Wandeffekte bei dispers-plastischen Materialien

Zusammenfassung Die Wandgleitgeschwindigkeit von dispers-plastischen Gemischen aus Kaolinpulver und Paraffinöl wird nach der Drei-Spalte-Methode für die Couette-Strömung mit einem Searle-Rheometer ermittelt. Sie steigt zunächst mit zunehmender Schubspannung an, erreicht ein Maximum, fällt mit weiter steigender Schubspannung wieder ab und wird schließlich sogar negativ. Eine negative Wandgleitgeschwindigkeit ist natürlich physikalisch unmöglich. Dispersplastische Gemische aus Kaolinpulver und Paraffinöl zeigen also ein komplizierteres Wandverhalten als reines Wandgleiten.Zur Deutung dieses komplizierten Wandeffektes wird eine Modellvorstellung entwickelt. Wichtig ist hierbei, daß eine zunehmende Wandgleitgeschwindigkeit auftritt, bevor eine starke Scherströmung im Innern des Strömungsfeldes einsetzt. Mit beginnender Scherströmung führen die plättchenförmigen dispersen Teilchen auf Grund von Zusammenstößen seitliche Schwankungsbewegungen um die makroskopisch wahrnehmbaren Bahnkurven aus.Diese Teilchenbewegungen führen zur Zerstörung der zunächst beim Wandgleiten sich ausbildenden Mikrostrukturen an der Wand. Daher kann die Wandgleitgeschwindigkeit trotz steigender Wandschubspannung abnehmen. Die Behinderung der seitlichen Partikelbewegungen an der Wand — die dispersen Teilchen können sich auf der Wand abstützen — führt bei weiter steigender Schergeschwindigkeit im Innern des Strömungsfeldes makroskopisch zu einer Versteifung des Materials in Wandnähe. Damit können negative Werte der sog. Wandgleitgeschwindigkeit — man spricht besser von einer integralen Wandfunktion — sowie bestimmte experimentelle Befunde bei der Druckabhängigkeit und bei der Temperaturabhängigkeit der rheologischen Eigenschaften und des Wandeffektes erklärt werden.Die experimentellen Untersuchungen beschränken sich im wesentlichen auf den Wandeffekt an schwach gekrümmten Wänden in Couette-Spalten, an denen ein Krümmungseinfluß auf den Wandeffekt mit großer Wahrscheinlichkeit vernachlässigbar ist. Die Auswirkung eines Krümmungseinflusses auf die rheometrischen Meßergebnisse wird jedoch diskutiert. Die aus rheometrischen Messungen bestimmbare integrale Wandfunktion liefert im Fall des komplizierten Wandeffektes noch keine vollständige Information über das Wandverhalten.

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The wall slip velocity of disperse plastic mixtures of kaolin powder and paraffin oil is determined by the so-called three-gap method for Couette flow with a Searle rheometer. At the start it grows with increasing shear stress, reaches a maximum, then decreases with further increases in shear stress and finally becomes negative. From a physical point of view, negative wall-slip-velocities are impossible. Thus it is concluded that disperse plastic mixtures of kaolin powder and paraffin oil show a more complicated wall effect than pure wall slip.In order to explain this complicated wall effect a model of the microstructure near the wall is developed: It is essential that increasing wall slip velocity occurs before the start of shear flow in the interior of the flow field. With shear flow the slab-like disperse particles perform lateral fluctuations around their macroscopically perceptible flow paths. These are caused by collisions between the particles. These lateral particle movements destroy the microstructure at the wall which was built up by pure wall slip. Therefore the wall slip velocity may decrease inspite of increasing wall shear stress. One may then assume a suppression of lateral particle movement at the wall with further increases in the shear in the interior of the flow field which will cause some kind of stiffening of the material near the wall. This assumption can explain the negative values of the so-called slip velocity (which is better termed an integral wall function) as well as some effects in connection with the pressure and temperature dependence of the flow function and integral wall function.The experimental investigations are confined to slowly curved walls in Couette gaps, where an influence of wall curvature on the wall effect may be neglected, but the influence of wall curvature on the wall effect is discussed theoretically. The integral wall function which can be determined from rheometric measurements does not yield complete information on the complicated wall effect.

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f() Schubspannungsfunktion - Schubspannungsfunktion in Wandnähe - h axiale Erstreckung eines Couette-Spaltes - M _d übertragenes Drehmoment in der Couette-Strömung - R kleinster Krümmungsradius einer Wand an einer Stelle - R _w Radius einer zylindrischen Wand - R _a, R_i Radien von Außen- und Innenzylinder eines Couette-Spaltes - R ₁, R₂, R₃ Radien eines Drei-Spalte-Couette-Systems - R _w1, R_w2 Radien von zwei Rohren - Volumenstrom in einer Rohrströmung - Volumenströme durch zwei verschiedene Rohre bei gleicher Wandschubspannung - v _w (_w) Wandgleitgeschwindigkeit - Winkel zwischen Wandschubspannung und der Richtung, in der die Wand am schwächsten gekrümmt ist - =(R_a/R_i)² quadratisches Radienverhältnis - (_w) Dicke der vom komplizierteren Wandeffekt beeinflußten Wandschicht - Dicke eines Gleitfilms bei Wandgleiten - _w Schubspannungsänderung in der Wandschicht (_w) - f(_w, ) Wandfunktion - Wandabstand - ø _w (_w) integrale Wandfunktion bei vernachlässigbarer Wandkrümmung und vernachlässigbarer Schubspannungsänderung in der Wandschicht (_w) - ø _Couette ( _w, ₂) integrale Wandfunktion der Couette-Strömung - ø _Rohr ( _w, R_w) integrale Wandfunktion der Rohrströmung - ø _Couette ^* ( _w, R₂) experimentell ermittelte Wandfunktion der Couette-Strömung - ø _Rohr ^* ( _w, R_w, R_w2) experimentell ermittelte Wandfunktion der Rohrströmung - ₁, ₂ größter bzw. kleinster Krümmungsradius einer Wand - _w Wandschubspannung - _a, _i Wandschubspannung am Außen- bzw. Innenzylinder eines Couette-Spaltes - ₂ Wandschubspannung in einem Drei-Spalte-Couette-System am mittleren RadiusR ₂ - Schubspannung - Winkelgeschwindigkeitsdifferenz zwischen Außen- und Innenzylinder eines Couette-Spaltes - _I (M_d), _II (M_d), _III(M_d) Winkelgeschwindigkeitsdifferenzen an einem Drei-Spalte-Couette-System als Funktionen des übertragenen Momentes     

Convective heat transfer within fibrous insulation slabs

A numerical study of convective heat flow within a fibrous insulating slab is presented. The material is treated as an anisotropic porous medium and the variation of properties with temperature is taken into account. Good agreement is obtained with available experimental data for the same geometry.

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Zusammenfassung Für den konvektiven Wärmestrom in einem faserförmigen Isolierstoff wird eine numerische Berechnung angegeben. Der Stoff wird als anisotropes poröses Medium mit temperaturabhängigen Stoffwerten angesehen. Die Übereinstimmung mit verfügbaren Versuchswerten ist gut.

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Nomenclature C_p specific heat of the gas at the mean temperature - Da Darcy number=k_y/H² - Gr^* modified Grashof number=gTHk_y/²= (Grashof number) × (Darcy number) - H thickness of the specimen - P gas pressure - Pr^* modified Prandtl number= C_p/_x - Ra^* modified Rayleigh number=Gr^* Pr^* - R_p ratio of permeabilities=k_y/k_x - R_k ratio of conductivities= _y/_x - T absolute temperature of the gas - t₁ absolute temperature of the hot face - T₂ absolute temperature of the cold face - T_m mean temperature of the gas=(T₁+T₂)/2 - k_x specific permeability of the porous medium along the x-direction - k_y specific permeability of the porous medium along the y-direction - p T/T_m - q exponent - r exponent - u gas velocity along the x-direction - v gas velocity along the y-direction - X_* distance along the x-direction - y_* distance along the y-direction - T temperature difference=t₁–T₂ - thermal coefficient of expansion of the gas - _m thermal coefficient of expansion of the gas at the mean temperature - _* T–T_m - dimensionless temperature= ^*/T - _a apparent thermal conductivity of the porous medium along the x-direction - _al local apparent thermal conductivity of the porous medium along the x-direction - _x thermal conductivity of the porous medium along the x-direction in the absence of convection - _y thermal conductivity of the porous medium along the y-direction in the absence of convection - dynamic viscosity of the gas - _m dynamic viscosity of the gas at the mean temperature - kinematic viscosity of the gas - _m kinematic viscosity of the gas at the mean temperature - density of the gas - _m density of the gas at the mean temperature - ^* stream function at any point - dimensionless stream function= ^*/( _m/_m)     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

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].     

The Preston tube in adiabatic compressible flow

A new procedure for the reduction of Preston tube data is introduced, based on the van Driest transformation. It appears to give results agreeing with the better calibration experiments, although a significant assumption in its derivation is violated.List of Symbols M _s Mach number sensed by Preston tube - M Friction Mach number (=u/wall sound speed) - R Gas constant - T _w Wall temperature - d Diameter of Preston tube - h Height of effective centre of Preston tube - p Preston tube pressure difference reading - p _i Equivalent incompressible Preston tube reading - p _w Wall pressure - r Recovery factor (=0.896) - u Friction velocity (=[_w/wall density]^1/2) - Empirical constant allowing for departure from Crocco temperature-velocity correlation (=0.975) - Specific heat ratio - Fluid kinematic viscosity - _w Wall shear-stress     

On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

In this paper, the compactness of quasi-conforming element spaces and the—convergence of quasi-conforming element method are discussed. The well-known Rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties, and the generalized Poincare inequality. The generalized Friedrichs inequality and the generalized inequality of Poincare-Friedrichs are proved true for them. The error estimates are also given. It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity, and satisfy the rank condition of element and pass the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L²²()-errors are O(h), O(h), O(h ² ), O(h ² ), O(h ), and O(h ⁴ ) respectively.     

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     

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     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

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Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

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Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

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With 6 figures     

Flow of a simple non-newtonian fluid past a sphere

Summary Creeping flow past a sphere is solved for a limiting case of fluid behaviour: an abrupt change in viscosity.List of Symbols d _ij Component of rate-of-deformation tensor - F _d Drag force exerted on sphere by fluid - G ^(d) Coefficients in expression for _ij in terms of d _ij - G _YOJK ^(d) Coefficients in power series representing G ^(d) - R Radius of sphere - r Spherical coordinate - V Velocity of fluid very far from sphere - v _i Component of the velocity vector - x Dimensionless radial distance, r/R - x _i Rectangular Cartesian coordinate - Dimensionless quantity defined by (26) - ^(d) Potential defined by (7) - Value of x denoting border between Regions 1 and 2 as a function of - ₁, ₂ Lower and upper limiting viscosities defined by (10) - Spherical coordinate - * Value of for which =1 - Value of denoting border between regions 1 and 2 as a function of x - Newtonian viscosity - _ij Component of the stress tensor - Spherical coordinate - ₁, ₂ Stream functions defined by (12) and (14) - Second and third invariants of the stress tensor and of the rate-of-deformation tensor, defined by (3)     

Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

Experimental study of a wall shear stress sensor based on a porous element

The paper presents a new type of local wall shear stress sensor made of a high-porosity material with filter grade 40 n. The pressure variation caused by the shear stress acting on the surface can be transferred in this porous material, while the effect of the momentum change of the fluid is eliminated. Having neither protrusions nor cavities on the wall surface, the sensor presents little disturbance of the measured boundary layer. A pressure difference reading of the sensor is directly proportional to the local wall shear stress and the wall shear stress can be written as = C P. The present investigation also deals with problems of sensor design and its influence on the performance of the sensor.List of symbols angle - density of fluid - kinematic viscosity of fluid - local wall shear stress - ₀ shear stress for the smooth surface - _P shear stress for the porous element - _po shear stress on the surface of the porous element - _p y ₀/2 _p at a certain depthy ₀/2 - pore-surface area ratio of the porous surface - dynamic viscosity - a length of the porous element surface - A height of the duct with rectangular cross-section - b width of the porous surface - B width of the duct with rectangular cross-section - c thickness of the porous element - C, C ₁ constants - D _h hydraulic diameter of the duct - L, l length or distance - P pressure, see Fig. 3 - P ₀ static pressure at wall - P differential pressure, P-P ₀ - P _l differential pressure over length l - F force - u velocity component parallel to surface at distance y - v _p velocity in the porous element - v _po velocity on the surface of the porous element - x, x ₀ distance - y ₀ height of the rectangular passage     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

The generation of turbulence from displaced cross-members in uniform flow

This paper presents details of an experimental investigation into the nature of turbulence generated in the wake of a single grid node. The latter has been considered as two members placed perpendicular to each other in the geometry of a cross, with square and circular sections representing bars and rods respectively. The effect of member spacing has been examined in an attempt to identify the complex flow phenomena associated with such a configuration, and in this respect a critical gap width has been found.List of symbols C _p pressure coefficient, (p — p ₀)/1/2 U ₀ ² - C _pb pressure coefficient measured on the base centre-line - C _ps pressure coefficient measured at stagnation point - D diameter/section depth of model - L distance between central axis of two cylinders or bars - n vortex shedding frequency - p local pressure on model's surface - p ₀ static pressure - R () autocorrelation coefficient - R _e Reynolds number, DU ₀/v - St Strouhal number, n D/U ₀ - U ₀ mean freestream velocity in X-direction - ovu local mean velocity in X-direction - u velocity fluctuation in X-direction - ovv local mean velocity in Y-direction - ovw local mean velocity in Z-direction - X cartesian co-ordinate in longitudinal direction - Y cartesian co-ordinate perpendicular to wind tunnel floor - Z cartesian co-ordinate in lateral direction - dynamic viscosity of fluid - v kinematic viscosity of fluid, / - _n spectral energy, 4 ₀ R () cos 2 n d - density of fluid - _x longitudinal component of vorticity, /y – /z This paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, Sept. 22–24, 1986     

Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)