-Grenzschicht ein Film, der Flüssigkeitsoder viskoelastische Eigenschaften aufweisen kann.Dieser Film wird durch zwei Diffusionsvorgänge von der Gesamtmasse zur Grenzschicht und durch zwischenmolekulare Vernetzung gebildet.Gelatinelösungen bis 2% (Gewicht/Volumen) wurden verwendet, um mit dünnflüssigem Paraffin über einen beschränkten p _h -Bereich eine Grenzschicht zu bilden. Für die 2% Gelatinelösung wurde der p _h -Wert im sauren Bereich zwischen p _h 2,5–3,2 gehalten, weil bei Erhöhung des p _h die gesamte wässerige Lösung bei Raumtemperatur zu gelieren begann. Bei geringeren Gelatinekonzentrationen, wie z. B. 0,5% (g/ml), konnte der brauchbare p _h -Bereich bis zu P _h 4 ausgedehnt werden.Die rheologischen Eigenschaften der Grenzschicht wurden mit dem Oberflächenrheometer untersucht, das in einem vorhergehenden Vortrag beschrieben wurde.Vorläufige Ergebnisse haben gezeigt, daß im Falle einer 1% Gelatinelösung mit p _h 3 der Grenzfilm bis gegen Ende der ersten halben Stunde nach seiner Bildung flüssig war. Dann zeigte dieser Festkörper-Eigenschaften. Nach 4 Std. konnte der Film in einem Kriechtest untersucht werden; wobei maximale Retardationszeiten von mehreren Minuten beobachtet wurden. Weitere Kriech- und Kriecherholungsversuche, die bis zu 200 Std. nach der Filmbildung durchgeführt wurden, zeigten deutlich, daß die Festkörper-Eigenschaften des Filmes weiterhin zunahmen.Gelatinegrenzflächenfilme waren für einen vergleichbaren Zeitraum und eine ähnliche Gesamtkonzentration beträchtlich nachgiebiger als ähnliche Filme mit Acacia Senegal. Dieses unterschiedliche Verhalten kann von den verschiedenen molekularen Grundstrukturen der beiden Polymeren herrühren.

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Paper presented to the British Society of Rheology Conference on Rheology in Medicine and Pharmacy, London, April 14–15, 1970.     

Experimental study of natural convection heat transfer between an outer horizontal cylindrical envelope and an inner concentric heated square cylinder with two slots

This paper presents the results of an experimental study of natural convection heat transfer between a horizontal cylindrical envelope and an internal concentric heated square cylinder with two slots. The internal cylinder was a hollow one with horizontal slots on its top and bottom surfaces. The ratio of slot widthS to the side heightH was 0.0612 and 0.3878. The ratio of the envelope inner diameterD _o to the side heightH was 2.653. Air was used as the working fluid. The range of Ray-leigh number was 1.77×10²8.72×10⁶ forS/H=0.0612 and 1.32×10²6.25×10⁶ forS/H=0.3878. The results show that there are three different heat transfer regimes in different Ray-leigh number regions, i.e. pure conduction regime, transition regime and convection regime. The average heat transfer results were correlated into two empirical equations. Comparison was made with the non-slotted case. It is found that slots of the internal cylinder can significantly enhance the heat transfer.

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Experimentelle Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem konzentrischen, beheizten, quadratischen Prisma mit zwei Schlitzen

Zusammenfassung In der Arbeit werden die Ergebnisse einer experimentellen Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem beheizten quadratischen Prisma mit zwei Schlitzen vorgestellt. Das Prisma selbst ist hohl und weist in der oberen und unteren Begrenzungsfläche je einen horizontalen Längsschlitz auf. Das Verhältnis von SchlitzweiteS zu SeitenhöheH beträgt 0,0612 und 0,3878, das des HülleninnendurchmessersD _o zur SeitenhöheH beträgt 2,653. Als Arbeitsmedium diente Luft. Die Rayleigh-Zahlen variierten zwischen 1,7·10² und 8,72·10⁶ fürS/H=0,0612 und zwischen 1,32·10² und 6,25·10⁶ fürS/H=0,3878. Die Ergebnisse belegen die Existenz dreier unterschiedlicher Wärmeübergangsregime in den verschiedenen Rayleigh-Zahl-Bereichen, und zwar reiner Leitungsbereich, Übergangsgebiet und Konvektionsbereich. Die Ergebnisse für den Wärmeübergang werden im Vergleich mit jenen für ein Prisma ohne Schlitze durch zwei Korrelationbeziehungen dargestellt. Es zeigt sich, daß durch Anbringung von Schlitzen am Innenprisma der Wärmeübergang wesentlich verstärkt werden kann.

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Nomenclature C _p specific heat at constant pressure, J/(kg·K) - D _i diameter of the related circular cylinder whose circumferential area is equal to that of the unslotted square cylinder, m - D _o internal diameter of the outer circular envelope, m - F _i surface area of the inner two slot cylinder, m - g gravitational acceleration, m/s² - H distance between the opposite sides of the square cylinder with two slots, m - K _eq dimensionless equivalent thermal conductivity - L axial length of the test section, m - m ratio of the area of the unslotted square cylinder surface to that of the slotted square cylinder - P pressure in the enclosure, Pa - Q total power input to the enclosure, W - Q _cond radial heat conduction, W - Q _conv convective heat transfer, W - Q _r radiation heat transfer, W - Q _los end heat dissipation, W - R air gas constant, J/(kg·K) - Ra Rayleigh number - S slot width, m - T _i wall temperature of the inner cylinder, K - T _o wall temperature of the outer envelope, K - T _m mean temperature, K - T temperature difference=T _i –T _o , K - W maximum gap width of the test annuli=(D _o –H)/2 for the square case, m Greek symbols ₀ black body radiation constant, W/(m²·K⁴) - _s equation system emissivity - air thermal conductivity, W/(m·K) - _eq equivalent thermal conductivity, W/(m·K) - air dynamic viscosity, kg/(m·s) This work was supported by the National Natural Science Foundation of China.     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o     

Fatigue gage utilizing slip-initiation phenomenon in electroplated copper foil

Fundamental experiments are carried out to examine the parameter that dominates the slip-band initiation in electroplated copper foil under the condition where the mean stress as well as the stress amplitude varies. In the case of constant-amplitude stressing, the relation between the critical stress for the slip-band initiation σ _p and the number of cyclesN is well represented by σ _p ^α N=C. In other words, the slip bands appear when the total hysteresis energy applied to the copper foil attains a critical value. In the case of variable stresses, the range-pair mainly dominates the occurrence of the slip bands, and Miner's linear cumulative damage rule holds for the accumulation of the fatigue damage for the slip-band initiation. Accordingly, the parameter (Σσ _i ^α n _i/Σn _i)^1/α is equivalent to the critical stress σ _p under constant amplitude stressing, where σ _i andn _i are the stress amplitude and the number of cycles counted by the range-pair method, respectively, and α is the exponent of the σ _p -N relation. Based on these results, the applicability of the copper foil to the fatigue gage that accumulates and indicates a load experience is discussed.     

Poroacoustic solitary waves under the unidirectional Darcy–Jordan model

Working in the context of poroacoustics, we present new, physically relevant, explicit solutions to the Cauchy problem for the model we term the (1D) damped Riemann equation. The solitary waveforms that evolve from both Lorentzian ($C^{\infty }$-smooth) and symmetric-exponential ($C^0$-smooth) initial conditions are analyzed, the focus being on wave overturning and the evolution/structure of the shocks which develop thereafter. In addition to those for both the multi- and single-valued forms of each solution, expressions for the shock amplitude, velocity, and critical values of the physical parameters are derived/compared. Lastly, links to other areas of continuum physics, and possible follow-on investigations, are noted.     

New Explicit Expression of Barnett-Lothe Tensors for Anisotropic Linear Elastic Materials

The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic elastic materials [1–3]. They also appear in certain three-dimensional problems [4, 5]. The algebraic representation of H, L, S requires computation of the eigenvalues p^v(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing p ^v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form. The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C^μ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′ _μv , the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of L, S that remain valid for the degenerate cases p ₁ p ₂ and p₁=p₂ = p ₃. One may want to compute H, L, S using either C _μv or s′ _μv v, but not both. We show how an expression in C^μ v can be converted to an expression in s′ _μv v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in C^μ v and s′ _μv v. This revised version was published online in August 2006 with corrections to the Cover Date.     

Similarities Between Rotation and Stratification Effects on Homogeneous Shear Flow

Linear theory is applied to examine rotation and buoyancy effects on homogeneous turbulent shear flows with given vertical velocity shear, S=d/dx ₃. In the rotating shear case (where the rotation vector is perpendicular to the plane of the mean flow, Ω _i =Ωδ _{i 2}), general solutions for the Fourier components of the fluctuating velocity are proposed. These solutions are compared with those proposed in the literature for the Fourier components of the fluctuating velocity and density in the case of a homogeneous stratified shear flow with vertical density gradient, S _ρ=d/dx ₃. It is shown that from the normal mode stability stand point the Bradshaw parameter B=2Ω/S(1+2Ω/S) (in the rotating shear case) and the Richardson number R _i (in the statified shear case) play similar roles in identifying the stability for all the wave components except in the case where Ω·k=0, for which rotation has no effects on the flow. Analysis of the long-time behavior of the non-dimensional spectral density of energy, e ^g , is carried out. In the stable case, e ^g has decaying oscillations or undergoes a power law decay in time. Analytical solutions for the streamwise two-dimensional energy ℰ _ii ¹/2 (i.e. the limit at k ₁=0 of the one-dimensional energy spectra) are proposed. At large time, ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (3R _i +1)/(4R _i ) except at R _i =1 it stays constant in time. Similar behavior for ℰ _ii ¹(t)/ℰ _ii ¹(0) is also observed in the rotating shear case (ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (1+4B)/(4B)). Due to the behavior of the dimensionless spectral density of energy in both flow cases, the turbulent kinetic energy, /2, the production rate, ?, and the rate due to the buoyancy forces, ℬ, are split into two parts, , ?=?₁+?₂, ℬ=ℬ₁+ℬ₂ (in the stratified shear case, both ?₁ and ℬ₁ vanish when R _i >?, while in the rotating shear case one has ℬ=0). It is shown that when rotation is “cyclonic” (i.e. Ω/S>0), part reaches maximum magnitudes at St ≈2, independent of the B value, and the first time to a zero crossing of ?₂ occurs at this particular value. When rotation is “anticyclonic” (i.e. Ω/S<0) one finds St ≈1.6 instead of St ≈2. In the stratified shear case, both ?₂ and ℬ₂ cross zero at Nt=St ≈2, and part reaches maximum magnitudes at this particular value. These results and in particular those for the turbulent kinetic energy are compared with previous direct numerical simulation (DNS) results in homogeneous stratified shear flows. Received 30 July 2001 and accepted 19 February 2002