Supersonic flow past axisymmetric flared cones

Systematic data on the determination of the aerodynamic characteristics of axisymmetric bodies with a break in the generating line (Fig. 1a, b) in supersonic flow at zero angle of attack are presented in [1, 2, and others]. A characteristic feature of the flow past such bodies is the appearance of an extensive separation zone dec in the region of the break in the generator when the break angle exceeds some minimum value _min, which for a turbulent boundary layer depends basically on the Mach number M at the body surface ahead of the separation zone. In this case, compression waves which change into the oblique compression shocks dc and cc, emanate both from the beginning of the separation zone (point c) and from the end of it (point d). These shocks, intersecting at the point c, form the triple shock configuration acd and acc for which we introduce the notationac[c, d]. The maximum value (_max) of the generator break angle is limited by the possibility of the existence of an attached compression shock, dc. According to these data a change in the generator break angle for the range _minmax of the angle does not disrupt the nature of the flow in the separation zone, but only alters the size of this zone.We shall examine the flow past cones with values of the generator break angles (_max) for which the attached shock dc cannot exist.     

Strömung und Wärmeübergang bei der Oberflächenverdampfung und Filmkondensation

Zusammenfassung Mit Hilfe der Mischungswegtheorie wurden Gleichungen zur Berechnung der Geschwindigkeitsprofile und des Druckabfalles bei der turbulenten, abwärtsterichteten Gas/Film-Strömung aufgestellt. Zur Berechnung des Wärmeübergangs wurde die turbulente Temperaturleitfähigkeit aus einem halbempirischen Ansatz bestimmt. Es konnte eine befriedigende Übereinstimmung zwischen den berechneten und gemessenen Nußelt-Zahlen bei der Oberflächenverdampfung erzielt werden. Zur Auslegung von Fallstromverdampfern wurde ein Computerprogramm erstellt. Damit lassen sich Einflußgrößen wie Wandtemperatur, Filmdicke, Verdampfungsrate usw. in Abhängigkeit von der Lauflänge bestimmen.

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Flow and heat transfer in surface evaporation and film condensation

Using the mixing length model, equations were established to calculate the velocity profiles and pressure drop in turbulent downward directed gas/film flow. The thermal diffusivity needed for the calculation of heat transfer was determined from a semiempirical model. The calculated Nußelt-numbers agreed very well with experiments. For the design of falling-film evaporators, a computer program was developed, which enables to evaluate wall temperature, film thickness, evaporation rate etc. as a function of flow-path length.

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Formelzeichen a Temperaturleitfähigkeit - c spez. Wärmekapazität - d Durchmesser - f_m bezogene mittlere turbulente Temperaturleitfähigkeit - Fi /(3²/g)^1/3) Filmkennzahl - Fr Froude-Zahl - g Fallbeschleunigung - Ka ³/g⁴ Kapitza-Zahl - L Rohrlänge - l Mischungsweg - m Massenstrom - Nu (²/g)^1/3/ Nußelt-Zahl - Nu / Nußelt-Zahl des Filmes - p Druck - Pr /a Prandtl-Zahl - q Wärmestromdichte - R Radius - Re Reynolds-Zahl - Re_ü Übergangs-Reynolds-Zahl - Re_w Schubspannungs-Reynolds-Zahl der Flüssigkeit - r radiale Koordinate - T Temperatur - u Geschwindigkeit - u_w Schubspannungsgeschwindigkeit der Flüssigkeit - u Grenzflächengeschwindigkeit - u_T Schubspannungsgeschwindigkeit des Gases - y Wandabstand - y^* y/ dimensionsloser Wandabstand - z axiale Koordinate Griechische Zeichen Wärmeübergangskoeffizient - Filmdicke - dyn. Viskosität - dimensionslose Temperatur - Wärmeleitfähigkeit - kin. Viskosität - Dichte - Oberflächenspannung - Schubspannung Zusatzzeichen und Indizes G Gas - K Kondensation - s Sättigung - t turbulent - w Wand - wi Welleninstabilität - Phasengrenze - - mittlere Größe     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Die in Teil I vorgestellten Reynolds 'schen Gleichungen und Transportgleichungen werden für Strömungen mit Grenzschichtcharakter angegeben. Weiter werden Integralbedingungen mitgeteilt. Nach einer Diskussion über die Schließung des Gleichungssystems werden Lösungsverfahren besprochen. Dabei wird speziell auf Integralverfahren eingegangen.

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About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart II: Thin shear flow layers

The Reynolds equations and transport equations given in part I are presented for thin shear flow layers. Integral relations are given. After a discussion of the closure problem methods of solution are described. Specially integral methods are discussed.

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Formelzeichen c Massenkonzentration der Komponente - c_t charakteristische Konzentrationsschwankung - c_o Bezugskonzentration - c spezifische Wärme bei konstantem Druck - c_f Reibungsbeiwert - c_D Dissipationsintegral - c_E Entrainment-Funktion - c Schubspannungsintegral - D binsrer Diffusionskoeffizient - H Formparameter - H₁₂ Formparameter - H₃₂ Formparameter - j Kassendiffusionsstrom - L Bezugslänge - p Druck - p_t charakteristische Druckschwankung - p_o Bezugsdruck - Pr Prandtl-Zahl - q Wärmestrom - q²/2 kinetische Energie der Schwankungsbewegung - Re_L mit L gebildete Reynolds-Zahl - Re mit gebildete Reynolds-Zahl - Re₂ mit ₂ gebildete Reynolds-Zahl - Sc Schmidt-Zahl - T absolute Temperatur - T_t charakteristische TemperaturSchwankung - T_o Bezugstemperatur - u,v,w Geschwindigkeitskomponenten - u_t charakteristische Geschwindigkeitsschwankung - u_o Bezugsgeschwindigkeit - U=/ü dimensionslose. x-Komponente der Geschwindigkeit - x,y,z Komponenten des Ortsvektors Griechische Symbole Grenzschichtdicke - ₁ Verdrängungsdicke - ₂ Impulsverlustdicke - ₃ Energieverlustdicke - _T Enthalpieverlustdicke - _c Konzentrationsverlustdicke - =d/dx Parameter für die Grenzschichtabsch:atzung - turbulente Impulsaustauschgröße - _D turbulente Stoffaustauschgröße - _q turbulente Energieaustauschgröße - Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - v=/ kinematische Viskosität - Dichte - Produktionsdichte - Schubspannung Indizes mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil - Außenrand der Grenzschicht - w Wand     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

The motion of a detaching elastic body

Conclusions The qualitative behavior of the displacement (t) and the radius R(t) during the different phases of the motion is illustrated in the diagram of Fig. 6.1.After the first impact at t = 0 the displacement (t) varies according to (5.2). If the first maximum of (t) is higher than ₁ then at time t ₁ the graph of (t) intersects the straight line = _cand detachment first occurs. In the second phase the dependance of on t is expressed by (5.6). The detachment will end at the instant t ₂ when vanishes.The radius R remains equal to R ₀ until (t) reaches the critical value ₁ = _c that is at t = t ₁. After t ₁, R(t) will decrease according to (4.4) up to its final value ₂.A rather unexpected property of the solution is that the greatest elongation is finite for every non-vanishing value of the ratio .To Jerry Ericksen for his 60^th birthday     

Hydrodynamic instability of the contact zone between accelerated gases

An experimental apparatus for investigating Rayleigh-Taylor instability in the transition layer between two gases at accelerations g 10⁵g₀ (g₀ is the acceleration of gravity) is described. The constantly acting acceleration is communicated to the contact zone by the compression wave formed ahead of a flame front. The linear stage of development is investigated together with the effect of the thickness of the contact zone. It is shown that on the interval 0.3 < <- ( is the wavelength of the disturbance at the edge of the contact zone) the rate of growth of the perturbation amplitude 0.5₀, where ₀ is the amplitude growth rate for media separated by an interface with a discontinuous change of density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–21, November–December, 1991.     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

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Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

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Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Control of low-speed turbulent separated flow using jet vortex generators

A parametric study has been performed with jet vortex generators to determine their effectiveness in controlling flow separation associated with low-speed turbulent flow over a two-dimensional rearward-facing ramp. Results indicate that flow-separation control can be accomplished, with the level of control achieved being a function of jet speed, jet orientation (with respect to the free-stream direction), and jet location (distance from the separation region in the free-stream direction). Compared to slot blowing, jet vortex generators can provide an equivalent level of flow control over a larger spanwise region (for constant jet flow area and speed).Nomenclature C _p pressure coefficient, 2(P-P_)/V ² - C _Q total flow coefficient, Q/ v - D ₀ jet orifice diameter - Q total volumetric flow rate - R Reynolds number based on momentum thickness - u fluctuating velocity component in the free-stream (x) direction - V free-stream flow speed - VR ratio of jet speed to free-stream flow speed - x coordinate along the wall in the free-stream direction - jet inclination angle (angle between the jet axis and the wall) - jet azimuthal angle (angle between the jet axis and the free-stream direction in a horizontal plane) - boundary-layer thickness - momentum thickness - lateral distance between jet orifices A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 Sept. 1990     

Some characteristic features of diffusion of a magnetic field into a moving conductor

The study of the diffusion of a magnetic field into a moving conductor is of interest in connection with the production of ultra-high-strength magnetic fields by rapid compression of conducting shells [1,2]. In [3,4] it is shown that when a magnetic field in a plane slit is compressed at constant velocity, the entire flux enters the conductor. In the present paper we formulate a general result concerning the conservation of the sum current in the cavity and conductor for arbitrary motion of the latter. We also consider a special case of conductor motion when the flux in the cavity remains constant despite the finite conductivity of the material bounding the magnetic field.Notation ₁, * flux which has diffused into the conductor - ₂ flux in the cavity - ₀ sum flux - r radius - r* cavity boundary - thickness of the skin layer - (r) delta function of r - t time - q intensity of the fluid sink - v velocity - flux which has diffused to a depth larger than r - x self-similar variable - dimensionless fraction of the flux which has diffused to a depth larger than r - * fraction of the flux which has diffused into the conductor - a conductivity - c electrodynamic constant - R_m magnetic Reynolds number - dimensionless parameter     

Approximate method for calculating turbulent boundary layer with positive pressure gradient

A single-parameter integral method is proposed for calculating the turbulent boundary layer with positive pressure gradient which makes it possible to calculate the friction, thermal flux, and layer thickness both ahead of the separation point and in some region behind the separation point.Notation u velocity - density - ^* displacement thickness - ^** momentum thickness - energy thickness - M Mach number - r radius - dynamic viscosity - c_p specific heat at constant pressure - Reynolds number based on initial boundary layer thickness - P Prandtl number - p₁ static pressure at point of initial interaction - p₂ static pressure at pressureplateau - p₀ stagnation pressure - T₀ stagnation temperature - I enthalpy - T_e recovery temperature - T_w ⁰ temperature factor - H form parameter - r₁ recovery coefficient Indices 0 denotes initial section of boundary layer - 1 parameters taken at edge of boundary layer - w parameters taken at the wall temperature - * parameters referred to flow on a flat plate with =0     

Schwere symmetrische Kreisel kleiner Nutationen

Zusammenfassung Zur Integration der Eulerschen Bewegungsgleichungen schwerer symmetrischer Kreisel werden der Winkel (t) (Abb. 1) durch (t)=₀+(t) ersetzt und in sämtlichen Reihenentwicklungen von abhängiger Funktionen die Potenzen höheren als zweiten Grades vernachlässigt. Dadurch ist es möglich, die Eulerschen Winkel (t), (t) und (t) durch elementare Formeln zu beschreiben und somit sind die wesentlichsten Erscheinungen im Bewegungsablauf der schweren symmetrischen Kreisel einfach zu übersehen.     

The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik      

Stability criterion for steady-state burning of powders

A model of a powder with a variable surface temperature is considered. A stability criterion for the steady-state burning of powders at constant pressure is found on the assumption that all the processes in the gas phase and the reaction layer of the condensed phase are inertialess. It is shown that the stability region is determined by two parameters: K=(T₁ – t₀) ( ln m/T₀)p and r=(T₁/T₀)p (here T₁ is the surface temperature, T₁ is the initial temperature of the powder, m is the mass burning rate, p is pressure). Burning is always stable if < 1. If > 1, the burning process is stable only when r > ( – 1)²/ / + 1.The author expresses his appreciation to A. S. Kompaneits, O. I. Leipunskii, A. G. Istratov, V. B. Librovich, and S. S. Novikov for discussing his work.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

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The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Untersuchung des lokalen Stoffüberganges am laminaren und turbulenten Rieselfilm

Zusammenfassung Der lokale Stoffübergang wurde in Abhängigkeit von der Meßlänge, dem Startort und der Zulaufhöhe gemessen. Der Gültigkeitsbereich der Theorie von Nusselt wird ermittelt. Die Reynolds-Zahl nahm Werte zwischen 3,86 und 2496 an. Die örtlich wirkende Hydrodynamik ist entscheidend für das Anwachsen der örtlichen Sherwood-Zahl. Die Genauigkeit aller Versuchsergebnisse kann auf ± 5% abgeschätzt werden.

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Investigation of the local mass transfer of a laminar and turbulent falling liquid film

The local mass transfer was measured as a function of the measuring length, the starting point and the liquid height above the ring-slot. The range of the Reynolds number was 3,86 Re 2496. The validity of the Nusselt theory and the range of it is shown. The local hydrodynamic is the most important factor of the increase of the local Sherwood number. The accuracy of the measurements is ± 5%.

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Bezeichnungen a Temperaturleitfähigkeit m²/s=/(c_p) - c Konzentration, c=¯c + c kmol/m³ - c_i0 Konzentration im Flüssigkeitskern kmol/m³ - D Diffusionskoeffizient m²/s - EL-NR Elektrodennummer - Fa Faraday-Konstante A s/kgäq=96,5·10⁶ - g Erdbeschleunigung m/s² - i_G Grenzstromdichte A/m² - u Geschwindigkeit in x-Richtung, u= + u - U Umfang des Rohres m - v Geschwindigkeit in y-Rich- m/stung, v=¯v + v - V^* Volumenstrom m³/s - x Lauflänge, Koordinate in m Strömungsrichtung - x_M Meßlänge für den Stoff-Übergang m - x_ST Startort für den Stoff-Übergang m - y Wegkoordinate senkrecht zur Rohroberfläche m - z Wertigkeit der Elektro-denreaktion kgäq/kmol - ZH Zulaufhöhe m - Wärmeübergangskoeffizient W/m²C - Stoffübergangskoeffizient m/s - Filmdicke m - Wärmeleitfähigkeit W/(mC) - kinematische Viskosität m²/s - Re=u/=V^*/U Reynolds-Zahl - Pr=/a=c_p/ Prandtl-Zahl - Sc=/D Schmidt-Zahl - Nu= / Nusselt-Zahl - Sh= /D Sherwood-Zahl - SHL lokale Sherwood-Zahl - SHM mittlere Sherwood-Zahl - - zeitlich gemittelt - örtlich gemittelt Die Durchführung der Arbeit am Institut für Verfahrens — und Kältetechnik der ETH Zürich bei Prof. Dr. P. Grassmann wurde ermöglicht durch Zuschüsse der Kommission zur Förderung der wissenschaftlichen Forschung und meiner Eltern.