Forced convective flow over a flat plate in non-Newtonian power law fluids

A new forced convection parameter

Mathematical models for mass transport through arcs and flames

Summary The transport of particles, caused by axial and radial diffusion and axial flow of convection, will be considered in a D.C. arc and in a laminar flame. The following mathematical model will be discussed. Assuming a steady state in both cases the mass transport may be described in cylindrical coordinates (r, z) by the following partial differential equation where C means the particle concentration, D the coefficient of diffusion, and W the axial velocity. D and W are taken to be constant; various boundary conditions, corresponding to different approximations of the physical situation, are considered. Solutions of (0.1) are obtained in an explicit form.     

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Transient response of a finite crack in an orthotropic strip under the in-plane shear impact

The transient response of a central crack in an orthotropic strip under the in-plane shear impact loading is studied by using the dual integral equation method proposed by Copson and Sih. The general formula for the shear stress intensity factor near the crack tip is derived. Numerical results of with in various cases are obtained by solving the second kind Fredholm integral equation and by performing the inverse Laplace transform.     

The double velocity correlation function of homogeneous turbulence with constant mean velocity gradient

In this article, as the velocity gradient is taken as a constant value, we obtain the solutions of the equation of fluctuation velocity after Fourier transform. When the mean velocity gradient is small, they represent the picture of eddies, of which the homogeneous turbulence (both isotropic and nonisotropic) of the final period is composed. By using the eddies of these types at different times, we may compose the steady turbulent field with the constant velocity gradient and this field may represent the turbulent field in the central part of the channel flow or pipe flow approximately. Then we may obtain the double velocity correlation function of this turbulent field, which involves both longitudinal correlation coefficient and the transversal correlation coefficient . We compare these theoretical coefficients with the experimental data of these coefficients at initial period and final period of isotropic homogeneous turbulence. And then we obtain the relationship between the turbulent double velocity correlation coefficient and the mean velocity gradient. Finally, we get the expressions of the Reynolds stress and the eddy viscosity coefficient.     

On the operator of symmetric differentiation on a compact Riemannian manifold with boundary

We state a particular case of one of the theorems which we shall prove. Let Ω be a bounded open set in ℝ ⁿ with smooth boundary and let σ=(σ _ij )be a symmetric second-order tensor with components σ _ij εH ^k(Ω) for some (positive or negative) integer k; H ^k are Sobolev spaces on Ω. Then we have for some u _i εH ^k +1(Ω),i=1,...,n, if and only if (if k<0, the integral is in fact a duality) for any symmetric tensor (ω with components and such that ). Some applications in the theory of elasticity are also given.     

Experimental GH singularity field around growing crack-tip

The displacement fieldsu _x ,u _y at growing crack tip of LY12-M specimens with double edge cracks are measured using moire method. The experimental singularity fields are compared with GH theoretical field [12–14]. The size and shape of the experimental GH singularity fields are obtained. The error in both the experimental and theoretical evaluations is controlled within ±10%. The experiments show that there is singularity dominant around a growing crack tip. The shape of this dominant region ranges from butterfly wing to oblate and circular. Inside GH-field, there is a 3-D deformed damage zone where no GH singularity exists. The project suppoted by National Natural Science Foundation of China     

The unsteady aerodynamics of a first stage stator vane row

The fundamental unsteady aerodynamics on a vane row of an axial flow research compressor stage are experimentally investigated, demonstrating the effects of airfoil camber and steady loading. In particular, the rotor wake generated unsteady surface pressure distributions on the first stage vane row are quantified over a range of operating conditions. These cambered airfoil unsteady data are correlated with predictions from a flat plate cascade inviscid flow model. At the design point, the unsteady pressure difference coefficient data exhibit good correlation with the nonseparated predictions, with the aerodynamic phase lag data exhibiting fair trendwise correlation. The quantitative phase lag differences are associated with the camber of the airfoil. An aft suction surface flow separation region is indicated by the steady state surface static pressure data as the aerodynamic loading is increased. This separation affects the increased incidence angle unsteady pressure data.List of symbols b airfoil semi-chord - C airfoil chord - C _p dynamic pressure coefficient, - _p static pressure coefficient, - i incidence angle - k reduced frequency, - N number of rotor revolutions - p dynamic pressure difference - static pressure difference, - S stator vane circumferential spacing - U _t rotor blade tip speed - u longitudinal perturbation velocity - V absolute velocity - V _axial absolute axial velocity - v transverse perturbation velocity - x _sep location of separation point - inlet angle - inlet air density - blade passing angular frequency     

Regularity for the stationary Navier-Stokes equations in bounded domain

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain ^N, 5N . If u, p satisfy the additional conditions

Numerical study of a nonlinear integrodifferential equation

I.IntroductionItiswell-knobal.nthatKorteweg-deVriesequationisacanonicalmodeltodescribethebalanceofthenonlineareffectandthedispersiveeffectofaphysicalsystem.Thisequationpossessestheso-called'soliton"solution,whichhasbeenfoundnumericallybyZabuskyandKruskall'].Ho-c'Jlever,sometimesthebalanceofnonlinearityanddispersionofasystemmayleadtoa,integroditTerentialequationinsteadofadifferentialequation.Forinstance,inthestudyofvortexbreakdownofanunboundedrotatingfluidLeibovich12]derivedfollowingnonline…     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, and , the size of the HRR dominant zone is much smaller than forV and , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( ) and over most of the region where measurements were obtained ( ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

About a condition for blow up of solutions of cauchy problem for a wave equation

ＣｏｎｓｉｄｅｒｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｆｏｒｔｈｅｗａｖｅｅｑｕａｔｉｏｎｉｎＲＮ×Ｒ＋（Ｎ≥２）：２ｕ（ｘ,ｔ）ｔ２－ｘｉａｉｊ（ｘ）ｘｊｕ＝｜ｕ｜ｐ－１·ｕ　　（（ｘ,ｔ）∈ＲＮ×（０,Ｔ））,ｕ（ｘ,０）＝ｇ（ｘ）　（ｘ∈ＲＮ）,ｕｔ（ｘ,０）＝ｈ（ｘ）　（ｘ∈ＲＮ）,（１）ｗｈｅｒｅｕ（ｘ,ｔ）ｉｓｎｏｎｔｒｉｖｉａｌｓｏｌｕｔｉｏｎｗｉｔｈｆｉｎｉｔｅｓｐｅｅｄｏｆｐｒｏｐａｇａｔｉｏｎａｎｄｉｓｓｕｐｐｏｒｔｅｄｏｎａｆｏｒｗａｒｄｃｏｎｅ（ｘ,ｔ）·ｔ≥０,｜…     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

Dynamics of compressible air flow with friction in a variable-area duct

An adiabatic compressible air flow with friction in a variable-area duct is investigated. A nozzle-flow apparatus is built in which compressed air flows in a frictional diverging conical duct. According to the experimental investigation together with the numerical solution of the governing differential equations it is colligated that the dominance of either factor, friction or variation of area effect, depends upon the product of the inequality .

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Dynamik einer kompressiblen Luftströmung mit Reibung in einem Kanal veränderlichen Querschnitts

Zusammenfassung Es wurde eine adiabate kompressible Luftströmung mit Reibung in einem Kanal veränderlichen Querschnitts untersucht. Dazu wurde ein düsenförmiger Kanal gebaut, in dem die komprimierte Luft im erweiterten Teil strömt. Sowohl die Versuche als auch die numerische Lösung der beschreibenden Differentialgleichungen ergaben, daß entweder der Reibungsbeiwert oder die Änderung des Strömungsquerschnittes von dem Produkt abhängen.

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Nomenclature A cross-sectional area of the duct - A _in inlet cross-sectional area of the duct - D diameter of a cross-sectional area - f friction coefficient - M Mach number - /D relative surface roughness of the duct wall - P local static pressure at any axial location - P₀ stagnation pressure at any location - P_{0 in} inlet stagnation pressure - mass flow rate - T static temperature - T_o stagnation temperature - R air gas constant - Re Reynolds number - ratio of specific heats - x measured from the duct inlet     

Laminar forced convection of power-law non-Newtonian fluids inside ducts

Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.

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Laminare Zwangskonvektion von nicht-Newtonschen Fluiden, die in Kanälen strömen und dem Potenzansatz folgen

Zusammenfassung Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.

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Nomenclature b radius of circular duct or half the spacing between parallel plates - C - D _h hydraulic diameter=4b for parallel plate, 2b for circular tube - f(r), F(r) temperature distribution at the inlet, dimensional and dimensionless, respectively - g (r), G (R) energy generation, dimensional and dimensionless, respectively - h (z) heat transfer coefficient - H _i(Z) defined by Eq. (11b) - k thermal conductivity - l₀,l _N reference lengths to nondimensionalizer andz coordinates respectively (l ₀=b andl _N= D_h) - n power-law index - Nu _av average Nusselt number=h _av D_h/k - Nu (Z) local Nusselt number=h (z)D _h/k - p 0 for parallel-plate duct, 1 for circular duct - r radial or normal coordinate - R = dimensionless radial coordinate - T(r, z) fluid temperature - T _av (z) average fluid temperature - T ^* reference temperature - T reference temperature difference - U(R) = normalized velocity profile - U ^*(R) - w (r) fully developed velocity profile - w _av average velocity - z axial coordinate - Z = dimensionless axial coordinate Greek letters thermal diffusivity - (R,Z) = dimensionless temperature - _w = dimensionless wall temperature - _i eigenvalues of the eigenvalue problem (10) - (_i,R) eigenfunctions of the eigenvalue problem (10)     

Thermal convection at a melting benzene surface

Summary Heat transfer by thermal free convection at the surface of a sphere has been studied experimentally by melting a sphere of solid benzene in a large quantity of liquid benzene of homogeneous temperature. The influence of cold liquid produced by the melting process is taken into account to yield results that are representative for the pure effect of heat transfer without melting. In the general formula for heat transfer by thermal convection, =C(GrPr)^1/4, we found C=0.525.     

On nonlinear hyperbolic equation in unbounded domain

The following nonlinear hyperbolic equation is discussed in this paper: where The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)