On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Laminar forced convection in elliptic ducts

The problem of laminar forced convection heat transfer in short elliptical ducts with (i) uniform wall temperature and (ii) prescribed wall heat flux is examined in detail with the well known Lévêque theory of linear velocity profile near the wall. Moreover, consideration is given to the variation of the slope of the linear velocity profile with the position on the duct wall. A correction factor for the temperature dependent viscosity is included. Expressions for the local and average Nusselt numbers and wall temperatures are obtained. For the case of constant heat flux the Nusselt numbers are higher than for constant wall temperature.The results corresponding to the classical Graetz and Purday problems are deduced as special cases.Nomenclature a, b semiaxes of ellipse, b Graetz number (average), Re Pr D _e/Z - h _i ^o local heat transfer coefficient - J _n(x) Bessel function of order n - K thermal conductivity of the fluid - [X] Laplace transform of X - N _u ^o local Nusselt number, h _i ^o D _e/K - perimeter average Nusselt number - overall average Nusselt number - Nu _w wall Nusselt number - Nu Nusselt number at large distance from the inlet - p Laplace transform parameter - Pr Prandtl number, C _a/K - Re Reynolds number, D _e / _a - T temperature of the fluid - T ₁, T _W inlet and wall temperatures, respectively - u _z local isothermal velocity along the axis of the duct - average fluid velocity - x, y, z Cartesian coordinates, z-axis parallel to the axis of the duct (z=0 at duct inlet) - Z length of the duct - thermal diffusivity, K/C - * correction factor for the temperature dependent viscosity - (x) gamma function - coordinate measured normal to the wall of the duct - _a, _w viscosity of fluid at average and wall temperatures - , , z elliptic cylindrical coordinates - density of fluid - (z) heat flux     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

The dynamic and steady shear properties of synovial fluid and of the components making up synovial fluid

Steady-shear and dynamic properties of a pooled sample of cattle synovial fluid have been measured using techniques developed for low viscosity fluids. The rheological properties of synovial fluid were found to exhibit typical viscoelastic behaviour and can be described by the Carreau type A rheological model. Typical model parameters for the fluid are given; these may be useful for the analysis of the complex flow problems of joint lubrication.The two major constituents, hyaluronic acid and proteins, have been successfully separated from the pooled sample of synovial fluid. The rheological properties of the hyaluronic acid and the recombined hyaluronic acid-protein solutions of both equal and half the concentration of the constituents found in the original synovial fluid have been measured. These properties, when compared to those of the original synovial fluid, show an undeniable contribution of proteins to the flow behaviour of synovial fluid in joints. The effect of protein was found to be more prominent in hyaluronic acid of half the normal concentration found in synovial fluid, thus providing a possible explanation for the differences in flow behaviour observed between synovial fluid from certain diseased joints compared to normal joint fluid.Nomenclature A Ratio of angular amplitude of torsion head to oscillation input signal - G Storage modulus - G Loss modulus - I Moment of inertia of upper platen — torsion head assembly - K Restoring constant of torsion bar - N ₁ First normal-stress difference - R Platen radius - S ⁽ⁱ⁾ Geometric factor in the dynamic property analysis - t ₁ Characteristic time parameter of the Carreau model - X, Y Carreau model parameters - Z () Reimann Zeta function of - Carreau model parameter - Shear rate - Apparent steady-shear viscosity - ^* Complex dynamic viscosity - Dynamic viscosity - Imaginary part of the complex dynamic viscosity - ₀ Zero-shear viscosity - ₀ Cone angle - Carreau model characteristic time - Density of fluid - Shear stress - Phase difference between torsion head and oscillation input signals - ₀ Zero-shear rate first normal-stress coefficient - Oscillatory frequency     

The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

Summary A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.Nomenclature c /, ft/sec - e base of natural logarithms - F(x, y) integration function, equation (38) - (x) mean value of F(x, y) - g local acceleration of gravity, ft/sec² - h width of conduit, ft - k (2m–1)^{2 2} L/h ² c, equation (35) - k* 12L/h ² c, frictional resistance parameter, equation (46) - L length of conduit, ft - m positive integer - n positive integer - p pressure, lb/ft² - p ₀ constant pressure at inlet of conduit, lb/ft² - P pressure plus elevation head, p+gz, equation (4) - mean value of P over the conduit width h - P ₀ p ₀+gz ₀, lbs/ft² - R frictional resistance coefficient for steady state wall shear, lb sec/ft⁴ - s positive integer; also, condensation, equation (6) - t time, sec - t ct/L, dimensionless time - u x component of fluid velocity, ft/sec - u _m mean velocity in conduit, equation (12), ft/sec - u ₀(y) velocity profile in Poiseuille flow, equation (19), ft/sec - transformed velocity - U initial mean velocity in conduit - x distance along conduit, measured from valve (fig. 1), ft - x x/L, dimensionless distance - y distance normal to conduit wall (fig. 1), ft - y y/h, equation (25) - z elevation, measured from arbitrary datum, ft - z ₀ elevation of constant pressure source, ft - isothermal bulk compression modulus, lbs/ft² - _n , equation (37) - _n (2n–1)/2, equation (36) - viscosity, slugs/ft sec - / = kinematic viscosity, ft²/sec - density of fluid, slugs/ft³ - ₀ density of undisturbed fluid, slugs/ft³ - ø angle between conduit and vertical (fig. 1) The research upon which this paper is based was supported by a grant from the National Science Foundation.     

Hydrodynamic instability of the contact zone between accelerated gases

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

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameter=(Ra) ^1/4/(Re)^1/2, with=Pr/(1+Pr) and=Pr/(1 +Pr)^1/2, is proposed to replace the conventional Richardson number, Gr/Re², for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra ^1/4/(Re ^1/2 Pr ^1/3) forPr 1, and (RaPr)^1/2/(RePr ^1/2 forPr 1. Furthermore, new coordinates and dependent variables are properly defined in terms of, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

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Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameter=( Ra) ^1/4 (Re)^1/2, mit=Pr/(1 +Pr) und=Pr/(1 +Pr)^1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re², ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra ^1/4/(Re ^1/2 Pr ^1/3) fürPr 1 und (RaPr)^1/4/(RePr)^1/2 fürPr 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion von definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Nomenclature C _f local friction coefficient - C _p specific heat capacity - f reduced stream function - g gravitational acceleration - Gr local Grashoff number,g T _w –T )x³/v² - Nu local Nusselt number - Pr Prandtl number,v/ - Ra local Rayleigh number,g T _w –T x ³/( v) - Re local Reynolds number,u x/v - Ri Richardson number,Gr/Re ² - T fluid temperature - T _w wall temperature - T free stream temperature - u velocity component in thex direction - u free stream velocity - v velocity component in they direction - x vertical coordinate measuring from the leading edge - y horizontal coordinate Greek symbols thermal diffusivity - thermal expansion coefficient - mixed convection parameter (Ra)1/4/Re)^1/2 - pseudo-similarity variable,(y/x) - ₀ conventional similarity variable,(y/x)Re ^1/2 - dimensionless temperature, (T–T T _W –T - unified mixed-flow parameter, [(Re) ^1/2 + (Ra)^1/4] - dynamic viscosity - kinematic viscosity - stretched streamwise coordinate or mixed convection parameter, [1 + (Re)^1/2/(Ra) ^1/4]^–1=/(1 +) - density - Pr/(1 + Pr) _w wall shear stress - stream function - Pr/(l+Pr)^1/3 This research was supported by a grand from the National Science Council of ROC     

Einfluß der Beheizungsart auf den Wärmeübergang im horizontalen Verdampferrohr

Zusammenfassung Durch die Phasenverteilung von Flüssigkeit und Dampf ergeben sich im horizontalen Verdampferrohr i.a. unterschiedliche lokale Wärmeübergangskoeffizienten (WÜK) am Rohrumfang. Die beiden technisch vorkommenden Randbedingungen — konstante Wärmestromdichte und konstante WandemperaturT _W am Rohrinnendurchmesser — können unterschiedliche Auswirkungen auf umfangsgemittelte WÜK haben. Während bei konstanter aufgeprägter Wärmestromdichte ein interner Transport der zu übertragenden Wärmemenge durch die Wärmeleitung der Wand erfolgen kann, ist dies beiT _W =konst. nicht möglich. Die Ergebnisse zeigen, daß sich der umfangsgemittelte WÜK bei konstanter Wärmestromdichte gegenüber dem Wert bei konstanter Wandtemperatur verringert. Dies ist jedoch nur dann der Fall, wenn der Rohrumfang unvollständig benetzt ist. Dabei hängt die Verminderung des umfangsgemittelten WÜK stark vom Wärmeleitvermögen der Wand, s, ab ( Wärmeleitfähigkeit,s Wandstärke). Es wurde festgestellt, daß die Verminderung sowohol aus der höheren Wandtemperatur am Rohrscheitel als auch aus der Änderung des benetzten Umfangs resultiert, da dann der Anteil des Rohrumfangs mit schlechtem Wärmeübergang vergrößert wird.

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Effect of thermal boundary conditions on heat transfer in a horizontal evaporator tube

As a result of the non uniform phase distribution of liquid and vapour local heat transfer coefficients vary considerably along the circumference in a horizontal evaporator tube. It was expected that both modes of heating, uniform heat flux ( = const) and uniform inside wall temperature (T _W =const), lead to different circumferential averaged heat transfer coefficients (HTC). The experiments show that the circumferential averaged HTC is lower in the case of uniform heat flux compared to the value of uniform wall temperature if the perimeter of the tube is only partly wetted. However, the reduction is affected by circumferential heat conduction. This reduction is a result of both, the higher wall temperature on the top of the tube and the reduction of the wetted perimeter.

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Formelzeichen a Verhältnis von -Werten - c _p spezifische Wärmekapazität - C _F Faktor - d Innendurchmesser - g Fallbeschleunigung - h _v spezifische Verdampfungsenthalpie - Massenstromdichte - M Rippenkenngröße - n Exponent der Wärmestromdichte - Nu Nusselt-Zahl - p Druck - p _k Kritischer Druck - p* reduzierter Druck (p/p _k ) - Pr _L Prandtl-Zahl Flüssigkeit ( c _p /) - Pr _G Prandtl-Zahl Dampf ( c _p /) - Wärmestromdichte - r Radius - Re Reynolds-Zahl - R _p Glättungstiefe nach DIN 4762 - s Wandstärke - T Temperatur - Strömungsdampfgehalt - z Koordinate in Strömungsrichtung - Wärmeübergangskoeffizient - Differenz - Dampfvolumenanteil - dynamische Viskosität - Faktor - Wärmeleitfähigkeit - Druckverlustbeiwert - Dichte - Oberflächenspannung - normierter unbenetzter Bogen - Zentriwinkel - unbenetzter Bogen - Funktion - Faktor Indizes B Blasensieden - G Dampf - G0 gesamte Massenstromdichte als Dampf - k konvektives Strömungssieden - kr kritische Größe - L Flüssigkeit - Lb mit Flüssigkeit benetzt - L0 gesamte Massenstromdichte als Flüssigkeit - s Siedezustand - W Wand - gesättigte Flüssigkeit - gesättigter Dampf - – Mittelwert Herrn Prof. Dr.-Ing. Dr. h.c./INPL E.-U. Schlünder zum 60. Geburtstag gewidmet     

Bestimmung der Schubspannungsfunktion des Blutes mit dem Couette-Rheometer unter Berücksichtigung des Wandverhaltens

Zusammenfassung Bei Blutergeben sich aus Messungen an verschiedenen Couette-Systemen verschiedene Verläufe für die Schubspannungsfunktion, wenn man Wandhaften annimmt. Es wird daher ein Wandeffekt angenommen, bei dem die unentmischte Blutsuspension unmittelbar auf der Wand gleitet. Die Wandgleitgeschwindigkeit wird als Funktion der Wandschubspannung angesetzt und aus Messungen an drei verschiedenen Couette-Systemen bestimmt.Aus der so ermittelten Wandgleitgeschwindigkeit kann die Dicke eines Blutplasmafilmes an der Wand abgeschätzt werden. Sie ergibt sich zu einigen µm. Dadurch wird die angenommene Modellvorstellung für den Wandeffekt bestätigt.Bei Berücksichtigung der ermittelten Wandgleitgeschwindigkeit ergibt sich für die Schubspannungsfunktion aus Messungen an verschiedenen Couette-Systemen derselbe Verlauf. Bei Annahme von Wandhaften ergeben sich dagegen deutlich zu hohe Werte für die Schubspannungsfunktion.

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Summary Measurements with blood in different Couette-Systems are resulting in different shear functionsf() if no wall-slip effect is assumed.Therefore we use the model that the homogeneous blood suspension is sliding directly on the wall. The wall-slip velocity is introduced as a function of the wall shear stress. This wall-slip function can be determined from measurements with three different Couette-Systems.After the wall-slip function is determined the thickness of a plasma film on the wall can be estimated. One gets a thickness of a few µms. Thus the assumed model for the wall effect is confirmed. Measurements with different Couette-Systems evaluated according to the wall-slip model, are leading to the same shear functionf().

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D Rohrdurchmesser - f() Schubspannungsfunktion - f() mittlerer Fehler vonf() - h Länge des Couette-Spaltes - M in der Couette-Strömung übertragenes Drehmoment - M _ij Meßwerte fürM - Mittelwert derM _ij - mittlerer Fehler von - r _a ,r _i äußerer bzw. innerer Begrenzungsradius eines Couette-Systems - r ₁,r ₂,r ₃ Begrenzungsradien der Couette-Systeme I, II, III - s Spaltweite im Couette-System - v _w Wandgleitgeschwindigkeit - Schergeschwindigkeit - Dicke des Wandfilms - ( _w) Funktion des Wandgleitens - Viskosität - Schubspannungsfeld im Couette-System - ₁, ₂, ₃ Schubspannungen an den Stellenr ₁,r ₂,r ₃ in den Couette-Systemen I, II, III bei gleichem Moment - _w Wandschubspannung - Winkelgeschwindigkeitsdifferenz zwischen äußerem und innerem Zylinder im Couette-System - _i vorgegebene Werte für - mittlerer Fehler von - _I, _II, _III für die Couette-Systeme I, II, III - _{I, II, III} mittlere Fehler von _I, _II, _III Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Aachen vom 5.–7. März 1979.Mit 5 Abbildungen     

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

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

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure     

Effect of nonlinear constitutive equation on vibrations of beams

Summary This paper is concerned with the analytical investigation of static and dynamic nonlinear behaviors of beams with different boundary conditions. While geometric type of nonlinearities on beams have been investigated extensively, material type nonlinearities have received very little attention. Therefore, material nonlinearities of the Ramberg-Osgood type are considered in this analysis. The use of Self-Generating functions for nonlinear beam problems is demonstrated for this type of nonlinearity. Transverse shear and rotatory inertia effects have been included in the formulation to study moderately thick beams. For all the cases investigated here nonlinear frequency ratios are calculated at various amplitudes of vibration and geometric parameters of beams. Numerical results indicate that the Ramberg-Osgood type nonlinearity produces softening-type responses. The study is limited to materials which are nonlinearly elastic and the effect of geometric nonlinearity is not considered in this paper.

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Sommario Questo lavoro riguarda lo studio dei comportamenti non lineari statici e dinamici di travi con diverse condizioni a contorno. Mentre le non linearità di tipo geometrico sono state studiate estesamente, quelle di tipo non lineare hanno ricevuto un'attenzione molto ridotta. Perciò in questa ultima analisi si considerano non linearità materiali del tipo di Ramberg-Osgood. Si dimostra uso delle funzioni autogeneratrici nei problemi non lineari per le travi con questo tipo di non linearità. Nella formulazione dello studio di travi moderatamente spesse si sono inclusi effetti di taglio e inerzia rotatoria. Per tutti i casi qui studiati si calcolano i rapporti di frequenza non lineare per varie ampiezze di vibrazione e parametri geometrici delle travi. I risultati numerici indicano che la non linearità del tipo di Ramberg-Osgood produce risposte del tipo ammorbidimento. Lo studio si limita a materiali con nonlinearità elastica e non si considera nel lavoro l'effetto della non linearitá geometrica.

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List of symbols T _s Transverse shear - k ₁ Shear connection factor - A, B, m Material constants - I Moment of inertia - G Shear modulus - Mass per unit length - w Lateral displacement - h Beam thickness - b Bredth of beam - t Time - R _i Rotatory inertia - a Area of cross-section of beam - q(x) Lateral load on beam - _x Stress - _x Strain - Length of beam - x Beam coordinate - - r Radius of gyration - w Nondimensional maximum deflectionw _max/r - q ₀ ^* Nondimensional load, (q ₀ ³/Al - Thickness parameter,h/ - (T _sK₁/Ga) - ₀ Linear frequency - Nonlinear frequency - w _max w measured at the point of maximum deflection.