Eine Erweiterung der Analogie zwischen der Gasdynamik und Flachwasserströmungen auf rotierende Systeme mit konservativen Massenkräften

Zusammenfassung Die bekannte Analogie zwischen den Theorien reibungsfreier flacher Wasserströmung und der isentropen idealen Gasdynamik wird im zweidimensionalen instationären Fall durch Zulassung rotierender Bezugssysteme und konservativer Massenkraftfelder erweitert. Damit kann man in einem rotierenden Wasserbehälter mit geeignet gestaltetem Boden u. a. atmosphärische Probleme näherungsweise studieren.

---

Summary The well known analogy of the theories of nonviscous shallow water flows and isentropic ideal gasdynamics is generalized in the two-dimensional nonstationary case to apply to rotating systems with conservative body forces. This enables one, in particular, to simulate atmospheric problems by use of a rotating water tank with suitable bottom shape.

---

Liste der verwendeten Symbole spezifische Wärme des Gases bei konstantem Druck - Fr in (2) definierteFroude-Zahl - Gravitationsbeschleunigung - H ortsabhängige Ruhetiefe bei Drehung des Wassers als starrer Körper - Druck - atmosphärischer Druck auf der freien Wasseroberfläche - Bezugslänge - Zeit - absolute Temperatur - Geschwindigkeitskomponenten parallel zu den Achsen - kartesische Koordinaten - durch Abweichung von der Drehung des Wassers als starrer Körper verursachte orts-und zeitabhängige Änderung der Wassertiefe - Verhältnis der spezifischen Wärmen des Gases - zu gehörende Wellenlänge - Dichte - Winkelgeschwindigkeit Mit 1 Textabbildung1 Boden des Wasserbehälters 2 freie Wasseroberfläche II freie Wasseroberfläche bei einer Drehung des Wassers als starrer KörperEin oberer Querstrich gibt an, daß die betreffende Größe eine physikalische Dimension hat; Größen ohne einen solchen Querstrich sind dimensionsfrei.     

Laminar thermal boundary layer on a continuous accelerated sheet extruded in an ambient fluid

Summary Exact boundary layer similarity solutions are developed for flow, friction and heat transfer on a continuously accelerated sheet extruded in an ambient fluid of a lower temperature.Melt-spinning, polymer and glass industries and the cooling of extruded metallic plates are practical applications of this problem.Results for skin-friction and heat-transfer coefficients are given. Larger acceleration is accompanied by larger skin-friction and heat-transfer coefficients. Rapid cooling of the sheet is accompanied by a larger Nusselt number.Nomenclature sheet width - c dimensionless constant - c _f local skin friction coefficient - F dimensionless transformed stream function - G dimensionless transformed temperature - local heat transfer coefficient - fluid thermal conductivity - length of deformation zone - m exponent of surface speed variation - q exponent of surface temperature variation - T dimensionless temperature - sheet surface temperature - solidification temperature - ambient temperature - sheet thickness - u velocity component along the sheet - u _s sheet surface velocity - wind up velocity - v velocity component normal to the sheet - x dimensionless coordinate along the sheet - y dimensionless coordinate normal to the sheet - Nu Nusselt number, - Pr Prandtl number, - Re Reynolds number, - =Re^–0.5 - dimensionless similarity coordinate - dynamic viscosity - kinematic viscosity - fluid mass density - sheet mass density - wall shear stress - dimensionless stream function With 3 Figures     

Nonlinear liquid oscillations in spherical systems under zero-gravity

Summary Nonlinear free oscillations of the interface of a concentric frictionless immiscible liquid system in a spherical container are investigated in a zero-gravity environment. The natural frequencies are determined for the axisymmetric and asymmetric oscillations of the interfacial surface with the diameter ratio and density ratio as parameters. It was found that for small outer- to inner liquid density ratio the oscillations exhibit softening, while for large density ratios it renders hardening oscillation. The asymmetric oscillations exhibit in the softening range softer and in the hardening range harder liquid oscillations. For a liquid layer around a rigid center sphere the oscillations of the free liquid surface yields softening behavior, where for thinner layers the softening effect is more pronounced.Nomenclature a radius of spherical container, or radius of rigid center sphere - b radius of undisturbed interfacial surface, or radius of undisturbed free liquid surface - k=a/b diameter ratio - pressure - pressure (dimensionless) - , , spherical coordinates - dimensionless radius - R _i main radii of curvaturei=1, 2 time - dimensionless time - v _i liquid velocity (j=1 spherical layer region,j=2 inner liquid sphere region) - V volume of the liquid - Y _nm tesseral surface harmonics - _i density of liquids - velocity potential - dimensionless velocity potential - interfacial surface- or free surface elevation - dimensionless interfacial surface- or free surface elevation - ₀ maximum elevation - circular frequency - circular frequency - _n0 axisymmetric natural frequency - _n1 asymmetric natural frequencym=1 - _nm ⁽⁰⁾ natural frequency of linearized liquid system - mean curvature - _nm Kronecker symbol With 10 Figures     

On the motion of a curved and twisted rod

Summary In this paper, the equations of motion of a curved and twisted rod are derived from the basic principles of dynamics. The set of equations gives the extensional, flexural and torsional motions of the rod. The coupling among these types of motion due to the curvature and tortuosity of the rod is shown explicitly in the case of a helical spring. By successive simplification of the equations, the equations of motion of a circular ring and those of a straight rod are obtained. In this respect, the derived equations can be considered as a generalization of the elementary theories of rod in extensional, torsional and flexural motion.The dispersion relation of a helical spring is calculated for the two lower frequency modes. It is shown that the frequency-wave length relationship is not monotonically decreasing as in the cases of uncoupled flexural or torsional motion. Finally, frequencies are calculated based on the approximate frequency expression ofLove to show that Love's frequency expression for a helical rod is accurate.

---

Zur Bewegung eines gekrümmten und verdrillten Stabes

Zusammenfassung In dieser Arbeit werden die Bewegungsgleichungen eines gekrümmten, verdrillten Stabes von den Grundgleichungen der Dynamik hergeleitet. Dieses System von Gleichungen legt die Bewegung des Stabes durch Längsdehnung, Biegung und Torsion fest. Die Kopplung dieser Bewegunsarten, durch Krümmung und Verdrillung, wird für den Fall einer Spiralfeder explizit gezeigt. Durch Vereinfachung der Gleichungen werden die Bewegungsgleichungen des Kreisringes und die des geraden Stabes erhalten. In dieser Hinsicht können die hergeleiteten Gleichungen als Verallgemeinerung der elementaren Theorein der Bewegung eines Stabes betrachtet werden.Die Dispersionsgleichung der Spiralfeder wird für die beiden niedrigsten Frequenzen berechnet. Es wird gezeigt, daß das Verhältnis Frequenz-Wellenlänge nicht wie in den Fällen der ungekoppelten Biege-oder Torsionsbewegung monoton abnimmt. Abschließend werden die Frequenzen nach dem Loveschen Näherungsausdruck berechnet, um zu zeigen, daß die Genauigkeit dieses Ausdruckes für Spiralfedern gut ist.

---

Nomenclature position vector of a point on axis of curved rod - position vector relating any pointB in a plane perpendicular to the axis of the rod to pointP where plane cuts axis of rod - R modulus of - linear momentum vector - density - axial force vector = - axial force vector at origin - transverse shear force vector - transverse shear force vector at origin - moment vector - moment vector at origin - s distance measured along axis of rod - t time - a radius of gyration of rod for circular sections - l total length of spring along axis - A cross-sectional area of rod - s/a — normalized coordinates measured along axis of rod - applied load vector - angular momentum vector - trihedral of space curve-unit vectors in tangential, normal and binormal directions - _t axial rotation - displacement vector - normalized displacement vector= - ₀,₀ curvature and tortuosity of rod - , normalized curvature and tortuosity of rod - , circular frequency and normalized frequency respectively - E, G Young's modulus and shear modulus - v Poisson's ratio - Wave number - pitch angle of helical spring - d wire diameter of helical spring - R coil radius of helical spring - N number of turns of helical spring With 5 Figures     

Viscous response of an anchored spinning liquid column to various axial excitations

Summary For a solidly rotating viscous cylindrical liquid column of finite length the response to axial synchronous, counter- and one-sided excitation is determined for anchored contact lines at the disc-rim. For a rotating column additional responses of inertial waves (hyperbolic range) appear for < 2 ₀, while in the elliptic range < 2 ₀ the sloshing response occurs. The various responses for the free surface displacement have been numerically evaluated. Only in the one-sided exitation case all resonance peaks appear, while for synchronous excitation only the odd resonances and for counter-excitation only the even resonance peaks occur.Notation a radius of column - h length of liquid bridge - I _n modified Bessel function - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution - Weber number - z ₀ excitation amplitude - liquid density - surface tension - surface tension parameter - Ohnesorge number - liquid surface displacement - kinematic viscosity - ₀ rotational speed - dimensionless rotational speed - forcing frequency - dimensionless forcing frequency - dimensionless forcing frequency for non-viscous liquid - a= root of bi-cubic Eq.(33) - root of bi-cubic Eq.(33)     

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.     

Damped response of an axially excited rotating liquid bridge in zero-gravity

Summary The response of a solidly rotating finite liquid bridge due to axial excitation exhibits for frictionless liquid at the resonances singularities. For the experimenter in a spacelabmission the actual resonance amplitude is of quite some importance. For this reason damping, that has to be measured in ground tests, has been introduced into the results of the response.Notation a radius of the liquid bridge - h length of the liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - r, ,z polar coordinates - t time - excitation amplitude - elliptic case - hyperbolic case - abbreviation - damping factor of liquid - (z, t) free surface displacement - =² – ² surface tension - surface tension - liquid density - ₀ rotational speed of liquid bridge - forcing frequency of axial excitation - natural frequency of liquid bridge With 2 Figures     

The laminar stagnation-point flow against a solidifying moving shell

Summary This paper considers the two-dimensional laminar stagnation-point flow due to a jet impinging onto a solidifying moving boundary. The flow is of interest in connection with the horizontal belt strip casting process. An exact solution to the Navier-Stokes equations is found that is shown to depend on a single ordinary differential equation. The solution is useful in the study of morphological and hydrodynamic instabilities within the impingement region. Solutions for the steady-state shape of the initial stages as well as the asymptotic behavior of the solidifying interface are also discussed in a perturbative manner.Nomenclature A suction velocity in boundary layer variables - a jet width [m] - c specific heat of the solid metal [J/m³K] - h Newtonian heat transfer coefficient [W/m²K] - k velocity gradient in units ofU/a - m dS ^*/dX ^* local inclination of the solidifying phase - S ^* (L)/L average slope of the solidifying phase - S ^* local thickness of the solidified phase [m] - S, S local thickness of the solidified phase in units ofL and , resp. - T absolute temperature [K] - T _f fusion temperature of metal [K] - T ₀ temperature of cooling water [K] - U jet velocity [m/s] - V belt velocity [m/s] - +i complex velocity potential in units ofUa - x coordinate tangential to the solidifying interface in units ofa - X ^* coordinate tangential to the belt [m] - X, X coordinates tangential to the belt in units ofL and , resp. - y coordinate orthogonal to the solidifying interface in units ofa - Y ^* coordinate orthogonal to the belt [m] - Y, Y coordinates orthogonal to the belt in units ofL and , resp. - z x+iy complex coordinates in units ofa - unit vector along the belt - unit vector orthogonal to the belt - local unit normal vector to the solidifying interface - h _f latent heat of fusion of metal [J/m³] - thermal diffusivity of solid metal [m²/s] - belt velocity in units ofU - { _n }, { _n } asymptotic sequences of the outer and inner expansion, resp. - m suction velocity outer variables - velocity potential in units ofUa - jet inclination relative to the local solidifying interface - coordinate orthogonal to the solidifying interface in units of - x c thermal conductivity of solid metal [W/mK] - displacement thickness in units of - v kimematic viscosity of liquid metal [m ²/s] - arctan (dS ^*/dX ^*) local angle of inclination of the solidifying interface - =(T–T ₀)/(T _f –T ₀) dimensionless temperature - perturbation parameter - coordinate tangential to the solidifying interface in units ofa/k - stream function in units ofUa - magnified stream function valid within the boundary layer - solidification constant Dimensionless parameter P _eL VS ^* (L)/ Peclet number - Q h/(cV) Heat flux number - R Ua/v Reynolds number - St Stefan number     

On primitive and free roots in a finite field

In this paper them-dimensional extension of the finite field of orderq is investigated from an algebraic point of view. Looking upon the additive group as a cyclic module over the principal ideal domain , we introduce a new family of polynomials over which are the additive analogues of the cyclotomic polynomials. Two methods to calculate these polynomials are proposed. In combination with algorithms to compute cyclotomic polynomials, we obtain, at least theoretically, a method to determine all elements in of a given additive and multiplicative order; especially the generators of both cyclic structures, namely the generators of primitive normal bases in over , are characterized as the set of roots of a certain polynomial over .     

Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden

Zusammenfassung Messungen des Anwachsens von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkav gekrümmten Wänden (Görtler-Taylor-Wirbel) ergaben drei deutlich getrennte Bereiche: Es traten zunächst Wirbel mit der Wellenläge 0,9 auf (=Grenzschichtdicke, =Höhe einer Zelle, die zwei gegensinnig drehende Wirbel enthält). Je nach Größe der mit der Verdrängungsdicke ₁ der Grenzschicht gebildeten Reynolds-Zahl erschienen dann kurze Zeit später Wirbel mit 2,5, wenn war. Im Bereiche dagegen traten stattdessen bei den hier durchgeführten Versuchen immer Wirbel mit der Wellenlänge 6,5 auf. Bei werden die ersten Tollmien-Schlichting-Wellen mit der Wellenlänge _TS 6· angefacht. In ihren wandnahen Bereichen der Wellentäler könnten sich dann die oben genannten Längswirbel der Wellenlänge 6,5· ausbilden, die die zwei-in eine dreidimensionale Störung allseits gleicher Größenordnung verwandeln können.

---

The development of longitudinal vortices in boundary layers growing with time along concave walls

Summary Measurements of the growth of longitudinal vortices in boundary layers growing with time along concave walls (Görtler-Taylor vortices) rendered three distinctly separated regions. First, vortices with a wave-length 0.9 appeared (-boundary layer thicness, =height of a cell containing two counterrotating vortices). Then, depending on the Reynolds number R _{a 1}/v ₁=displacement thickness), vortices with 2.5 appeared shortly afterwards, provided . In the region , however, the wave-length was 6.5. For the first Tollmien-Schlichting waves with _TS 6 were excited. In the wave-throughs close to the wall the abovementioned longitudinal vortices with wave length 6.5 may then be formed. This might transform the two-dimensional into a three-dimensional flow of equal order of magnitude in all directions.

---

Zeichenerklärungen R _a Innenradius - Re _a Reynolds-Zahl gebildet mit dem InnenradiusR _a - Reynolds-Zahl gebildet mit der Verdrängungsdicke ₁ - kritische Taylor-Zahl - h Standhöhe der Flüssigkeit im Zylinder - t Zeit - z Anzahl - Steigungswinkel der Geraden - Grenzschichtdicke - ₁ Verdrängungsdicke - Wellenlänge (enthält ein gegensinnig rotierendes Längswirbelpaar) - v kinematische Zähigkeit - Winkelgeschwindigkeit Indizes K Knickpunkt der Geradensteigung - L unterhalb des Knickpunktes der Geradensteigung - TS Tollmien-Schlichting - e Einsatz der Wirbelentstehung     

Stresses in a cylindrical shell supported by a core of a different material

Summary The stress problem of a thin cylindrical shell supported by an elastic core of a different material and subjected to arbitrary loading on its curved surface is considered. The problem is solved by applying the three-dimensional theory of elasticity to the core and using membrane or bending solutions for the shell. Equilibrium and compatibility equations are satisfied at the junction of the shell and the core. It is pointed out that the procedure can easily be extended to the case of a hollow core with or without another shell of another material in it. Numerical results are presented to illustrate the effectiveness of even a weak core in reducing the shell stresses.

---

Zusammenfassung Gegenstand der Untersuchung ist eine dünne Kreiszylinderschale, die durch einen elastischen Kern aus einem anderen Werkstoff gestützt ist und eine beliebige Belastung trägt. Die Lösung verbindet die strenge, dreidimensionale Theorie des zylindrischen Kerns mit der Membran- oder Biegetheorie der Schale. An der Grenze zwischen beiden Teilen müssen die Verschiebungen und gewisse Spannungskomponenten stetig übergehen. Es wird darauf hingewiesen, daß die Lösung leicht auf den Fall ausgedehnt werden kann, daß der Kern ein Hohlzylinder ist, der möglicherweise auf der Innenseite mit einer zweiten Zylinderschale verbunden ist. Zahlenergebnisse zeigen, daß selbst ein verhältnismäsig nachgiebiger Kern einen großen (und günstigen) Einfluß auf die Spannungen in der Schale ausübt.

---

Principal Symbols a Radius of the middle surface of the shell - t Thickness of the shell - =1–t/2a - u _c,v _c,w _c Displacements respectively in the axial, circumferential and radial directions of a point in the core - X(x), (), (r/a) 3×3 square matrices - ,m Parameters - l Length of the cylinder - c A vector containing constantsc ₁,c ₂ andc ₃ - =r/a - =m+4(1–v _e) - E _c,v _e Elastic constants for the core material - Stresses at a point in the core - D _c - A vector containing _rx , _r and _r - (r/a) A 3×3 matrix - Displacements at the surfacer=a of the core - A vector containing - Amplitudes of displacements - A vector containing - =(x, ,a) - _ij Constants - A A square matrix containing constants _ij - Stress resultants in the shell as defined in reference [3] - p _x,p ,P _r Components of applied loading per unit area of shell's middle surface - () - ()· - u, v, w Displacements of a point on the middle surface of the shell - E _s,v _s Elastic constants for the shell material - D _s - K - k - p _xmn,p _mn,p _rmn Amplitudes of loadsp _x,p , p_r - u _mn, v_mn,w _mn Amplitudes of displacementsu, v, With 1 Figure     

Free vibrations of a circular plate with attached concentrated mass,spring and dashpot

Summary The natural vibrations of a circular plate with attached concentrated mass, spring and dashpot have been obtained by means ofYoung's analysis [1]. The results are presented in terms of eigen-functions of the plate alone. The case of a plate carrying two masses and resting on elastic foundation has also been studied. Some particular cases have been deduced.

---

Zusammenfassung Die Eigenschwingungen einer Kreisplatte mit lokal befestigter Einzelmasse, Feder und Dämpfer werden nach derYoungschen Methode [1] ermittelt. Die Ergebnisse werden als Entwicklung nach den Eigenfunktionen der reinen Plattenschwingung dargestellt. Der Fall der elastisch gebetteten Platte mit zwei Einzelmassen wird ebenfalls studiert. Einige Sonderfälle werden hergeleitet.

---

Nomenclature a radius of circular plate - h plate thickness - k ₁ spring constant - k _c generalized spring constant - modulus of elastic foundation - decay constant - c dashpot strength - D , flexural rigidity of plate - E Young's modulus - v Poisson's ratio - p natural frequency of plate alone - natural frequency of composite system - w deflection mode of plate - r, cylindrical coordinates - mass density - r - (l/D)^1/4 - - - F _m ,L _m ,G _m ,M _m unknown constants With 5 Figures     

Higher approximations for supersonic flow past slowly oscillating bodies of revolution

Summary Supersonic flow past slowly oscillating pointed bodies of revolution is studied. Starting from the complete nonlinear potential equation an elementary linearized solution is discussed and it is shown how this solution together with the method of matched asymptotic expansions can be used to derive an elementary second-order slender body theory. This approach is further demonstrated for the oscillating cone and its range of validity is evaluated by comparison with other theoretical methods.

---

Zusammenfassung Es wird die Überschallströmung um langsam schwingende spitze Rotationskörper untersucht. Ausgehend von der vollständigen nichtlinearen Potentialgleichung wird zuerst eine elementare linearisierte Lösung besprochen und gezeigt, wie diese Lösung im Verein mit der Method of matched asymptotic expansions zur Herleitung einer elementaren Schlankkörpertheorie zweiter Ordnung verwendet werden kann. Die Theorie wird am Beispiel des schwingenden Kegels näher erläutert und mit anderen Methoden verglichen.

---

Symbols a Velocity of sound - c _N Normal force coefficient - Damping coefficient - F _(x) Dipole distribution - k Reduced frequency - M Mach number - R (x) Meridian profile - t Time - x, r, Cylindrical coordinates - - Ratio of specific heats - Amplitude of oscillation - Thickness ratio - Perturbation potential - Zero angle of attack potential - æ - Velocity potential - Out-of-phase potential - - In-phase potential - - Source coordinate With 4 Figures     

Tunneling with very long relaxation times in glasses,organic materials,and Nb-Ti-H (D)

The long-time (t=10–200 h) heat release from glasses, from organic materials, and from Nb-Ti-H (D) was measured at 30T70 mK. For Suprasil W glass, Dimethyl-Siloxan, Stycast 1266, Stycast 2850 FT, Vespel, and for Nb-Ti-H (D) with various Ti and D concentrations, we found . Typical values are = 0.05 nW/g for the organic materials and for Nb-Ti-H (D) and = 0.005 nW/g for the glass att=100 h after cooldown from room temperature. For charging temperaturesT _i <5 K, we find the predicted dependence (investigated for Suprasil W glass and for Nb-Ti-D). The observed time and temperature dependences agree with predictions of the conventional two-level tunneling model for amorphous materials even at these very long times. No heat release was observed for Teflon, graphite, and Al₂O₃.     

On a stability theory of non-selfadjoint mechanical systems

Summary The concept of the Hamiltonian functional is generalized in such a way that a bilinear functional results, which plays the role of the Hamiltonian for non-selfadjoint systems. For this generalized Hamiltonian the condition leads to the so called hybrid Galerkin's equations, and the condition , to the load-frequency reationship. This relationship can be interpreted as a surface in the load-frequency space, the projection of which on the load-planes yields the stability boundaries, i.e. the buckling loads.

---

Zu einer Stabilitätstheorie nicht-selbstadjungierter mechanischer Systeme

Zusammenfassung Der Begriff des Hamiltonschen Funktionals wird in solcher Weise verallgemeinert, daß ein bilineares Funktional bei nicht-selbstadjungierten Systemen an seine Stelle tritt. Für dieses verallgemeinerte Hamiltonsche Funktional führt die Bedingung auf die sogenannten hybriden Galerkinschen Gleichungen und die Bedingung auf die Last-Frequenz-Funktion. Diese Funktion kann im Last-Frequenz-Raum als eine Fläche aufgefaßt werden, deren Projektion auf die Last-Ebenen die Stabilitätsgrenzen und damit die Knicklasten liefert.

---

---

Dedicated to Professor Kurt Magnus in honor of his sixtieth birthday.     

Load relaxation studies of germanium

A series of compressive load relaxation experiments were conducted on germanium single crystals in the temperature range 400 to 885° C. The curvature of the log-log data obtained from load relaxation tests changes from concave upward to concave downward as the test temperature increases at fixed stress level, or as the strain level increases at fixed temperature. At intermediate temperatures, 600° C, the transition from concave upward to concave downward curvature happens on a single relaxation curve. These observations are consistent with the two-branch rheological model proposed by Hart to explain the deformation behaviour of metals and were analysed in terms of this model. The transition from concave upward to concave downward curvature could be moved to higher temperature by doping germanium with gallium, which decreases the dislocation glide velocity relative to that in pure germanium. The transition could be shifted to lower temperature by compressing samples along [1 1] rather than [1 0] because the [1 1] orientation favours cross-slip while the [1 0] orientation does not. Dislocation dipoles and straight dislocations dominated the microstructure of samples which had concave upward log-log curves, while well-developed dislocation cell structures dominated the microstructure of samples which yielded concave downward curves. The observed changes in the curvature of the load relaxation curves and the dislocation structure both indicate the increased importance of dislocation climb with increasing temperature. When compared through the Orowan equation, the load relaxation results are in good agreement with published stress-dislocation velocity data.     

A note on the turbulent energy as a parameter of turbulent flow

The art of modeling turbulence is a needed tool in the construction of computer codes for turbulent flows. The state to which this art has been developed is inadequate, and quotations from authoritative sources support this point of view. The energy contained in the turbulent fluctuations, i.e., the turbulent energy, is often used as a parameter in the modeling process. The present article attempts to examine this quantity as it is being created, transported, and dissipated. For this purpose experimental evidence from the author's own experiments (free jets), as well as theoretical conclusions from the elementary deductions of the basic equations, the concept of turbulent potential flow, and a general solution to the Navier-Stokes-Reynolds equations, is drawn to attention. Recirculating flow is given special attention. The paper concludes with recommendations for principles that must be satisfied if improved modeling is to be achieved. These principles are necessary; whether they are also sufficient is open to question.Nomenclature A ₀ Constant - b _1/2 Jet's half-width - b ₁ ^2/(0) Jet's half-width at z=z(in0) - E _z Kinetic energy contained in the jet's axial velocity at a given profile - E _r Kinetic energy contained in the jet's radial velocity at a given profile - f() Dimensionless velocity profile [f(0)=1] - F(), H() Defined functions - L _char Jet's characteristic length - m, n Exponents - p Pressure - q Kinetic energy in the turbulent fluctuations - Heat flux - q ² - r, , z Cylindrical coordinates - t Time - û Internal energy - u, v, w Velocity components - Mean velocity components - Mean velocity components - U ₀ Constant - U _plate Plate's velocity - Uskc/(0) Centerline velocity at z=z₀ - X, Y, Z Components of body force - W Total work done by surface stresses - W ₁ Recoverable work done by surface stresses - W ₂ Dissipated work - z ₀ Downstream distance from the nozzle beyond which self-similar velocity profiles occur - Fluid's kinematic viscosity - Fluid's density - Normal stresses - Shear stresses - Normal stresses with the pressure removed - Dimensionless Crossflow coordinate - ₀ Constant - Stress functions - Stress potential Paper dedicated to Professor Joseph Kestin.Definitions of symbols are given under Nomenclature.     

Computer-aided design of small patch antenna using (Al, Mg)TaO2 ceramics

The microwave dielectric properties of (AMT) ceramics and the design of small coplanar waveguide fed antenna (CPWFA) have been investigated. ( and ( have orthorhombic and tetragonal structure, respectively. As ( concentration increased, AMT ceramics transformed into the tetragonal structure. Specimens having tetragonal single phase could be obtained above x=0.6. As ( concentration increased, the grain size, dielectric constant and quality factor (Q) significantly increased and the temperature coefficient of resonant frequency changed from negative to positive. The of was realized at x=0.65 and the Q · f _O value and for this composition were 112 470 GHz and 26.1, respectively. Newly developed dielectric materials were used for 1.5 GHz band CPWFA design and fabrication. The size of the CPWFA can be reduced by using high dielectric constant AMT ceramics, insetting slits into the patch, and fabricating CPW feed line in the ground plane. The slits play a role in not only lowering a center frequency but also fine tuning for the proposed antenna together with the open stub of CPW feed line. The CPWFA with slits has a lower center frequency than the conventional CPWFA, which suggests that the antenna size can be reduced by as much as 16.3%. The structure simulations of the CPWFAs have been performed to obtain impedance matching and to investigate the effects of slits. Experimental results of the fabricated device were in good agreement with the simulation.     

Free convection from a heated vertical cylinder embedded in a fluid-saturated porous medium

Summary We consider the free convection boundary layer flow induced by a heated vertical cylinder which is embedded in a fluid-saturated porous medium. The surface of the cylinder is maintained at a temperature whose value above the ambient temperature of the surrounding fluid varies as then ^th power of the distance from the leading edge. Asymptotic analyses and numerical calculations are presented for the governing nonsimilar boundary layer equations and it is shown that, whenn<1, the asymptotic flowfield far from the leading edge of the cylinder takes on a multiple-layer structure. However, forn>1, only a simple single layer is present far downstream, but a multiple layer structure exists close to the cylinder leading edge. We have shown that the fully numerical and asymptotic calculations are in stisfactory agreement, especially for exponentsn close to zero. Comparisons of the present numerical solutions obtained using the Keller-box method with previous numerical solutions using local methods are also given.List of symbols a radius - scaled streamfunctions - f ₀,f ₁,f ₂ inner zone streamfunctions whenn<1 - leading order streamfunctions inn>1, 1 asymptotic solution - F ₀,F ₁ outer zone streamfunctions whenn<1 - G large parameter satisfyingG=X ² lnG - g gravitational acceleration - K permeability of the porous medium - n exponent in prescribed temperature law - r radial co-ordinate - r rescaled radial co-ordinate - R Darcy-Rayleigh number - T temperature of convective fluid - T _w temperature of cylinder at leading edge - T ambient temperature of fluid - u velocity in axial direction - v velocity in azimuthal direction - w velocity in radial direction - x axial co-ordinate - x escaled axial co-ordinate - X dimensionless axial co-ordinate - thermal diffusivity of the saturated medium - coefficient of thermal expansion - constant in the boundary conditions forF ₀ - dimensionless radial co-ordinate - co-ordinate for the outer zone in then<1 solution - scaled radial co-ordinates - scaled fluid temperature - similarity variable for then=1 problem - nondimensionalisation constant (Eq. (9)) - viscosity of fluid - scaled axial co-ordinates - density of fluid - co-ordinate for the inner zone in then<1 solution - azimuthal co-ordinate - similarity variables for then>1 problem - streamfunction     

Dynamic response of a beam on elastic foundation of finite depth under a moving force

Summary In this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.List of symbols b width of the beam - C _b coefficient of viscous damping for the beam - C _f coefficient of viscous damping for the foundation - E Young's modulus - f frequency - H foundation depth - I moment of inertia - i =(–1)^0,5 - K, k indexing variables - k _f foundation modulus - m mass per unit length of the beam - N total number of frequency points in Eqs. (25) and (26) - n indexing variable - P moving force - Q axial force on the beam - q(x, t) foundation pressure per unit length of the beam - q() foundation pressure in the moving co-ordinate system - t time variable in sec. - U _j () generalized coordinate in Eq. (4) - U _j ^*(f) Fourier transform ofU _j - u(y, t; x) axial displacement in the foundation at a particularx value - u(y, ) foundation displacement in the moving coordinate system - , nondimensionalized foundation deflection - v velocity in meters/sec. - v _cr critical velocity corresponding to massless foundation - w(x, t) beam deflection - w() beam deflection in the moving coordinate system - =w()/L nondimensionalized beam deflection - Fourier transform of - x, y coordinate axis - velocity parameter - _cr critical velocity parameter - mass parameter - moving coordinate - _b beam damping parameter - _f foundation damping parameter - (y, t, x) vertical stress in the foundation - () Dirac delta function - foundation mass per unit depth per unit length of the beam