Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Relations between steady flow and oscillatory shear measurements

Summary Results are given of a comparison between dynamic oscillatory and steady shear flow measurements with some polymer melts. Comparison of the steady shear flow viscosity,, with the absolute value of the dynamic viscosity, ¦¦, at equal values of the shear rate,q, and the circular frequency,, has shown the relation thatCox andHerz had found empirically to be substantially correct.Further, the coefficients of the normal stress differences obtained by streaming birefringence techniques have been compared with 2G () · ^{– 2} in the same range of shear rates as covered by the viscosity measurements (G is the real part of the dynamic shear modulus). Two polystyrenes with narrow molecular weight distribution showed the same shift factor along the orq axis for the normal stress coefficients with respect to 2G () · ^{– 2} and the steady shear flow viscosities with respect to the real part of the dynamic viscosity,. For two polyethylenes the results are not so conclusive owing to the smallness of the shift factor found. An empirical equation is proposed predicting the main normal stress difference from dynamic measurements only.

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Zusammenfassung Die Ergebnisse von Messungen unter erzwungenen Schwingungen und stationärer Scherströmung an einigen Polymerschmelzen werden miteinander verglichen. Der Vergleich der stationären Viskosität mit der absoluten dynamischen Viskosität ¦¦ bei gleichen Werten des Strömungsgradientenq und der Kreisfrequenz zeigt die Gültigkeit der empirischen Beziehung vonCox undHerz.Weiter wurden die Koeffizienten der Normalspannungsdifferenzen, welche durch Messung der Strömungsdoppelbrechung erhalten wurden, mit 2G() · ^–2 verglichen, und zwar wiederum bei gleichen Werten vonq und, wobeiG die Speicherkomponente des dynamischen Schubmoduls ist. Zwei Polystyrole mit enger Molekulargewichtsverteilung zeigen die gleiche Verschiebung entlang der-oderq-Achse für die Normalspannungskoeffizienten in bezug auf2G()· ^–2 und für die stationären Scherviskositäten in bezug auf den Realteil der dynamischen Viskosität. Für zwei Polyäthylene sind die Ergebnisse weniger signifikant, da die entsprechenden Verschiebungen zu klein waren. Eine empirische Beziehung zwischen den Hauptnormalspannungsdifferenzen und den dynamischen Meßwerten wird vorgeschlagen.

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Paper presented at the British Society of Rheology Conference, held at Shrivenham, from 9th–12th September, 1968.     

Shear softening and thixotropic properties of wheat flour doughs in dynamic testing at high shear strain

Shear softening and thixotropic properties of wheat flour doughs are demonstrated in dynamic testing with a constant stress rheometer. This behaviour appears beyond the strictly linear domain (strain amplitude ₀ 0.2%),G,G and |^*| decreasing with ₀, the strain response to a sine stress wave yet retaining a sinusoidal shape. It is also shown thatG recovers progressively in function of rest time. In this domain, as well as in the strictly linear domain, the Cox-Merz rule did not apply but() and | ^*())| may be superimposed by using a shift factor, its value decreasing in the former domain when ₀ increases. Beyond a strain amplitude of about 10–20%, the strain response is progressively distorted and the shear softening effects become irreversible following rest.     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Pseudoähnlichkeitslösung des Rayleighschen Problems für reinviskose nicht-newtonsche Flüssigkeiten

Zusammenfassung Die exakte Ähnlichkeitslösung des Problems der nichtstationären Strömung einer hypothetischen Potenzflüssigkeit in der Umgebung einer ruckartig beschleunigten Platte (Rayleighsches Problem) wird zur Konstruktion einer Näherungslösung des analogen rheodynamischen Problems für reinviskose nicht-newtonsche Flüssigkeiten benutzt. Die Überprüfung der Genauigkeit der genäherten Pseudoähnlichkeitslösung basiert auf Berechnungen der Residuen der integralen Bilanzen des Impulses und der mechanischen Energie. Numerische Ergebnisse dieses Problems werden für das Powell-Eyringsche Modell der Viskositätsfunktion angegeben.

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Summary The exact similarity solution of the problem of the unsteady flow of a hypothetic power-law liquid near a suddenly started plate (Rayleigh's problem) is employed for the construction of an approximative solution of the same problem for arbitrary purely viscous non-Newtonian liquids. The testing of accuracy of this approximative pseudosimilarity solution is based on calculation of residua in the macroscopic balances of momentum and mechanical energy. Numerical results are reported for the Powell-Eyring model of the viscosity function.

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Symbole a Parameter der Ähnlichkeitslösung, definiert durch die Gln. [21] und [22],a =C ₀/B ₀ - B ₀ Parameter der Ähnlichkeitslösung, definiert durch Gl. [22] - B ₁ Parameter der Ähnlichkeitslösung, definiert durch Gl. [37] - C ₀ Parameter der Ähnlichkeitslösung, definiert durch Gl. [21] - D Differentialoperator, Gl. [10a, b] - Ey Kennzahl der rheologischen Ähnlichkeit für das Powell-Eyringsche Modell der Viskositätsfunktion, Gl. [43]G = / _I G _w = _w/ _I - K Konsistenzkoeffizient, Parameter des Potenzmodells der Viskositätsfunktion [7] - K scheinbarer Konsistenzkoeffizient, Gl. [23b] - n Fließindex des Potenzmodells der Viskositätsfunktion [7] - n scheinbarer Fließindex - n Index der logarithmischen Konvexität der Viskositätsfunktion - r unabhängige Veränderliche in der Ähnlichkeitslösung des Rayleighschen Problems, Gl. [17] - r unabhängige Veränderliche der Pseudoähnlichkeitslösung, Gl. [26]S = / _I S _w = _w/_I - t Zeit - T normierte Zeitvariable, Gl. [9c] - u ₀ Geschwindigkeit der Platte - U (Y, T) Pseudoähnlichkeitsnäherung des FeldesV (Y, T) - V (Y, T) normiertes Geschwindigkeitsfeld, Gl. [9a — c] - Y normierte Entfernung von der Platte, Gl. [9b] - W(r; n) Ähnlichkeitsdarstellung des Geschwindigkeitsfeldes, Gl. [20a, b] - Schergeschwindigkeit, Gl. [4] - _I Schergeschwindigkeits-Parameter der Viskositätsfunktion (Stoffkonstante) - _w momentane Schergeschwindigkeit an der Wand - _M normiertes Residuum der Impulsbilanz - _E normiertes Residuum der Bilanz der mechanischen Energie - dimensionsloser Parameter, definiert durch die Gln. [13] und [14] - dimensionsloser Parameter, definiert durch Gl. [23c] - Dichte - ₁₂, Schubspannung bei einer viskometrischen Strömung - [] Viskositätsfunktion - _I Schubspannungs-Parameter der Viskositätsfunktion (Stoffkonstante) - _w momentane Schubspannung an der Wand Mit 2 Abbildungen     

A note on the tan δ(T) peak as a glass transition indicator in biosolids

The plasticization of many biosolids can take place over a fairly broad temperature range. The resulting loss of stiffness is primarily expressed by a drastic drop of G(T) whose magnitude is usually higher than G(T) by one or two orders of magnitude. Both G(T) and G(T) have characteristic properties that can vary widely among biomaterials. Consequently, the tan (T) peak need not be a mark of the transition center and it can be observed at temperatures where different materials have undergone a very different degree of plasticization as judged by the magnitude of G(T). This is demonstrated by computer simulations using typical functions that describe G(T) and G(T) at the glass transition region and with published data on the dynamic mechanical behavior of a variety of biosolids.     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

Symplectic approach to the theory of quantized fields. II

We prove that the set D of vector fields on the configuration space B of a field whose 1-parameter groups locally associated are groups of fibre-preserving transformations of B that leave invariant that field in the sense of variational theory, is a Lie algebra with respect to ordinary addition, multiplication by real numbers and Lie brackets. We see that this Lie algebra structure can be carried over to the corresponding set of Noether invariants, which then becomes a Lie algebra in a natural way.Further, we define the n-form of Poincaré-Cartan of a field, and we use it to generalize the Lie algebras D and in a reasonable way. The algebras D and are subalgebras of the new Lie algebras D and introduced. A main result in this connection is the following: the differential d of the n-form of Poincaré-Cartan is –(d+f), where (, d+f) are the field equations on the vertical bundle B.The symplectic manifold of solutions associated with a field is introduced in a formal way and the former Lie algebras D, , D, are interpreted on this manifold. In imitation of the case of analytical dynamics, the main results in this direction are: a) Every vector field of the Lie algebra D defines, in a canonical way, a vector field on the manifold of solutions such that its polar 1-form with respect to the symplectic metric ₂ is the differential of its corresponding Noether invariant, and b) the Lie bracket [, ] of two Noether invariants , is the Noether invariant given by ₂(D, D), where D, D are the vector fields on the manifold of solutions defined, in the sense a), by two infinitesimal generators of , , respectively. This will allow us to regard the Lie algebra as the analogous object in field theory to the Poisson algebra of analytic dynamics.We apply the general formalism to the relativistic theory of non-linear scalar fields, and we compare our results with the formalism developed by I. Segal for this case.     

Die Kennzeichnung der Gelbildung bei PVC-Pasten durch Viskositätsmessung

Zusammenfassung Für hochviskose Pasten und raschen Temperatur-wechsel wird ein Kapillarviskosimeter aus Glas beschrieben, das bei Treibdrucken bis 60 atü einen Bereich der Schubspannung von 100–10000 Dyn cm^–2 und der Schergeschwindigkeit von 2–100 sec^–1 aufweist. Damit wird die Umwandlung des bei 160° gebildeten Solzustandes bei PVC-Pasten in den Gelzustand durch Abkühlung verfolgt. Die Fließkurven lassen sich durch dieOstwald-de Waelesche Beziehung beschreiben, wobei die Konstanten undn bezüglich Zeit, Temperatur und Konzentration über die einfache Gleichung log=an+b (a=A _t , bzw.A _t ,A _c ,b=log fürn=1 bzw. log fürt=160° bzw. log des Weichmachers) zusammenhängen.Die vorliegende Arbeit wurde im Institut für technische Chemie der Technischen Hochschule Hannover angefertigt. Die Verfasser danken herzlich Herrn Prof. Dr.G. Schiemann für das gewährte Gastrecht.     

Bursting and structure of the turbulence in an internal flow manipulated by riblets

The buffer layer of an internal flow manipulated by riblets is investigated. The distributions of the ejection and bursting frequency from the beginning to the middle part of the buffer layer, together with high moments of the fluctuating streamwise velocity,u, and its time derivative are reported. The profiles of the ejection and bursting frequency are determined and compared using three single point detection schemes. The effect of the riblets on the bursting mechanism is found confined in a localized region in the buffer layer. The multiple ejection bursts are more affected than the single ejection bursts. The skewness and flatness factors of theu signal are larger in the manipulated layer than in the standard boundary layer. That, also holds true for the flatness factor of the time derivative, but the Taylor and Liepman scales are not affected. The spectrum of theu signal is altered at the beginning part of the viscous sublayer.Nomenclature u Friction velocity - Viscosity - l _v ;f _v wall scalesv/u ;u ² /v - y Vertical distance to the wall - z Spanwise extent - (⁺) Variable normalized with wall scales - u Velocity;u=Turbulence intensity - h, s Height and width of the riblets - f _e Ejection frequency - f _b Bursting frequency - f _BME Frequency of the Bursts with Multiple Ejection - f _BSE Frequency of Single Ejection Bursts - S andS _du/dt Skewness factor ofu and its time derivative - F _u andF _du/dt Flatness factor ofu and its time derivative Abbreviations SBL Standard (non-manipulated) Boundary Layer - MBL Manipulated Boundary Layer - BME Bursts with Multiple Ejections - BSE Bursts with Single Ejections - VITA Variable Interval Time Averaging technique - u–l u-level technique - mu Modifiedu-level technique     

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     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Messung und Berechnung des örtlichen und mittleren Stoffübergangs an stumpf angeströmten Kreisscheiben bei unterschiedlicher Turbulenz

Zusammenfassung In einer vergleichenden Literaturübersicht zu Umströmung, Druck- bzw. Geschwindigkeitsverteilung sowie Wärme- und Stoffübergang werden bislang vorliegende Angaben zu stumpf angeströmten Kreisscheiben und -Zylindern zusammengefaßt. Wenige und zudem divergierende Ergebnisse zum Wärme- und Stoffübergang machen grundlegende experimentelle und theoretische Untersuchungen notwendig, wie sie in [l, 2] für die Eichung von Stoffübergangsmeßmethoden benötigt werden.Unter Einbeziehung des quer angeströmten Kreiszylinders wird gezeigt, daß genaue Angaben zum Wärme- und Stoffübergang bei zwei- wie dreidimensionalen Staupunktströmungen bislang nur über die Messung möglich sind. Über gemessene Geschwindigkeitsverteilungen berechnete Stoffübergangskoeffizienten werden von der Messung nicht bestätigt. Sie liegen gegenüber dem Experiment zu niedrig.Die Messungen wurden bei Turbulenzintensiten 0,8%Tu6%, Reynolds-Zahlen 2·10³5 und Scheibendurchmessern 9,3mmd73,7mm durchgeführt. Der Einfluß der Turbulenz auf den Stoffübergang im Staupunkt von Kreisscheiben kann nur näherungsweise über den Smith-Kuethe-Parameter Tu · Re/100 erfaßt werden. Differenzen zwischen Theorie nach Smith und Kuethe für Tu· Re<5 und Messung lassen sich über die Stabilitätstheorie erklären. Für eine genauere Erfassung des Stoffübergangs muß den unterschiedlichen Transportvorgängen über Turbulenzballen oder Längswirbeln sowie der Struktur der Turbulenz Rechnung getragen werden.

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Measuring and computation of local and average mass transfer to disks in cross flow at different turbulence intensities

The results of different publications concerning the flow, pressure and velocity distributions as well as the heat and mass transfer of disks and cylinders in cross flow are compared by a literature review. A few diverging results for heat and mass transfer require new experimental and theoretical approaches. The calibration of recently developed techniques for the determination of mass transfer rates as published in [1, 2] make these investigations expecially necessary. Including the cylinder in cross flow the authors show, that up to now exact data of heat and mass transfer for two- or three-dimensional flow at a forward stagnation region can be obtained by direct measuring only.Mass transfer coefficients computed from measured velocity distributions are not confirmed by the experimental results. Compared to the experimental data they are too low. The measurements were accomplished for turbulence intensities 0.8%Tu6%, Reynolds-numbers 2· 10³5 and disk diameters 9.3 mm d 73.7 mm.The influence of the turbulence on the stagnation point mass transfer of disks can be obtained only approximately by the Smith-Kuethe-parameter Tu·Re/100. Differences between theoretical results of Smith and Kuethe and experimental ones for Tu·Re/100<5 may be explained by the stability theory. For a more accurate determination of the mass transfer the different transport mechanisms of the scale of turbulence or the tree-dimensional flow pattern like Taylor-Görtler-vortices as well as the structure of the turbulence itself have to be regarded.

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Bezeichnungen a Temperaturleitkoeffizient - C_p Beiwert für den statischen Druck - C₂, C₃ Gradient der bezogenen Geschwindigkeit U⁺ am Staupunkt bei ebener, räumlicher Strömung - D_A Diffusionskoeffizient von Ammoniak in Luft - d Durchmesser - Fr=Sh/Re Frössling-Zahl für den Stoffübergang - Fr=Nu/Re Frössling-Zahl für den Wärmeübergang - Le=a/D_A Lewis-Zahl - L Bezugslänge - M Maschenweite von Turbulenzgittern - Nu=·d/ Nußbelt-Zahl - n Exponent der Prandtl-bzw. Schmidt-Zahl - Pr=/a Prandtl-Zahl - p Druck, Partialdruck - p_x statischer Druck an der Stelle x am Rand der Grenzschicht - Re=U · d/ Reynolds-Zahl - r Radius - r(x) radiale Distanz von der Rotationsachse eines Körpers zu einem Oberflächenelement - Sc=/D_A Schmidt-Zahl - Sh= _A ·d/D_A Sherwood-Zahl - T absolute Temperatur - Tu Turbulenzintensität (Turbulenzgrad) in% - U Strömungsgeschwindigkeit in x-Richtung am Rand der Grenzschicht - U Hauptströmungsgeschwindigkeit im freien Kanalquerschnitt - U⁺=U/U bezogene Geschwindigkeit in x-Richtung am Rand der Grenzschicht - u Strömungsgeschwindigkeit in x-Richtung, tangential zur Oberfläche - mittlere turbulente Geschwindig-keitsschwankung in x-Richtung - v Strömungsgeschwindigkeit in y-Richtung, normal zur Oberfläche - x Koordinate in Strömungsrichtung, tangential zur Oberfläche - x_G Entfernung vom Turbulenzgitter in Strömungsrichtung - x⁺ bezogene Länge x/r - y Koordinate normal zur Oberfläche - Wärmeübergangskoeffizient - _A Stoffübergangskoeffizient (Ammoniak) - dimensionsloses Temperaturgefälle an der Wand - Keilvariable - Wärmeleitkoeffizient - Wirbelweilenlänge (mm) - kinematische Zähigkeit - transformierte bezogene Länge - _A Partialdichte von Ammoniak Indices B mit Korrektur aufgrund der Verengung - m mittel - S bezogen auf die Kreisscheibe - Z bezogen auf den Kreiszylinder Herrn Prof. Dr.-Ing. habil. Josef Ipfelkofer zum 70. Geburtstag am 7. April 1977 gewidmet.     

On the indentation of an inhomogeneous plastic solid

Summary The problem considered here is that of the indentation of a semi infinite, inhomogeneous rigid-plastic solid by a smooth, flat ended punch under conditions of plane strain. It is assumed that the yield stress of the solid k(x, y) has the form k ⁰+k(x, y) where k ⁰ is a constant and is small. A perturbation method of solution developed by Spencer [1] is used, and general results are obtained for arbitrary values of k(x, y). Some particular cases are then considered.     

Rheological properties of polyethylene oxide solutions

Summary Measurements were made on solutions of Polyethylene oxide (WSR-301) varying in concentration from .0511 g/dl to 4.014 g/dl, prepared from two samples of dry material of different ages (I, II), using aWeissenberg Rheogoniometer with cone-and-plate and parallel-plate geometries, and also using capillary viscometers. Steady shear data were obtained for eight decades of strain-rates (10^–3 <k < 10⁵ sec^–1), and oscillatory data for over four decades of frequency (10^–3 <f < 10¹ Hz). Results are presented for the shear-dependent viscosity,(k), normal stress differences, ₁(k), ₂(k), and the complex viscosity, ^*(f). It was found that characteristic fluid times obtained from continuum arguments correlated the experimental(k), (f) andG(f) data.Using the ^*(f) data, the stress relaxation function,(), was calculated, from which the second-order fluid coefficients and ₀ were obtained, and compared to the directly measured values.Evidence is given to show that the sign of ₁(k) varies both with concentration and strainrate.Using solutions prepared from sample II, correlations with the material properties of solutions of sample I were found which indicated the effect of aging on stored dry samples.

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Zusammenfassung Es wurden Viskositätsmessungen an Polyäthylenoxid-(WSR-301)-Lösungen mit Konzentrationen zwischen 0,0511 g/dl und 4,014 g/dl ausgeführt, die aus zwei Proben von Trockenmaterial verschiedenen Alters (I, II) genommen waren. Verwendet wurde einWeissenberg-Rheogoniometer mit Kegel-Platte- und Parallel-Platten-Geometrie sowie verschiedene Kapillarviskosimeter. Werte für die stationäre Scherung wurden über acht Dekaden der Schergeschwindigkeit (10^–3 <k < 10⁵ s^–1) erhalten, solche für periodische Beanspruchung über mehr als vier Dekaden der Frequenz (10^–3 <f < 10 Hz). Es werden die Werte der Scherviskosität(k), der Normalspannungsdifferenzen ₁(k) und ₂(k) sowie der komplexen Viskosität ^*(f) mitgeteilt. Man findet, daß die experimentell ermittelten Werte von(k), (f) undG(f) mit Hilfe charakteristischer Zeitkonstanten, die man aus kontinuumsmechanischen Überlegungen gewinnt, korreliert werden können.Aus dem Verlauf von ^*(f) wurde die Spannungsrelaxationsfunktion() berechnet, woraus sich die Koeffizienten zweiter Ordnung und ₀ bestimmen lassen. Diese wurden mit den auf direkte Weise gewonnenen Werten verglichen. Es wird nachgewiesen, daß das Vorzeichen von ₁(k) sowohl bei der Veränderung der Konzentration als auch der Deformationsgeschwindigkeit wechselt.Durch Vergleich der an den Proben I und II erhaltenen Ergebnisse wird auf Alterungserscheinungen bei der trocken gelagerten Probe geschlossen.

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With 19 figures and 1 table     

Mass transfer effects on flow past a vertical oscillating plate with variable temperature

An exact solution to the flow of a viscous incompressible fluid past an infinite vertical oscillating plate, in the presence of a foreign mass has been derived by the Laplace-transform technique when the plate temperature is linearly varying as time. The velocity profiles are shown on graphs and the numerical values of the skin-friction are listed in a table. It is observed that the skin-friction increases with increasingSc, t orGr but decreases with increasingGm ort, whereSc (the Schmidt number), (frequency),t (time),Gr (the Grashof number) andGm is (the modified Grashof number) andt.

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Einfluß von Stoffübergang auf die Strömung entlang einer senkrechten oszillierenden Platte veränderlicher Temperaturen

Zusammenfassung Mit Hilfe der Laplace-Transformation wird eine exakte Lösung für die Strömung einer zähen, inkompressiblen Flüssigkeit entlang einer unendlich ausgedehnten, senkrechten, oszillierenden Platte gewonnen, wobei die Einwirkung eines Feststoffs Berücksichtigung findet und die Plattentemperatur linear mit der Zeit veränderlich sein soll. Die Geschwindigkeitsprofile sind in Diagrammen dargestellt und die numerischen Werte der Reibungsschubspannung in einer Tabelle. Letztere wächst mit der Schmidt-ZahlSc, der Grashof-ZahlGr und dem Produkt aus Frequenz und Zeitt; sie nimmt ab, wennGm (die modifizierte Grashof-Zahl) undt zunehmen.

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Nomenclature C species concentration in the fluid near the plate - C ^– species concentration in the fluid away from the plate - C _W species concentration at the plate - C _p specific heat at constant pressure - D chemical molecular diffusivity - Gm modified Grashof number - Gr Grashof number - g acceleration due to gravity - K thermal conductivity - P Prandtl number - Sc Schmidt number - T Temperature of the fluid near the plate - T _W temperature of the plate - T temperature of the fluid far away from the plate - t time - u velocity of the fluid in the upward direction - U ₀ amplitude of oscillation - x coordinate axis along the plate in the vertically upward direction - y coordinate axis normal to the plate Greek symbols skin-friction - viscosity - coefficient of volume expansion - ^* coefficient of species expansion - density - frequency     

Relations between loss angles in isotropic linear viscoelastic materials

Summary Starting from the relations between complex dynamic moduli simple diagrams are deduced connecting the locus of complex Poisson's ratiov ^* v +iv in the complexv ^* plane with differences between various loss angles. From these diagrams the sequence of magnitudes of several loss angles appearing in linear viscoelastic theory is deduced. Although theoretically this sequence depends on the values ofv andv, it is found experimentally that for polymeric materials, due to the fact that the values ofv andv are constrained to limited ranges, general rules can be given. The sequences deduced are compared with experimental data. Finally some relations are used to illustrate the phase relationships between stress and deformations in an uniaxial stress experiment. From these relations a new method for measuring the loss angle in compression is suggested.

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Zusammenfassung Mit Hilfe der Beziehungen zwischen komplexen dynamischen Moduln wurden einfache Diagramme konstruiert, in denen der geometrische Ort des komplexen Poisson-Verhältnissesv ^* v +iv in der komplexenv ^*-Ebene mit Differenzen zwischen verschiedenen Verlustwinkeln verbunden werden. Aus diesen Diagrammen kann auf die Reihenfolge der Größe mehrerer in der linearen viskoelastischen Theorie auftretender Verlustwinkel gefolgert werden. Obwohl diese Reihenfolge theoretisch von den Werten vonv undv abhängig ist, wird experimentell gefunden, daß für polymere Substanzen allgemeine Regeln gegeben werden können. Dies findet seine Ursache in der Tatsache, daß die Werte vonv undv auf begrenzte Gebiete beschränkt sind. Zum Schluß werden für einen einachsigen Spannungsversuch die Phasenbeziehungen zwischen den Spannungen und den Verformungen illustriert. Mit Hilfe dieser Phasenbeziehungen wird eine neue Methode für die Messung des Verlustwinkels in Kompression vorgeschlagen.

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Part of this study was performed while the author was an employee of TNO, Delft.

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With 16 figures and 1 table     

On the dynamic response of viscoelastic fluids

Summary To define the dynamic shear behaviour of a viscoelastic fluid we require two functions — one elastic and one viscous. There are two simple alternatives based on the Voigt and Maxwell concepts which are mathematically interrelated. Current practice interprets the dynamic response of fluids as a function of angular frequency () in terms of the storage (G) and loss (G) moduli: the loss function is commonly converted to a viscosity = G/. As is well known the parameters andG are the elements of a Voigt model whereas it is near universal practice to interpret steady flow in terms of the Maxwell model. This paper shows how the interpretation of dynamic experiments on fluids in terms of the apparent Maxwell parameters is more simple, more sensitive, more consistent with steady flow behaviour, and physically more realistic.

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Zusammenfassung Zur Festlegung des dynamischen Scherverhaltens einer viskoelastischen Flüssigkeit benötigt man zwei Funktionen, eine für das elastische und eine für das viskose Verhalten. Dafür gibt es zwei einfache, mathematisch miteinander verknüpfte Alternativen, die entweder auf dem Voigtschen oder dem Maxwellschen Konzept gründen. Die übliche Praxis beschreibt das dynamische Verhalten der Flüssigkeiten als Funktion der Winkelgeschwindigkeit durch den SpeichermodulG und den VerlustmodulG bzw. die zugeordnete Viskosität = G/. Die Parameter undG sind bekanntlich die Elemente eines Voigt-Modells, wohingegen es nahezu allgemein üblich ist, das stationäre Fließen durch ein Maxwell-Modell zu beschreiben. Diese Arbeit zeigt nun, daß die Beschreibung dynamischer Versuche mit Hilfe von apparenten Maxwell-Parametern einfacher, empfindlicher, mit dem stationären Verhalten konsistenter und in physikalischer Hinsicht realistischer ist.

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Paper presented at British Society of Rheology conference. Rheometry: Methods of measurement and analysis of results, Shrivenham, April 8–11, 1975.

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With 13 figures and 2 tables     

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