On the propagation of elastic waves through composite media

Summary With the aid of an ultrasonic pulse technique, the propagation of elastic waves (longitudinal as well as transverse) through polyurethane rubbers filled with different amounts of sodium chloride particles was studied. The velocity of both longitudinal and transverse waves was found to increase with filler content. From the measured wave velocities, the effective modulus for longitudinal waves,L, bulk modulus,K, and shear modulus,G, were calculated according to the relations for a homogeneous isotropic material. All three moduli appear to be monotonously increasing functions of the filler content over the whole experimentally accessible temperature range (–70 °C to + 70 °C forL andK;}-70 °C to about –20 °C forG) and they, moreover, reflect the glassrubber transition of the binder.Poisson's ratio,, was found to decrease with increasing filler content and show a rise at the high temperature side of the experimentally accessible temperature range (about –20 °C) as a result of the approach of the glass-rubber transition.In addition to the velocities, the attenuation of both longitudinal and transverse waves was measured in the temperature ranges mentioned. It was found that in the hard region tan _L as well as tan _G are independent of the filler content within the accuracy of the measurements. In the rubbery region, however, tan _L, increases with increasing filler content.Finally, the experimental data are compared with a simple macroscopic theory on the elastic properties of composite media.     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Structure of permanent waves in density-stratified media

Non-linear steady waves in density-stratified channels are discussed. Under the assumption of non-diffusive incompressible flow, a method to classify all waveforms is given for moderate wave amplitudes, when the densityq at infinity decreases with altitude (q<0). The classification is worked out explicitly for the two highest critical Froude numbers. Furthermore, special density distributions are constructed with the purpose of showing that, for example, solitary waves exist even ifq>0.Dedicated to Professor Gaetano Fichera on the occasion of his 70th birthday. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, September 1992.     

Effect of lubricant inertia in bearings with a non-Newtonian lubricant

Summary The inertia effects in externally pressurized and squeeze film bearings with lubricants obeying a power law are considered. It is found that the inertia forces decrease the load capacity of the externally pressurized bearing with a given flow rate and the inertia effect increases with the flow behaviour index. At a given feeding pressure, on the other hand, the inertia increases or decreases the load capacity when the flow behaviour index is smaller than or greater than 3, respectively. For squeeze films between circular plates and rectangular plates, the rate of squeeze is slowed down by the inertia and the inertia effect is larger in dilatant lubricants than in pseudoplastic lubricants.Nomenclature 2a diameter of the bearing, width of rectangular plates - 2b diameter of the recess - 2h film thickness - 2h ₀ initial thickness of squeeze films - l length of the rectangular plates - m consistency index - n flow behaviour index - p pressure - p _e external pressure - p _i feeding pressure - q flow rate - r radial distance - t time - u velocity of the lubricant - v squeeze velocity - w load capacity - W dimensionless load capacity - axial distance - viscosity of the lubricant - density of the lubricant     

Longitudinal volume viscosity of poly (vinyl chloride)

The volume flow of poly (vinyl chloride) ( = 45,000,T _g = 350 K) has been measured in an Instron Capillary Rheometer.The elastic modulus in longitudinal compression, the longitudinal volume viscosity and initial longitudinal volume viscosity, and retardation times were determined at temperatures both below (324 – 343 K) and above (403 – 453 K) the glass transition temperatureT _g , and at compression rates between approximately 10^–5 and 200 · 10^–5 s^–1.An increase in the longitudinal volume viscosity was observed for decreases in the volume deformation, increases in the compression rate and increases in temperature.T _g decreased at 0.16 K/MPa. The volume flow activation energy was found to be equal to that for shear flow with a constant value of 91.37 kJ/mol.     

Nonlinear theory of the propagation of pulses of electromagnetic waves through a plasma boundary

A solution is obtained for the problem of the propagation of electromagnetic waves of arbitrary form through a plasma boundary on condition that the length of the wave train is much greater than the wave length. A solution is found both for the case of a wide spectrum of width much greater than the plasma frequency ₀, as well as for a narrow spectrum. The results obtained enable us to draw conclusions about the time and space variation of the shape of electromagnetic pulses in a plasma.The passage of high frequency electromagnetic waves through a plasma is similar to that of a beam of charged particles [1, 2]. This is associated with the fact that decay processes are similar to Cerenkov radiation effects. The dynamics of the development of transverse wave instabilities in a uniform Isotropic plasma were studied in [2] assuming that the wave phase behaves stochastically. It was calculated here that instabilities develop quite differently in the case of a wide frequency spectrum than in the case of a narrow monochromatic spectrum. If we can speak of transverse quanta diffusion effects in the field of the generated longitudinal quanta in the first case, and if the resulting effects are closely similar to the nonlinear effects arising when beam instability develops [3, 4], then the development of instabilities in the case of a narrow spectrum leads to the appearance of red satellites in the transverse wave spectrum differing from the basic frequency by a quantity ₀ (=1, 2, 3,...). In this case the development of the instability corresponds to a tendency for a plateau over the satellites to appear.Attention should however be drawn to the fact that the dynamics of instability development in a semibounded plasma may be quite different. This is associated first with the different values of group velocities of transverse and longitudinal waves, and what is also important, with the effect of longitudinal wave accumulation in the boundary region if the length of the wave train is sufficiently large. The treatment of a similar problem for beam instabilities in paper [5] showed that a narrow transition layer may arise with a transverse wave energy density greatly in excess of the energy density of the injected beam. In what follows we examine the part played by boundary effects in the passage of pulses of electromagnetic waves through the boundary of the plasma. The cases of both narrow and wide spectra are considered. We note that in the case of narrow spectra the wave train must necessarily be greatly in excess of ^–1, and the effects of the accumulation of oscillations will be appreciable.The phases of both transverse waves, and also generated longitudinal waves are assumed to be stochastic quantities. The boundary effects which have been treated may be applied both in the generation of longitudinal waves necessary for the effective acceleration of particles in a plasma as well as in the modulation and alteration of the initial transverse wave spectrum. It should also be stressed that these effects which have been considered could be applied for turbulent plasma diagnostics, as has already been pointed out in [2].The authors are grateful to Ya. B. Fainberg, M. S. Rabinovich, I. S. Danilkin, and M. D. Raizer for their interest in the paper and for valuable criticisms.     

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.     

Experimental investigations of fluid flow and heat transfer characteristics of a slot jet impinging on a rectangular cylinder

Experimental investigations on flow characteristics and average heat transfer due to slot jet impinging on a rectangular cylinder have been carried out for different non-dimensional parameters. The minimum value of the pressure coefficient is found on the lower face of the rectangular cylinder at an angle of inclination of 15°. Drag coefficient calculated from the measured pressure distribution is found to be maximum within a range of breadth/width ratio of 0.67 to 1.5 of rectangular cylinders. The maximum value of heat transfer rate is obtained at the angle of inclination of 15° of the cylinder to the jet axis. An increasing trend of heat transfer rate is observed with higher Reynolds numbers. A correlation of average Nusselt number is presented for rectangular cylinders.

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Experimentelle Untersuchung der Strömungs- und Wärmeübergangs-charakteristik eines auf einen rechteckigen Zylinder auftreffenden Strahls aus einer Schlitzdüse

Zusammenfassung Es wurden experimentelle Untersuchungen des Strömungs- und Wärmeübergangsverhaltens an einem rechteckigen, durch einen Strahl aus einer Schlitzdüse beaufschlagten Zylinders für verschiedene dimensionslose Parameter durchgeführt. Der Kleinstwert des Druckbeiwertes tritt an der Unterfläche des rechteckigen Zylinders bei einem Neigungswinkel von 15° auf. Der ausder gemessenen Druckverteilung berechnete Widerstandsbeiwert erreicht bei einem Breiten-Dicken-Verhältnis des Zylinders zwischen 0,67 und 1,5 Maximalwerte. Den maximalen Wärmestrom erhält man bei einem Neigungswinkel zwischen Zylinder und Strahlachse von 15°. Mit steigenden Reynoldszahlen erhöht sich der abgeführte Wärmestrom. Eine Korrelation für die mittlere Nusseltzahl an rechteckigen Zylindern wird mitgeteilt.

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Nomenclature A surface area of the rectangular cylinder - a width of the rectangular cylinder - b breadth of the rectangular cylinder - C _D drag coefficient =D/ - C _p pressure coefficient = (p – p _a )/ - C _pb base pressure coefficient = (p _b –p _a )/ - D drag force - h _f free convection heat transfer coefficient - average heat transfer coefficient - k thermal conductivity of air - L distance of the axis of the square cylinder from the nozzle exit - l length of the rectangular cylinder - Pr Prandtl number - p static pressure - P _a atmospheric pressure - P _b base pressure on the rear face - Nu _f free convection Nusselt number - average Nusselt number - q heat loss - q _f heat loss due to free convection - Re Reynolds number =u _j W/ _a - T _a ambient air temperature - average surface temperature - u _j average jet velocity at the nozzle exit - W nozzle width - angle of inclination of the rectangular cylinder to the jet axis in degrees - _a kinematic viscosity of air - _a density of air     

Experimental investigations of fluid flow and heat transfer characteristics of a slot jet impinging on a square cylinder

Experimental investigations on pressure distributions and average heat transfer on square cylinders due to slot jet impingement have been carried out for different parameters such as, slot jet-width, distance of the square cylinder from the nozzle exit, angle of inclination of the cylinder to the jet axis and Reynolds numbers. The minimum value of the pressure coefficient is obtained on the lower face at an angle of inclination of 15° for all distances of the square cylinder from the nozzle exit. At the lowest Reynolds number the maximum average heat transfer rate is obtained at a distance of eight times the jet width from the nozzle exit. An increasing trend of the heat transfer rate is observed for higher Reynolds numbers. The maximum value of the heat transfer rate is obtained between the angles of inclination of 15° and 30° of the square cylinder to the jet axis. A correlation for the average Nusselt number is proposed in terms of the relevant non-dimensional parameters.

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Experimentelle Untersuchung der Strömungs- und Wärmeübergangscharakteristik bei Schlitzdüsenanblasung eines quadratischen Zylinders

Zusammenfassung Druckverteilung und gemittelter Wärmeübergang bei Schlitzanblasung eines quadratischen Zylinders wurden experimentell für folgende Parameter untersucht: Schlitzbreite; Abstand Düsenmündung vom Zylinder; Neigungswinkel des Zylinders zur Strahlachse; Reynoldszahl. Den Minimalwert des Druckkoeffizienten erhält man für alle Abstände an der Unterseite (bei einem Neigungswinkel von 15°). Bei der niedrigsten Reynoldszahl tritt der höchste Wert des gemittelten Wärmestroms in einem Abstand Düsenmündung/Zylinder von 8 Strahlbreiten auf. Mit steigender Reynoldszahl nimmt der Wärmestrom zu. Dessen höchster Wert tritt im Bereich 15 bis 30° des Neigungswinkels zwischen Zylinder und Strahlachse auf. Eine die Meßwerte korrelierende Nusscltbeziehung als Funktion dimensionsloser Parameter wird angegeben.

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Nomenclature A surface area of the square cylinder - a width of the square cylinder - C _p pressure coefficient=(p–p _a )/ - C _pb base pressure coefficient=(p _b –p _a )/ - h _f free convection heat transfer coefficient - average heat transfer coefficient - k thermal conductivity of air - L distance of the axis of the square cylinder from the nozzle exit - l length of the square cylinder - Pr Prandtl number - p static pressure - p _a atmospheric pressure - p _b base pressure on the rear face - Nu _f free convection Nusselt number - average Nusselt number - q heat loss - q _f heat loss due to free convection - Re Reynolds number=u _j W/v _a - T _a ambient air temperature - average surface temperature - u _j average jet velocity at the nozzle exit - W nozzle width - angle of inclination of the square cylinder to the jet axis in degrees - _a kinematic viscosity of air - _a density of air     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Surface waves under constrained deformation

The possibility of the existence of surface waves in a range of velocities greater than the velocity of transverse waves, but smaller than the velocity of longitudinal waves is shown. It turns out that, in the boundary value problem for an elastic half-space in this velocity range, there are the surface waves whose velocity is constant and equal to \(\sqrt 2 \) b, where b is the velocity of transverse waves. These waves as well as the Rayleigh surface waves have no dispersion. Their velocity is specified only by the elastic constants and density of the medium. It is also shown that the existence of such a velocity is possibly related to the velocity of surface waves that appear as unloading waves under constrained deformation.     

Temperature distribution in a thin rotating disk

The problem of heat conduction in a thin rotating disk with heat input at a fixed point is considered. The disk is cooled by forced convection from its lateral surfaces. By defining a complex temperature, the temperature throughout the disk is presented as a series of Bessel functions of complex argument. Results are given for a range of rotational speeds.Nomenclature R radial coordinate - angular coordinate - a radius of disk - b thickness of disk - T temperature - T ambient temperature - rotational speed of disk - q heat flux into disk - k thermal conductivity of disk - density of disk - c specific heat of disk - h coefficient of convective heat transfer - r dimensionless radial coordinate, R/a - T* characteristic temperature, q ₀ a/ k - t dimensionless temperature, (T–T )/T* - C ₁, C ₂ dimensionless parameters defined in (3)     

Performance prediction of dual cylindrical wave laser devices for flow measurements

The dual cylindrical wave system is a variant of laser Doppler velocimetry, in which two cylindrical waves of laser light are used to illuminate a moving particle. This instrument is being used for local measurement of the unsteady skin friction in turbulent boundary layers, as well as droplet sizing in spray flows. In the present work, performance of these new devices is examined using the electromagnetic theory of light. Various requirements for the design and operation of these instruments have been further elaborated and extended. The accuracy of the previous experimental results has also been considered. The optics-related errors are shown to be negligible in the measurements of streamwise as well as spanwise wall velocity gradients. However, rigorous simulations appear to be essential for proper calibration of the particle sizing device.List of symbols A, B, C three particle positions - a half-width of an optical slit - a _gm amplitude of a plane wave in the spectrum of a cylindrical wave - d _f fringe spacing - d _p particle diameter - E amplitude of the electric oscillation in the optical field - E _c combined electric field of two cylindrical waves - E _o maximum strength of the electric field at the source of a cylindrical wave - E _s electric field of a scattered wave - E _y time-dependent electric field in the case of electric polarization - f characteristic length for the phase of the scattering amplitude - f _a anisotropic frequency - f _D Doppler frequency - F _DCW transfer function of DCW system for particle sizing - F _pDA Phase Doppler transfer function - g wall velocity gradient - g _m measured wall velocity gradient - I ₀, I₂ integrals in the asymptotic expansion of the scattering amplitude - I _s intensity of the scattered light - k wave number of laser light in the fluid medium - m refractive index of the particle relative to the surrounding medium - N ₀ nominal number of fringes resulting from interference of two cylindrical waves - P phase of a plane wave - P ₁, P₂ phases of plane waves from downstream and upstream cylindrical waves respectively - P _s scattered light power at a receiving aperture - r unit vector in the direction of light scattering - r _D distance of the signal detector from the particle center - S scattering amplitude of a cylindrical wave - S ₁, S₂ Scattering amplitudes of the cylindrical waves emanating from S₁ and S₂ respectively - magnitude of the scattering amplitude for a plane wave - S _c combined scattering amplitude of two cylindrical waves - S₁, S₂ downstream and upstream sources of cylindrical waves, respectively - S scattering amplitude of a plane wave - s half-spacing between sources of the cylindrical waves - t time - u velocity along x-axis - w ₀ 1/e half-width of the field distribution at the waist of a laser sheet - X ₀ nominal width of the fringe volume along the particle path - X particle position in the measuring volume - x, y, z Cartesian coordinates Greek symbols angle of the direction of wave propagation from x-axis - coefficient of the second-order term in the phase function of a cylindrical wave - angular size of the signal receiving aperture - incremental used for numerical differentiation the ratio of to _p - — ₀ integration parameter for I₀ and I₂ - half-angle between the directions of propagation of two waves - wavelength of laser in the fluid medium - ₀ wavelength of laser in vacuum - parameter defining the direction of propagation of a plane wave - 1/e 1/e half-width of the function A - ₀ direction of propagation of the dominant plane wave in the spectrum - _0s the direction of propagation of the plane wave that contributes predominantly to scattering in a particular direction - _p the value of . corresponding to one cycle of P - _s change in corresponding to a lobe of the scattering amplitude - a dimensional form of that determines lobes in the scattered field - signal phase, ₀ + _a - _a anisotropic phase shift - ₀ phase difference between two indicent waves - off-axis angle - elevation angle - circular frequency of laser light     

Influence of superimposed steady shear flow on the dynamic properties of non-Newtonian fluids

Summary Experiments in which an oscillatory shear flow is superimposed on a steady-state circular shear flow between a cone and a plate were performed on non-Newtonian solutions by means of aWeissenberg Rheogoniometer. The steady-state shear stress and in a first approximation also the normal stress difference arising from the steady shear flow appear not to be influenced by the superimposed oscillatory flow. On the other hand, the dynamic moduli as obtained from the oscillatory parts of shear stress and shear flow are highly dependent on the superimposed steady rate of shear. The absolute value of the complex shear modulus decreases and the phase difference between oscillatory shear stress and shear flow increases in all cases and for all frequencies if the superimposed shear rate is increased. Consequently, this phase difference can become equal to and even larger than /2. Between the angular frequency ₀ at which the phase difference is /2 and the steady shear rateq the relation ₀= 1/2,q was experimentally found to exist in most cases. These dynamic results cannot be described by the current theories of viscoelasticity. The large and fast deformations imposed on the material should explicitly be taken into account.     

Heat transfer and pressure drop for purely viscous non-Newtonian fluids in turbulent flow through rectangular passages

Experimental measurements of friction factor and heat transfer for the turbulent flow of purely viscous non-Newtonian fluids in a 21 rectangular channel are compared with results previously reported for the circular tube geometry. Comparisons are also made with available analytical and empirical predictions.It is found that the rectangular duct fully established friction factor measurements are within ± 5% of the Dodge-Metzner prediction if the Kozicki generalized Reynolds number is used. A modified form of the simpler explicit equation proposed by Yoo, [i.e.f=0.079n ^0.675(Re ^*)^–0.25], is found to yield predictions for both the rectangular duct and the circular tube geometries with approximately the same accuracy as the Dodge-Metzner equation.Fully developed Stanton numbers for the rectangular duct are in good agreement with the circular tube data over a range ofn from 0.37 to 0.88 for a given Prandtl number,Pr _a , when compared at a fixed value of the Reynolds number based on the apparent viscosity evaluated at the wall shear stress. In general, the experimental data are within ± 20% of Yoo's equation,St=0.0152Re _a ^–0.155 Pr _a ^–2/3 . A new equation is proposed to bring the prediction for circular pipes as well as rectangular channels into better agreement with generally accepted Newtonian heat transfer results.

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Wärmeübergang und Druckverlust für viskose nicht-Newtonsche Fluide in turbulenter Strömung durch rechteckige Kanäle

Zusammenfassung Es werden Messungen des Reibungsfaktors und des Wärmeübergangs bei turbulenter Strömung viskoser nicht-Newtonscher Fluide in einem rechteckigen Kanal mit dem Seitenverhältnis 21 verglichen mit früheren Ergebnissen, die an runden Rohren gewonnen wurden. Weiterhin werden Vergleiche mit aus der Literatur verfügbaren analytischen und empirischen Beziehungen gemacht.Es zeigte sich, daß die Messungen des Reibungsfaktors im rechteckigen Kanal bei vollausgebildeter Strömung auf ± 5% mit der Vorhersage von Dodge-Metzner übereinstimmen, wenn die von Kozicki verallgemeinerte Reynolds-Zahl verwendet wird. Eine modifizierte Form der einfachen von Yoo vorgeschlagenen einfachen Gleichung in explizierter Form (f=0,079n ^0,675(Re ^*)^–0,25) bewies, daß sie sowohl für den rechteckigen Kanal als auch das runde Rohr die Werte mit fast der gleichen Genauigkeit wie die Methode von Dodge-Metzner vorhersagen kann.Die Stanton-Zahlen für den rechteckigen Kanal bei vollausgebildeter Strömung sind in guter Übereinstimmung mit den Werten für das runde Rohr in einem Bereich vonn= 0,37 – 0,88 für eine gegebene Prandtl-Zahl, wenn man den Vergleich bei einem vorgegebenen Wert der Reynolds-Zahl anstellt, die auf die scheinbare Viskosität — abgeleitet aus der Wandschubspannungbezogen ist. Generell läßt sich sagen, daß die Werte auf ± 20% mit der Gleichung von Yoo (St=0,0152Re _a ^–0,155 )Pr _a ^–2/3 ) übereinstimmen. Es wird eine neue Gleichung vorgeschlagen, welche sowohl die Werte für runde Rohre als auch die für rechteckige Kanäle in bessere Übereinstimmung bringt mit den in der Literatur üblichen Ergebnissen für den Wärmeübergang an Newtonsche Fluide.

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Nomenclature a constant in Eq. (8) - A area of cross-section of channel [m²] - b constant in Eq. (8) - c _p specific heat of test fluid [J kg^–1 K^–1] - d capillary tube diameter [m] - D _h hydraulic diameter, 4A/P[m] - f Fanning friction factor, _w/(g9 V²/2) - h axially local (spanwise averaged) heat transfer coefficient,q _w /(T_wi-T_b) [Wm^–2 K^–1] - k _f thermal conductivity of test fluid [Wm^–1K^–1] - K consistency index of power law fluid - n power law index - Nu fully established, local (spanwise averaged) Nusselt numberh D _h /k _f - P perimeter of channel [m] - Pr _a Prandtl number based on apparent viscosjity, c _p /k _f - Pr ^* defined as (Re _a Pr _a )/Re ^* - q _w wall heat flux [Wm^–2] - Re _a Reynolds number based on apparent viscosity, VD _h/ - Re Metzner's generalized Reynolds number in Eq. (2) - Re ^* Reynolds number defined in Eq. (8) - St Stanton number,h/( V c_p) - T _b local bulk temperature of the fluid [K] - T _wi local inside wall temperature [K] - T _wo local outside wall temperature [K] - V bulk flow velocity [m s^–1] - x distance from the inlet of channel along flow direction [m] Greek symbols shear rate [s^–1] - apparent viscosity [Pa s] - density of test fluid [kg m^–3] - shear stress [Pa] - _w shear stress at the wall [Pa] Dedicated to Prof. Dr.-Ing. U. Grigull's 75th birthday     

Parametric excitation of corkscrew instability of a θ-pinch by a high-frequency longitudinal current

Equilibrium conditions are given for a thin annular -pinch with high in a magnetic field, transverse to the plane of the ring, exciting a high-frequency longitudinal current in the plasma. The parametric buildup of small corkscrew perturbations is studied on the basis of the model of a flexible straight plasma filament. Conditions under which parametric excitation may be suppressed are discussed. There is a brief discussion of forced oscillations of the radii of the annular -pinch caused by the alternating fields.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 2, pp. 21–27, March–April, 1970.In conclusion, the author thanks M. L. Levin for interest in this study and valuable advice.     

Band radiation heat transfer in open system filled with isothermal,emitting and absorbing gas

The new method of band calculation of radiation heat transfer in open enclosures filled wihth isothermal gas is presented. Introduced generalized band configuration factors enable the adaptation of existing methods systems without active gases for systems filled with optically active media. The new procedure allows the calculation of complex systems in equilibrium state to be performed.

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Band- Wärmeübertragung durch Strahlung in einem offenen System mit isothermem Gas emittierender und absorbierender Eigenschaften

Zusammenfassung In der Arbeit ist eine neue Methode zur Bandberechnung der Wärmeübertragung durch Strahlung in Systemen mit isothermem Gas vorgestellt. Der vorgeschlagene General-Band-Konfigurationskoeffizient ermöglicht die Nutzung der bisher eingesetzten Rechenmodelle ohne aktives Gas für Systeme mit optisch aktivem Medium. Der dargestellte Rechenalgorithmus macht es möglich, Berechnungen von komplexen Systemen anzustellen, die sich im thermischen Gleichgewicht befinden.

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Nomenclature matrixA defined by Eq. (8) - A _l average absorption coefficient or area - a _l band average absorptivity - a _kj (l) matrix elements defined by Eq. (9) - matrixB defined by Eq. (10) - _l matrix C defined by Eq. (11) - e _{lc k} band emissive power - e _c hemispherical emissive power - F configuration factor - generalized band configuration factor - h_l,k band effective energy flux - j, k indices denoting individual surfaces - l band - N number of real surfaces in a spatial system - n total number of surfaces (real and abstract) - Q _{s, p} dissipation loss - q _k energy flux - q _{l, k} band energy flux - T absolute temperature - _{l, k} selective emission coefficient - _l ,k–j transmissivity - _l ,k-j emissivity of medium - _k -j Kronecker delta - wave number - wavelength     

Exact analysis of laminar dispersion in non-Newtonian flow

An exact analysis is presented for dispersion of a solute in an Eyring model fluid flowing between two parallel plates under uniform pressure gradient. Using a generalised dispersion model, which is valid for all time after injection of the solute, expressions have been developed for the time-dependent longitudinal dispersion coefficient as well as of the area-mean concentration. It is observed that the dispersion coefficient markedly decreases as the fluid parameterG increases, which is attributable to the gradual flattening of the velocity profile. Consequently the concentration distribution becomes steeper at largerG. It is also established that forG<1, the extent of dispersion is practically same as in the case of Newtonian flow.

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Exakte Analyse einer laminaren Dispersion in einer Nicht-Newtonschen Strömung

Zusammenfassung Es wird eine exakte Analyse für eine Dispersion in einer Eyring-Modellflüssigkeit vorgestellt, die zwischen zwei parallelen Platten unter gleichmäßigem Druckgefälle fließt. Es wurden sowohl Formeln für den zeitabhängigen Longitudinaldispersionskoeffizienten, als auch für die mittlere Umgebungskonzentration entwickelt, wobei ein allgemeines Dispersionsmodell benutzt wurde, welches für den ganzen Zeitraum nach dem Einleiten der Lösung gültig ist. Es wird beobachtet, daß der Dispersionskoeffizient deutlich abnimmt, wenn der FlüssigkeitsparameterG sich erhöht, was bezeichnend für die allmähliche Abflachung des Geschwindigkeitsprofils ist. Folglich wird die Konzentrationsverteilung mit steigendemG steiler. Es wurde bewiesen, daß für G 1 die Ausdehnung der Dispersion praktisch mit der Newtonschen Strömung identisch ist.

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Nomenclature a half of channel width - A, B Eyring model parameters - C local concentration of the solute - C ₀ concentration of uniform slug - ¯ C dimensionless local concentration - ¯ C _m area-average concentration defined by (8) - D molecular diffusivity - G dimensionless Eyring model parameter - K ₁() time-dependent dispersion coefficient - Pe Peclet number - t time - u axial velocity - U average axial velocity - x axial coordinate - x _s length of uniform slug - X dimensionless axial coordinate - X _s dimensionless slug length - y transverse coordinate - Y dimensionless transverse coordinate Greek symbols dimensionless Eyring model parameter - coefficient of viscosity - dimensionless time     

Treatment of moisture condensation on fins

An analysis of natural convection from a vertical plate fin when the fin base temperature is below the dew point of the surrounding air is presented in this paper. The analytical solution derived is based upon a constant heat and mass transfer coefficient and is also valid for forced convection. The results of this simplified theory are compared with a numerical solution where the coupling of convection and conduction is taken into account. An experimental verification of the results is also shown.

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Aus Kondensation von Feuchtigkeit an Rippen

Zusammenfassung Es wird eine Analyse der freien Konvektion an einer vertikalen plattenförmigen Rippe dargestellt, bei der die Temperatur im Anfangsbereich der Rippe unterhalb des Taupunktes der umgebenden Luft liegt. Die abgeleitete analytische Lösung beruht auf einem konstanten Wärme- und Stoffübergangskoeffizienten und gilt auch für die erzwungene Konvektion. Die Resultate dieser vereinfachten Theorie werden mit einer numerischen Lösung verglichen, in der die Verbindung von Konvektion und Wärmeleitung in Betracht gezogen wird. Angeführt wird auch eine experimentelle Bestätigung der Resultate.

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Nomenclature a _f thermal diffusivity of air - A, B constants in Eq. (7) - c constant defined in Eq. (3) - D diffusion coefficient - f an arbitrary function ofT andx in Eq. (12) - F ₁,F ₂ coefficients in differential Eq. (13) - g gravitational acceleration - h heat transfer coefficient - h _m mass transfer coefficient - k thermal conductivity of fin - k _f thermal conductivity of air - l latent heat of moisture condensation - L total length of fin - L _w length of wet fin - m parameter, (h/kt)^1/2 - m _l dimensionless parameter, 1+ B/T _r - m _y parameter,m m _l ^1/2 - p pressure of surrounding air - p _ws saturation pressure of water vapor - p _w partial pressure of water vapor in air - Pr Prandtl number,/a _f - q total heat fluxl - q _c convective heat flux - q _m heat flux - q _r radiative heat flux - R parameter in Eq. (14) - R _w specific gas constant of water vapor - t half thickness of fin - T temperature - T _b base temperature of wet fin - T _c base temperature of dry fin=saturation temp. of vapor - T _r reference temperature defined in Eq. (15) - T temperature of surrounding air - T temp, difference between fin surface and surroundings - v initial temperature for quasilinearization - x vertical coordinate, see Fig. 1 - y horizontal coordinate, see Fig. 1 - coefficient of thermal expansion - emissivity - dimensionless parameter in Eq. (14) - ø _d heat flux of dry fin - ø _tot total heat flux of dry-wet fin - kinematic viscosity - Stefan-Boltzman coefficient - relative humidity of air     

On Small-Displacement Waves in Prestressed Bodies with Isotropic Incremental Elasticity Tensor

We consider a body at rest in a prestressed configurationwhich responds elastically to small incremental displacements fromthe incremental elasticity tensor is supposed isotropic. On the basis of the paper [1] we characterize the conditions for the propagation of longitudinal, transverse, and oblique small-displacement waves superimposed toFormulae for the propagation speeds of these waves are written in terms of the prestress components and Lamparameters. The amplitudes of longitudinal and transverse waves are eigenvectors for the prestress.