Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

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

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

Zum Mischen rheologisch inhomogener Stoffsysteme auf Einschneckenmaschinen

Zusammenfassung Das Mischen von Stoffen mit unterschiedlichen rheologischen Eigenschaften in Schneckenmaschinen ist in der Kunststoffauf- und -verarbeitung eine Standardaufgabe. Trotzdem gibt es hierfür kein zufriedenstellendes mathematisch-physikalisches Modell. Daher werden zunächst einfache Mischmodelle diskutiert. Auf der Basis dieser Modelle wird dann unter Berücksichtigung der Besonderheiten des Plastifizierextruderprozesses eine Mischgütebeziehung mathematisch formuliert. Die experimentelle Überprüfung erfolgt mit Hilfe der Grauwertanalyse extrudierter Zweistoffsysteme, bei denen ein Stoff mit Ruß eingefärbt war. Da der Mischprozeß hochgradig stochastisch ist, streuen die Meßergebnisse. Unter Berücksichtigung dieses Tatbestandes ist der theoretische Ansatz zufriedenstellend.

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Mixing of polymer resins with different rheological properties is a usual demand in plastics processing using screw extruders. A mathematical model describing this processing problem sufficiently is not known, however. Therefore, simple mixing models will be discussed. Based on these, a concept for the calculation of mixing homogeneity will be presented, including the particular requirement of the plasticating screw process. An experimental investigation utilizes the grey-value analysis of extruded two-component materials, which in one phase is carbon-black filled. Considering the fact that the mixing process is highly random, the theoretical model leads to a good level of aggreement with the scattering measurement data.

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b Schneckenkanalbreite - B Bandbreite der Grauwerte - c Konstante - mittlere Konzentration, bezogen auf die Grauwertbandbreite - h Höhe, Gangtiefe, Schneckenkanalhöhe - h ₀ Gangtiefe der Einzugszone - h ₁ Gangtiefe der Ausstoßzone - L Länge - gemittelte Schmelzebettlänge - n Exponent des Potenzfließgesetzes - s Standardabweichung der Grauwerte bezogen auf die Grauwertbandbreite - S Standardabweichung der Grauwerte - t Verweilzeit - t ₁ kürzeste Verweilzeit - mittlere Verweilzeit - ₀ Umfangsgeschwindigkeit - mittlere Geschwindigkeit - V Volumenstrom - w Dicke eines Kontrollelements - w Ausstreichdicke eines Kontrollelements - x Koordinate - Mittelwert der Grauwerte - y Koordinate - Scherdeformationswinkel - Scherdeformation - mittlere Scherdeformation - Schergeschwindigkeit - Viskosität - ₁ dimensionslose kürzeste Verweilzeit - dimensionsloser Volumenstrom - _LSM laminarer Schermischgrad - _{LSM, the} theoretischer laminarer Schermischgrad - _{LSM, exp} experimenteller laminarer Schermischgrad - ² Varianz der Verweilzeit im Schmelzebett - Schubspannung - Gangsteigungswinkel der Schnecke - ø Volumenanteil - dimensionslose Kennzahl     

On the effects of uniform high suction on the rotationally symmetric flow of a conducting liquid near a stationary disc in the presence of a transverse magnetic field

In the present paper an attempt has been made to find out effects of uniform high suction in the presence of a transverse magnetic field, on the motion near a stationary plate when the fluid at a large distance above it rotates with a constant angular velocity. Series solutions for velocity components, displacement thickness and momentum thickness are obtained in the descending powers of the suction parameter a. The solutions obtained are valid for small values of the non-dimensional magnetic parameter m (= 4 _e ² H ₀ ² /) and large values of a (a2).Nomenclature a suction parameter - E electric field - E _r , E , E _z radial, azimuthal and axial components of electric field - F, G, H reduced radial, azimuthal and axial velocity components - H magnetic field - H _r , H , H _z radial, azimuthal and axial components of magnetic field - H ₀ uniform magnetic field - H* displacement thickness and momentum thickness ratio, */ - h induced magnetic field - h _r , h , h _z radial, azimuthal and axial components of induced magnetic field - J current density - m nondimensional magnetic parameter - p pressure - P reduced pressure - R Reynolds number - U ₀ representative velocity - V velocity - V _r , V , V _z radial, azimuthal and axial velocity components - w ₀ uniform suction through the disc. - density - electrical conductivity - kinematic viscosity - _e magnetic permeability - a parameter, (/)^1/2 z - a parameter, a - * displacement thickness - momentum thickness - angular velocity     

Discretized ordinary differential equations and the Conley index

LetN be a compact isolating neighborhood of an isolated invariant setK with respect to an ODEx=f(x) (C) and(h) x=x + h(x, h) be a consistent one-step-discretization of (C). It is proved in this paper that for someh ₀ > 0 and allh ]0, h₀[, the setN isolates an invariant setK(h) of(h) and the discrete Conley index ofK(h) coincides with the continuous Conley index ofK.     

A noninvasive method for the measurement of flow-induced surface displacement of a compliant surface

A noninvasive optical method is described which allows the measurement of the vertical component of the instantaneous displacement of a surface at one or more points. The method has been used to study the motion of a passive compliant layer responding to the random forcing of a fully developed turbulent boundary layer. However, in principle, the measurement technique described here can be used equally well with any surface capable of scattering light and to which optical access can be gained. The technique relies on the use of electro-optic position-sensitive detectors; this type of transducer produces changes in current which are linearly proportional to the displacement of a spot of light imaged onto the active area of the detector. The system can resolve displacements as small as 2 m for a point 1.8 mm in diameter; the final output signal of the system is found to be linear for displacements up to 200 m, and the overall frequency response is from DC to greater than 1 kHz. As an example of the use of the system, results detailing measurements obtained at both one and two points simultaneously are presented.List of symbols C _t elastic transverse wave speed = (G/)^1/2 - d ⁺ spot diameter normalized by viscous length scale - G frequency average of G() - G() shear storage modulus - G() shear loss modulus - l. viscous length scale = v/u _* - N total number of sampled data values - r separation vector for 2-point measurements = (, ) - rms root-mean-square value - R momentum thickness Reynolds number = U _t8/v - t time - u (y) mean streamwise component of velocity in boundary layer - u _* friction velocity = (t _w/)^1/2 - U free-stream velocity - x, y, z longitudinal, normal and spanwise directions - y _o undisturbed surface position - vertical component of compliant surface displacement - ₉₉ boundary layer thickness for which u(y) = 0.99 U _t8 - _l viscous sublayer thickness 5 l _* - frequency average of G()/ - boundary layer momentum thicknes = - fluid dynamic viscosity - v fluid kinematic viscosity = / - , longitudinal, spanwise components of separation vector r - fluid density - time delay - _w wall shear stress     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

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     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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.     

Motion equations for a discrete structure fluid

The Maxwell method allows us to carry out a transition to the equations of hydrodynamics at any function of a molecule distribution with respect to velocities. This is the advantage of the above method. However, that the Navier-Stokes equations be obtained, Maxwell had to assume that in a physically small volume, over which the averaging is made, the transport velocities of two impinging molecules are equal. So Maxwell identified a hydrodynamic velocity with a mean velocity of atoms or molecules. — This Maxwell hypothesis may appear to be doubtful in a rarefied gas where the mean free path of molecules is commensurable with the dimensions of a physically small volume and a body in a flow and in eddy flows.In the present paper the equations of hydrodynamics with arbitrary transport velocities of two impinging molecules are derived. The formulae for a stress tensor in this case have the form: i, j=1, 2, 3.From formula (1), in a particular case, it is possible to obtain the equations of hydrodynamics in the form proposed by A. S. Predvoditelev and N. P. Kasterin. — The apparent motions may make a certain contribution to thermal motions of atoms or molecules. Then formulae (1) allow generalization of the state equation of Clapeyron as follows: p=(1–) RT here is the Predvoditelev parameter for non-ideal continuity. With the help of the above formula the Maxwell formula for a slip coefficient was generalized. — Moreover, an attempt was made to develop these ideas conformably to the non-Newtonian fluids. In this case the model of the general Stokes fluid which was for the first time proposed by Truesdell was taken as a basis.

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Zusammenfassung Nach der Maxwell-Methode können wir eine Umformung der hydrodynamischen Gleichungen hinsichtlich der Geschwindigkeiten durchführen für beliebige Funktionen der Molekülverteilung. Das ist der Vorteil dieser Methode. Um aber die Navier-Stokes-Gleichungen zu erhalten, mußte Maxwell annehmen, daß in dem kleinen Teilvolumen, über das gemittelt wird, die Geschwindigkeiten zweier sich stoßender Moleküle gleich sind. So setzte Maxwell eine hydrodynamische Geschwindigkeit gleich einer mittleren Atom-oder Molekülgeschwindigkeit. — Diese Maxwell-Hypothese kann zweifelhaft erscheinen für verdünnte Gase, für welche die mittlere freie Weglänge der Moleküle vergleichbar ist mit den Abmessungen des Teilvolumens und eines Körpers in der Strömung und in Wirbelströmungen. — Hier werden die hydrodynamischen Gleichungen für beliebige Geschwindigkeiten zweier sich stoßender Moleküle hergeleitet. Die Gleichungen für den Spannungstensor haben die Form i,j=1,2,3.Aus den Gln. (1) lassen sich für einen besonderen Fall die hydrodynamischen Beziehungen in der von A. S. Predvoditelev und N. P. Kasterin vorgeschlagenen Form erhalten. - Die sichtbaren Bewegungen können gewisse Zuschläge zu den thermischen Bewegungen der Atome oder Moleküle verlangen. Die Gln. (1) gestatten dann die Verallgemeinerung der Clapeyron-Gleichung nach p=(1–)RT wobei den Predvoditelev-Parameter darstellt für nichtideale Kontinuität. — Mit obiger Gleichung wurde die Maxwell-Beziehung für einen Schlupfkoeffizient verallgemeinert.Darüberhinaus wurde versucht diese Gedanken entsprechend für Nicht-Newtonsche Flüssigkeiten zu entwickeln. Dafür wurde das Modell der Stokesschen Flüssigkeit, das erstmalig von Truesdell vorgeschlagen wurde, herangezogen.

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Nomenclature V hydrodynamic velocity vector - density m mass of a molecule - Q quantitative expression of any quality - ¦¦=Pd Predvoditelev number - Tr Truesdell number - adiabatic index - V_m volume of one mole of a gas - m₀ molecular weight of a gas - R universal gas constant - U_i i-th velocity component of molecular motion - W_il i-th velocity component of the first molecule where W_il=v _i +u_i - W_i2 i-th velocity component of the second molecule when W_i2=V _i + u_i - V _i ^t transport velocity of the first molecule - V _i ^t transport velocity of the second molecule Dedicated to Prof. Dr.-Ing. U. Grigull on his 60th birthday.     

A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

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Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

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A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Numerical calculation of creep compliance from dynamic data for linear viscoelastic materials

Summary Numerical formulae are given for calculation of creep compliance from the known course of the storage and loss compliance with frequency for linear viscoelastic materials. These formulae involve values of the storage compliance and/or loss compliance at frequencies which are equally spaced on a logarithmic frequency scale. The ratio between successive frequencies corresponds to a factor of two.A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tan, at the angular frequency, ₀, at which the calculation is performed. The lower this damping, the easier is the calculation of the creep compliance. This calculation involves either the value of the storage compliance at a frequency ₀ = 1/t, and the values of the loss compliance in a rather narrow frequency region around ₀; or the value of the storage compliance at frequency ₀, the value of the loss compliance at frequency ₀/2, and the derivative of the storage compliance with respect to the logarithm of frequency in a frequency region around ₀.

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Zusammenfassung Numerische Formeln werden gegeben, die die Berechnung der Kriechfunktion aus der dynamischen Nachgiebigkeit ermöglichen. In diesen Formeln treten Werte der Speicher- bzw. Verlustkomponente der dynamischen Nachgiebigkeit auf, die bei logarithmisch äquidistanten Frequenzen gemessen wurden. Das Verhältnis zweier aufeinanderfolgender Frequenzen entspricht stets einem Faktor 2.Für alle Formeln werden obere und untere Schranken für den relativen Fehler abgeleitet. Diese Schranken hängen vom Werte der Dämpfung (tan) ab, die bei der Kreisfrequenz ₀ auftritt, für die die Berechnung erfolgt. Die Berechnung der Kriechfunktion ist desto leichter, je niedriger der Wert der Dämpfung ist. Zu dieser Berechnung benötigt man entweder den Wert der Speicherkomponente der dynamischen Nachgiebigkeit bei der Kreisfrequenz ₀ = 1/t und die Werte der Verlustkomponente der dynamischen Nachgiebigkeit in einem ziemlich engen Frequenzintervall um ₀; oder den Wert der Speicherkomponente bei der Kreisfrequenz ₀, den Wert der Verlustkomponente bei der Kreisfrequenz ₀/2 und den Wert der logarithmischen Frequenzableitung der Speicherkomponente in einem Frequenzintervall um ₀.

     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation

Summary Previous work on the creeping flow of viscoelastic fluids past a sphere is reviewed. Theoretical analyses available in the literature were obtained for weakly elastic fluids and therefore they predict only a small influence of fluid elasticity on the drag. In this paper, an approximate theoretical analysis is given for the creeping flow past a rigid sphere in an unbounded medium. The analysis uses a variational principle to solve the equations of motion and continuity in conjunction with the Carreau constitutive equation. The theoretical results are presented in terms of a correction factor to the Newtonian drag coefficient. The correction factor is a function of the power law flow behaviour indexn, the ratio of limiting viscosities ( ₀ – )/₀ and a dimensionless time which reflects the elastic nature of the fluids. The results are presented in graphical form covering a realistic range of these dimensionless groups.In order to verify the theoretical predictions, the drag coefficient of a number of spheres was measured in a series of shear thinning elastic test fluids. The flow properties of the test fluids were independently measured with a Weissenberg Rheogoniometer. The power law index of the test fluids varied between 1.0 and 0.4. Particle Reynolds number based on ₀ was in the range of 410^–6 to 410^–2. The difference between theoretically predicted values of drag coefficient and the experimentally measured values is less than ±7.5%. In addition, it is found that the Carreau viscosity equation can be used to predict the elastic parameter of primary normal stress difference with moderate to good accuracy for all the polymer solutions used in this work.

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Zusammenfassung Einleitend wird ein Überblick über die früheren Untersuchungen betreffend die schleichende Strömung um eine Kugel gegeben. Die in der Literatur vorliegenden theoretischen Analysen sind auf schwach viskoelastische Flüssigkeiten beschränkt und sagen deshalb nur einen geringen Einfluß der Elastizität auf den Widerstand voraus. In dieser Veröffentlichung wird dagegen eine genäherte theoretische Analyse für die schleichende Strömung um eine starre Kugel in einem unendlich ausgedehnten Medium gegeben, bei welcher zur Lösung der Bewegungsgleichungen und der Kontinuitätsgleichung in Verbindung mit den rheologischen Stoffgleichungen vonCarreau ein Variationsprinzip verwendet wird. Die theoretischen Ergebnisse werden mittels eines Korrekturfaktors zum newtonschen Widerstandskoeffizienten beschrieben. Dieser Korrekturfaktor ist eine Funktion des Potenz-Gesetz-Exponentenn, des Verhältnisses der Grenzviskositäten ( ₀ – )/₀ und einer dimensionslosen Zeit, welche das elastische Verhalten kennzeichnet. Die Ergebnisse werden in graphischer Form unter Zugrundelegung eines realistischen Wertebereichs dieser dimensionslosen Gruppen dargestellt.Um diese theoretischen Voraussagen zu verifizieren, wurde der Widerstandskoeffizient für eine Anzahl von Kugeln in einer Reihe von Scherentzähung aufweisenden elastischen Probeflüssigkeiten gemessen. Die Fließeigenschaften dieser Flüssigkeiten wurden zusätzlich mit dem Weissenberg-Rheogoniometer bestimmt. Der Potenz-Gesetz-Exponent variierte dabei zwischen 1,0 und 0,4. Die auf den Kugeldurchmesser und die Nullviskosität bezogenen Reynolds-Zahlen lagen zwischen 410^–6 und 410^–2. Der Unterschied zwischen theoretisch vorausgesagten und experimentell bestimmten Widerstandskoeffizienten war kleiner als ±7,5%. Außerdem wurde noch gefunden, daß die Viskositätsgleichung vonCarreau dazu verwendet werden kann, den elastischen Parameter erste Normalspannungs-Differenz für alle in dieser Untersuchung verwendeten Polymerlösungen mit mäßiger bis guter Genauigkeit vorauszusagen.

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Notation C _d drag coefficient - d diameter of sphere - f external body forces in equation of motion [2] - F _d drag force - g acceleration due to gravity - J integral defined in eq. [3] - n a parameter in the Carreau viscosity eq. [6] - p isotropic pressure term in equation of motion [2] - r,, spherical coordinates - R radius of sphere - Re ₀, Re₁ Reynolds numbers defined in eq. [16] - t time - u ⁱ ,u ^j velocities in equation of motion [2] - u _r ,u r and components of velocity - V terminal velocity of sphere in unbounded medium - V volume, in eq. [3] - X correction factor to the drag force, eq. [14] - y,z dimensionless spherical coordinates, eq. [9] - ratio of two Reynolds numbers given by eq. [16] - shear rate - apparent viscosity - ₀, zero shear rate and infinite shear rate viscosities respectively - a parameter in the Carreau viscosity eq. [6] - the dimensionless time, defined in eq. [11] - second invariant of the rate of deformation tensor - a parameter in the stream function, eq. [8] - stream function - _p,_f densities of sphere and fluid respectively With 7 figures and 1 table     

Dehnströmungen mit verdünnten Polymerlösungen: Ein theoretisches Modell und seine experimentelle Verifikation

Zusammenfassung Es werden theoretische Überlegungen zusammenfassend dargestellt, welche die Streckung und Ausrichtung von flexiblen Makromolekülen in stationären einfachen Dehnströmungen beschreiben. Die Makromoleküle werden hierbei als EDNE-(endlich dehnbare, nichtlinear elastische) Hanteln modelliert. Für den Fall niedriger bzw. hoher Dehnungsraten werden Dehnviskositätsgleichungen für Strömungen mit verdünnten Polymerlösungen angegeben.Die Arbeit vergleicht die abgeleiteten theoretischen Gleichungen mit experimentellen Ergebnissen, welche für Porenströmungen erhalten wurden; Porenströmungen weisen Dehnströmungen auf. Anhand der durchgeführten experimentellen Untersuchungen, in denen alle die den Druckverlust maßgebend beeinflussenden strömungsmechanischen und physikalisch-chemischen Parameter variiert wurden, kann gezeigt werden, daß sich die aufgezeigten theoretischen Zusammenhänge quantitativ bestätigen lassen.Schlüsselwörter Dehnströmung, Makromolekülmodell, Porenströmung, EDNE-Hantelmodell, Polymerlösung

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Summary The present paper summarizes theoretical considerations regarding the elongation of flexible macromolecules in simple steady elongational flows. The macromolecules are treated as FENE(finite extensible, nonlinear elastic)-dumbbells. Equations for extensional viscosity are given for flows of dilute polymer solutions applicable at low and high elongation rates.The present paper compares the derived theoretical relationships with experimental results. These results were obtained in porous media flows, which exhibit strong elongational rates. It can be shown on the basis of the experimental investigations, that all fluid mechanic and physico-chemical parameters that influence the measured pressure losses responded as predicted by the theory.

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a Mark-Houwink-Exponent - A Avogadro-Konstante - b Verhältnis von Molekülzeitkonstanten - c Polymergewichtskonzentration - d Kugeldurchmesser der Schüttung - D Diffusionskonstante - De Deborahzahl - f Reibungsbeiwert der Porenströmung - F Kraftvektor des Hantelmodells - g Erdbeschleunigung - H Hookesche Federkonstante des Makromoleküls - k Boltzmann-Konstante - k _1,2,3 empirische Konstanten - K Mark-Houwink-Konstante - l ₀ Länge des Monomeren - L Länge des statistischen Fadenelementes - L ₀ Maximallänge des gestreckten Polymermoleküls - L Bezugslänge für den Druckverlust der Porenströmung - m Masse des statistischen Fadenelementes - m ₀ Masse des Monomeren - Molarität - M Molekulargewicht des Polymeren - n Porosität der Kugelschüttung - n ₀ Hantelkonzentration - N Anzahl der statistischen Fadenelemente - p Druckverlust der Porenströmung - P Polymerisationsgrad - R Endpunktabstand des Makromoleküls - R ₀ maximaler Endpunktabstand des gestreckten Moleküls - mittlerer Endpunktabstand des Moleküls - Orientierungsvektor des Hantelmodells - Re Reynoldszahl der Porenströmung - t Zeit - T Temperatur - mittlere Filtergeschwindigkeit der Porenströmung - v Strömungsfeld - Aufweitungsparameter - Bindungswinkel zweier Kohlenstoffatome - Dehnungsrate - Stokesscher Reibungsfaktor - dynamische Viskosität - ^* reduzierte Viskosität - [] Grenzviskositätszahl - Dehnviskosität - ^* reduzierte Dehnviskosität - Widerstandskennzahl der Porenströmung - v kinematische Viskosität - Dichte des Fluids - _H Hookesche Relaxationszeit des EDNE-Hantelmodells - _H,e Hookesche Relaxationszeit des linear elastischen Hantelmodells - _R Relaxationszeit des starren Hantelmodells - _zz , _yy Normalspannungen - Volumenkonzentration - _fl. dimensionsloser Faktor des Strömungsfeldes - ₀ Konstante der Flory-Fox-Gleichung - Verteilungsfunktion des Hantelmodells - _eq. Gleichgewichtsverteilungsfunktion - a aufgeweitet - e effektiv - max maximal - p polymer - s solvent, Lösungsmittel - Theta-Zustand Mit 12 Abbildungen und 2 Tabellen     

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