Der Spannungs- und Verschiebungszustand drehsymmetrischer Membrane mit beliebig gekrümmter Meridiankurve

Zusammenfassung Die Einführung von Zylinderkoordinaten (x, r, ) in die Gleichgewichtsbedingungen der Schnittkräfte bzw. in die Beziehungen zwischen Verzerrung und Verschiebungen am differentialen Schalenabschnitt ermöglicht die Berechnung des Spannungs- und Verschiebungszustandes von drehsymmetrischen Membranen mit beliebig gekrümmter Meridiankurve auf die Integration einer einfachen, linearen partiellen Differentialgleichung zweiter Ordnung für eine charakteristische FunktionF bzw. zurückzuführen. Eine geschlossene Lösung und damit eine Darstellung der Schnittkräfte und Verschiebungen durch explizite Formeln ist bei harmonischer Belastung cosn für zwei Funktionsgruppen=x ² und=x ^–3 möglich. Im Sonderfall der drehsymmetrischen und der antimetrischen Belastung mitn=0 undn=1 gelten die Gleichungen der Schnitt- und Verschiebungsgrößen für eine beliebige Meridianfunktion=(). Die Betrachtungen der Randbedingungen offener Schalen bei harmonischer Belastung geben über die infinitesimalen Deformationen einer drehsymmetrischen Membran mit überall negativer Krümmung Aufschluß.     

Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection)

An analysis is carried out to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. The localized heating or cooling introduces a finite discontinuity in the mathematical formulation of the problem and increases its complexity. In order to overcome this difficulty, a non-uniform distribution of wall temperature is taken at finite sections of the plate. The nonlinear coupled parabolic partial differential equations governing the flow have been solved by using an implicit finite-difference scheme. The effect of the localized heating or cooling is found to be very significant on the heat transfer, but its effect on the skin friction is comparatively small. The buoyancy, magnetic and suction parameters increase the skin friction and heat transfer. The positive buoyancy force (beyond a certain value) causes an overshoot in the velocity profiles.A mass transfer constant - B magnetic field - C_fx skin friction coefficient in the x-direction - C_p specific heat at constant pressure, kJ.kg^–1.K - C_v specific heat at constant volume, kJ.kg^–1.K^–1 - E electric field - g acceleration due to gravity, 9.81 m.s^–2 - Gr Grashof number - h heat transfer coefficient, W.m².K^–1 - Ha Hartmann number - k thermal conductivity, W.m^–1.K - L characteristic length, m - M magnetic parameter - Nu_x local Nusselt number - p pressure, Pa, N.m^–2 - Pr Prandtl number - q heat flux, W.m^–2 - Re Reynolds number - Re_m magnetic Reynolds number - T temperature, K - T_o constant plate temperature, K - u,v velocity components, m.s^–1 - V characteristic velocity, m.s^–1 - x,y Cartesian coordinates - thermal diffusivity, m².s^–1 - coefficient of thermal expansion, K^–1 - , transformed similarity variables - dynamic viscosity, kg.m^–1.s^–1 - ₀ magnetic permeability - kinematic viscosity, m².s^–1 - density, kg.m^–3 - buoyancy parameter - electrical conductivity - stream function, m².s^–1 - dimensionless constant - dimensionless temperature, K - w, conditions at the wall and at infinity     

Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

This paper presents a study on the deformation of anisotropic fibrous porous media subjected to moistening by water in the liquid phase. The deformation of the medium is studied by applying the concept of effective stress. Given the structure of the medium, the displacement of the solid matrix is not taken into account with respect to the displacement of the liquid phase. The transport equations are derived from the model proposed by Narasimhan. The transport coefficients and the relation between the variation in apparent density and effective stress are obtained by test measurements. A numerical model has been established and applied for studying drip moistening of mineral wool samples capable or incapable of deformation.Nomenclature D mass diffusion coefficient [L²t^–1] - e void fraction - g gravity acceleration [Lt^–2] - J mass transfer density [ML^–2t^–1] - K hydraulic conductivity [Lt^–1] - K _s hydraulic conductivity of the solid phase [Lt^–1] - K ^* hydraulic conductivity of the deformable porous medium [Lt^–1] - P pressure of moistening liquid [ML^–1 t^–2] - S degree of saturation - t time [t] - V speed [Lt^–1] - X horizontal coordinate [L] - Z vertical coordinate measured from the bottom of porous medium [L] - z z-coordinate [L] Greek Letters porosity - ₁ total hydric potential [L] - _g gas density [ML^–3] - ₁ liquid density [ML^–3] - ₀ apparent density [ML^–3] - _s density of the solid phase [ML^–3] - density of the moist porous medium [ML^–3] - external load [ML^–1t^–2] - effective stress [ML^–1t^–2] - bishop's parameter - matrix potential or capillary suction [L] Indices g gas - 1 moistening liquid - p direction perpendicular to fiber planes - s solid matrix - t direction parallel to fiber planes - v pore Exponent * movement of solid particles taken into account     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Viscous properties of soil

Dynamic problems connected with the wave propagation in soils not saturated with water and with wave interaction with obstacles and structural elements at the present time are solved on the basis of models in which plastic but not viscous soil properties are taken into account [1–5]. An analysis of experimental data and their comparison with the calculated results [4, 5] confirms that it is permissible to apply the model of an elasticplastic medium to soils in problems concerning the interaction of waves and structures. At the same time plane-wave damping in soils takes place more intensively than would follow from calculations carried out on the basis of models of an elastic-plastic medium. For example, if in a section of a poured sandy soil, taken as the initial section, the maximum stress in the wave is _m=ll kgf/cm² and its duration is 6=8 msec, then at a distance of 25 cm the calculations give _m=9.5 kgf/cm², while the experiment gives _m= 5 kgf/cm². If in the initial section _m= 20 kgf/cm² and =6 msec, then at a distance of 35 cm the calculation gives _m= l7 kgf/cm², while the experiment gives _m= 9 kgf/cm². In the calculations it was assumed that unloading takes place with a constant strain. This deviation of the calculated results from the experiment can be explained, in the first place, by the dependence of the () on the strain rate , which is not taken into account in the model of an elastic-plastic medium. The viscous properties cause additional energy losses and a more intensive damping of the waves. Experimentally the dependence of the () curves on the strain rate has been investigated for many soils [5–8]. The dynamic load on the test sample was produced by a body falling from a height or being accelerated by some method. Below we present test results of viscous soil properties when the test sample is compressed by an air shock wave. Compression curves and approximate numerical values of the coefficient of viscosity are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 68–71, July–August, 1968.The author thanks A. I. Shishikin for his participation in the experiments.     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.     

Plastic deformation of metals subjected to intensive sources

The elastoplastic strain of metals being formed when they melt under the effect of a point heat source with a pulse duration greater than 10^–6 sec is considered in this paper. The time development of the plastic strain and pressure domains in the melt is investigated. It is shown that two plastic strain domains occur during the interaction under consideration: a relatively broad domain of mechanical influence and a narrow domain of thermal influence. The stress-strain distributions as well as the hydrostatic pressure in the fluid are determined by a quasistationary temperature distribution starting with times corresponding to half (of the quasistationary) the value of the melt radius X 0.5. It is shown that the dimensions of the weak and strong plastic strain domains formed by heat and acoustic waves grow continuously to the quasistationary values, while the hydrostatic pressure in the fluid reaches the maximum value for X 0.3...0.4. The ratio between the radii of the plastic strain zones and of the liquid bath for a quasistationary temperature distribution in the first domain lies within the range 10–50, and does not exceed 1.7 for Cu, Ni, and Fe in the second. The anomalous nature of the development of the strong plastic strain domain in Al, because of migration of the metal grain boundaries to result in collapse of the domain for the values X 0.5 accompanied by a jumplike diminution in the hydrostatic pressure in the fluid, is noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–140, May–June, 1976.     

Effect of the grain size on the wavelength of localized strain in aluminum specimens in tension

This paper studies the dependence of the wavelength of localized plastic strain at the parabolic stage of strain hardening on the grain size in polycrystal aluminum. This dependence is determined in the grainsize range 10^–2 – 10 mm. The effect of the grain size on the character of the plasticflow curve is studied.     

Nicht-isotherme Rieselfilmströmung einer hochviskosen Flüssigkeit mit stark temperaturabhÄngiger ViskositÄt

Zusammenfassung Es werden Geschwindigkeitsverteilungen und Filmdickenabnahmen von nichtisothermen NEWTONschen und nicht-NEWTONschen (Potenzansatz) Rieselfilmen mit temperaturanhÄngiger ViskositÄt berechnet, wobei die Temperaturverteilung im Film als linear vorausgesetzt wird. An dicken Rieselfilmen mit Re=10^–4... 10^–2 sind Geschwindigkeitsprofile, Filmdicken und OberflÄchentemperaturen gemessen und daraus die thermische EinlauflÄnge bestimmt worden. Ausgehend von der Penetrationstheorie für eine endlich dicke Platte kann man für diese EinlauflÄnge eine Approximationsformel erhalten, die für Strömungen mit Re < 1000 verwendet werden kann.

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Non-isothermal filmflow of a highly viscous liquid, the viscosity strongly depending on temperature

Velocity distributions and film thicknesses of nonisothermal NEWTONIAN and non-NEWTONIAN (power-law) falling films are computed assuming that the temperature across the film varies linearly. Experimental studies on thick falling films of Re=10^–4...10^–2 had been carried out to measure velocities, film thickness and surface temperature and to calculate the thermal entrance length. One can get for this entrance length a approximation formula which is valid for flows with RePr <1000 by applying the results for the thermal penetration into a material finite plate.

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Bezeichnungen B dimensionsloser Temperaturkoeffizient - ¯B [K] Temperaturkoeffizient (ln)/(1/T) - c_p [J/kgK] spezif. WÄrme bei konst. Druck - Fo FOURIER-Zahl - g [m/s²] Erdbeschleunigung - H dimensionslose Filmdicke - h [m] Filmdicke - m [Pas^2–n] ViskositÄtskoeffizient im Potenzansatz von OSTWALD-DE WAELE - Nu NUSSELT-Zahl - n Flüssigkeitsexponent im Potenzansatz von OSTWALD-DE WAELE - Pr PRANDTL-Zahl (Gl.3.5) - q [W/m²] WÄrmestromdichte - Re REYNOLDS-Zahl (Gl.3.4) - T [K] Temperatur - t [s] Zeit - U dimensionslose Geschwindigkeit (X-Komponente) - u [m/s] Geschwindigkeitskomponente in x-Richtung - X dimensionslose Koordinate (X=x/h₀) - x [m] LÄnge, Koordinate - Y dimensionslose Koordinate (Y=y/h₀) - y [m] Höhe, Koordinate - [W/m²K] WÄrmeübergangskoeffizient - Plattenneigungswinkel gegen Horizontale - [s^–1] Schergeschwindigkeit - dimensionslose Temperatur (Gl.3.3) - [m²/s] TemperaturleitfÄhigkeit (Gl.3.3) - [W/mK] WÄrmeleitfÄhigkeit - [Pas] ViskositÄt - [kg/m³] spezif. Dichte - [Pa] Schubspannung Indizes a scheinbar (apparent) - 0 bei x=0, auch: isotherm - P auf die Penetrationszeit bezogen - s an der OberflÄche - T bei linearer Temperaturdifferenz T - w an der Wand - 99 auf =0,99 bezogen - gemittelt, Mittelwert - thermisch ausgebildet, bei x - proportional - ¯t ungefÄhr - kleiner oder gleich ungefÄhr     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

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Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

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Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

The wedge subjected to tractions: a paradox resolved

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2 of the wedge satisfies the equation tan 2 = 2. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^–) as r0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^–), O(r ^– ln r), or O(r ^– ln² r) as r0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r0 for the half-plane or crack geometry. At the critical opening angle 2, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2.     

Wall pressure fluctuations of a turbulent separated and reattaching flow affected by an unsteady wake

The spatio-temporal characteristics of the wall-pressure fluctuations in separated and reattaching flows over a backward-facing step were investigated through pressure-velocity joint measurements carried out using multiple-arrayed microphones and split-film probes. A spoke-wheel-type wake generator was installed upstream of the backward-facing step. The flow structure at the effective forcing frequency (St _f=0.2) was found to be well organized in terms of wall pressure spectrum, cross-correlation, wavenumber-frequency spectrum, and wavelet auto-correlation. Introduction of the unsteady wake (St _f=0.2) reduced the reattachment length by 10%. In addition, the unsteady wake enhanced the turbulence intensity near the separation edge and, as a consequence, enhanced the quadrupole sound sources; however, the turbulence intensity near the reattachment region was weakened and the overall flow noise was attenuated. The greater organization of the flow structure induced by the unsteady wake led to a weakening of the dipole sound sources, which are the dominant sound sources in this system. The dipole sound sources generated by wall pressure fluctuations were calculated using Curles integral formula.Abbreviations AR Aspect ratio - SBF Spatial box filtering Roman symbols C _p Wall pressure fluctuation coefficient, p/0.5U ² - H Step height of backward-facing step (mm) - H _s Shape factor (H _s = ^*/) - R _s Distance from acoustic source point to observation point (m) - Re _H Reynolds number, U H/ - St The reduced frequency, fH/U - St _f Normalized forcing frequency by unsteady wake, f _p H/U - T Vortex shedding period (s) - U Free-stream velocity (m/s) - a Speed of sound (m/s) - f Frequency (Hz) - f _p Wake passing frequency (Hz) - k Turbulent kinetic energy (m²/s²) - k _x Streamwise wave number (1/m) - k _z Spanwise wave number (1/m) - l _j Cosine of angle - p Instantaneous wall pressure (Pa) - p _rms Root-mean-square of wall pressure (Pa) - p _SBF Spatial box filtered wall pressure (Pa) - p _d Dipole sound source (Pa) - p _w Conditionally-averaged wall pressure (Pa) - q Dynamic pressure, 0.5U ² (Pa) - r Distance from origin to observation point (mm) - u _c Convection velocity (m/s) - u_max Root-mean-square of streamwise velocity (m/s) - x _R Time-mean reattachment length (mm) Greek symbols _p Forward-flow time fraction - Auto-correlation of pressure at x ₀ - Two-dimensional cross-correlation of pressure with streamwise separation interval , spanwise separation interval , and time delay , at (x ₀, z ₀) - Boundary layer thickness (mm, _99%) - ^* Displacement thickness (mm, ) - _ij Kroneckers delta function - Phase angle (°) - Wavelength (mm) - Momentum thickness (mm, ) - Angle between vertical axis and observation point (°) - Density (kg/m³) - Time delay (s) - Streamwise separation interval (m) - Spanwise separation interval (m) - _p (f; x ₀) Autospectrum of pressure measured at x ₀ (Pa² s) - _pp (, ; x ₀) Streamwise cross spectrum of pressure at x ₀ (Pa² s) - _pp (, , ; x ₀, z ₀) Streamwise and spanwise cross spectrum of pressure at (x ₀, z ₀) (Pa² s) - _pp (kx, ; x ₀) Streamwise wavenumber-frequency spectrum of pressure at x ₀ (Pa² s) - _pp (kx, kz, ; x ₀, z ₀) Two-dimensional wavenumber-frequency spectrum of pressure at (x ₀, z ₀) (Pa² s)     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

StationÄrer und instationÄrer Vorgang der Filmkondensation auf der ebenen Platte,dem senkrechten Kegel,dem horizontalen Rohr und der Kugel

Zusammenfassung Die vorliegende Arbeit untersucht die Filmkondensation auf verschiedenen KörperoberflÄchen. Dabei wird sowohl der instationÄre Anlaufvorgang als auch der stationÄre Proze\ betrachtet. Die Ergebnisse für die Schichtdicke des abflie\enden Kondensates werden eingehend diskutiert. Ist die Schichtdicke als Funktion des Ortes und der Zeit bekannt, ist die Berechnung des kondensierenden bzw. abflie\enden Volumenstromes, sowie die Berechnung des lokalen bzw. für die Praxis bedeutungsvolleren globalen WÄrmeübergangs möglich.

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Steady and unsteady process of film condensation on a flat plate, a vertical coin, a horizontal pipe and a sphere

This paper investigates film condensation on different surfaces of geometric bodies. In this connection the unsteady starting process and the steady process are considered. The results for the thickness of layer of the flowing-off condensate are discussed detailed. If the thickness of layer is given as a function of time and location the computation of the condensing, respective flowing-off volume stream and the computation of the local, respective global heat transfer is possible.

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Bezeichnungen C Konstante - R Rohr- bzw. Kugelradius [m] - T Temperatur [K] - kondensierender Volumenstrom pro LÄngeneinheit [m² s^–1] - abflie\ender Volumenstrom pro LÄngeneinheit [m² s^–1] - kondensierender Volumenstrom [m³ s^–1] - abflie\ender Volumenstrom [m³ s^–1] - a Kegelachse - c spez. WÄrme der kondensierenden Flüssigkeit [J kg^–1 K^–1] - e ErzeugendenlÄnge des Kegels, an der die Randbedingung vorgeschrieben ist [m] - g Erdbeschleunigung [m s^–2] - l Platten- bzw. KegellÄnge [m] - p Druck [Nm^–2] - q WÄrmestromdichte [J m^–2 s^–1] - r VerdampfungswÄrme der Flüssigkeit [J kg^–1] - t Zeit [s] - u örtliche Geschwindigkeit des Fluids [m s^–1] - x, y kartesische Ortskoordinaten - r, Zylinder bzw. Kugelkoordinaten - WÄrmeübergangszahl [J m^–2 s^–1] - Neigungswinkel der Platte - öffnungswinkel des Kegels - Schichtdicke der kondensierten Flüssigkeit [m] - WÄrmeleitzahl der kondensierten Flüssigkeit [J m^–1 s^–1] - Dichte der kondensierten Flüssigkeit [kg m^–3] - OberflÄchenspannung der kondensierten Flüssigkeit [Nm^–1] - Schubspannung in der kondensierten Flüssigkeit [Nm^–2] - v kinematische ZÄhigkeit [m² s^–1] - dynamische ZÄhigkeit [kg m^–1 s^–1] - Winkelkoordinate (Rohr, Kugel), bei der eine Randbe-dingung vorgeschieben ist Indizes g gasförmige Phase - m mittlere - s SÄttigungszustand des gasförmigen Mediums - w auf die OberflÄche der Wand (Platte, Kegel, Rohr,Kugel) bezogen - 0 Ursprung der jeweiligen Störungsausbreitung Dimensionslose Kennzahlen Nu Nu\elt-Zahl - Pr Prandtl-Zahl - Re Reynolds-Zahl Kurzfassung der bei Prof. Dr. W. Schneider, Institut für Strömungslehre und WÄrmeübertragung TU Wien, angefertigten Diplomarbeit     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.