Temperament Development Modeled as a Nonlinear Complex Adaptive System

Using approach-withdrawal (AW) as a specific instance of temperament, a theoretical model of temperament as a complex dynamic system is proposed. Developmental contextualism (Lerner, 1998) serves as a guiding theory in determining the structural components of the system and Kauffman's (1993) Boolean models of self-organization are adapted to estimate the parameter functions. In this model P(AW) = f(, ) where P(AW) is the probability density function of an approach or a withdrawal response, ( is a standardized parameter estimate of the biological sensitivity to stimulation, and is a standardized parameter estimate of the contextual response to an approach or withdrawal response. It is theorized that the functions of ( and follow a Hill function of the forms: d /dt = (²/c² + ²) – K₁ d /dt = ( ²/c² + ²) – K₂, where K₁, K₂, and c are system constants. This results in a double sigmoid function in which at extreme values of and the system stabilizes on a steady state of either approach or withdrawal response patterns. At intermediate parameter values the probability density functions of approach and withdrawal responses are wider. Thus, AW can be modeled as representing two basins of attraction. In addition, considerations are given to the systems sensitivity to initial conditions.     

Topological entropy of a dynamical system on the space of one-dimensional maps

We study properties of the topological entropy of the map F: f , C(I), generated by a fixed continuous map f C(I) of an interval of the straight line. In particular, we show that the topological entropy h(F) > 0 if and only if h(f) > 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 180–187, April–June, 2004.     

A new interpretation of viscosity and yield stress in dense slurries: Coal and other irregular particles

Illinois coal was ground and wet-sieved to prepare three powder stocks whose particle-size distributions were characterized. Three suspending fluids were used (glycerin, bromonaphthalene, Aroclor), with viscosities _s that differed by a factor of 100 and with very different chemistries, but whose densities matched that of the coal. Suspensions were prepared under vacuum, with coal volume fractions that ranged up to 0.46. Viscosities were measured in a cone-and-plate over a shear rate range 10^–3–10² s^–1. Reduced viscosity _r = /_s is correlated in the high-shear limit ( ) with/ _M, where _M is the maximum packing fraction for the high-shear microstructure, to reveal the roles of size distribution and suspending fluid character. A new model that invokes the stress-dependence of _M is found to correlate _r well under non-Newtonian conditions with simultaneous prediction of yield stress at sufficiently high; a critical result is that stress and not governs the microstructure and rheology. Numerous experimental anomalies provide insight into suspension behavior.     

Bemerkungen über die Auswertung von Dämpfungsschleifen. II

Zusammenfassung Für eine lineare Beanspruchung erhält man eine elliptische Hysterese, deren Auswertung nachKemmnitz mit Hilfe der konjugierten Durchmesser vorgenommen werden kann. Konjugierte Durchmesser haben im Gegensatz zu den Hauptachsen eine einfache physikalische Bedeutung.Für die nicht lineare Beanspruchung kann nachKemmnitz in guter Näherung ein elliptischer Ansatz zur Auswertung der Hysterese beibehalten werden, wenn man diese aus der Summe oder Differenz von zwei Ellipsenhälften zusammensetzt.VonKemmnitz undEspanion ist ein graphisches Verfahren ausgearbeitet worden, mit dessen Hilfe der Phasenwinkelverlauf aus dem Kraft- und Längenverlauf gefunden wird.Dieser Phasenwinkelverlauf ist nur für die lineare Beanspruchung konstant. Wie sich aus Meßergebnissen vonMeumann ergibt, die nach diesem Verfahren ausgewertet wurden, erhält man für die nicht lineare Beanspruchung je nach dem Probenmaterial oder der Probenform den Verlauf= _a (1+cos t) oder= _a (1-cos t).     

Dynamic-mechanical properties of a bitumen-silica composite

Summary Dynamic-mechanical properties of bitumen-silica composite materials, measured at room temperature, do not vary with the volume ratio () in a simple manner as do usual bituminous concretes. However,E is a linear function of the interfacial area () between the filler and the binder per unit volume. ThusE = E ₀ +a(), wherea is a constant related to the storage modulus, in the absence of voids and with a void ratio factor. The loss moduli, plotted against () go through a maximum in a similar way as when plotted versus decreasing temperatures.

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Zusammenfassung Die bei Zimmertemperatur gemessenen dynamischmechanischen Eigenschaften von Bitumen, welches mit Siliziumdioxid gefüllt worden ist, ändern sich nicht in einer so einfachen Weise mit dem Volumenanteil des Füllstoffs, wie es bei gewöhnlichen Asphaltbetonen der Fall ist. Dagegen ergibt sich der SpeichermodulE als eine lineare Funktion der Grenzfläche zwischen Bindemittel und Füllstoff pro Volumeneinheit:E = E ₀ +a(), wobeia eine Konstante bedeutet, die zum einen von dem Speichermodul bei Abwesenheit von Hohlräumen und zum andern von einem durch solche Hohlräume bedingten Faktor abhängt. Der Verlustmodul als Funktion von zeigt ein Maximum ähnlich wie bei der Auftragung gegen die Temperatur.

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Résumé Les propriétés mécaniques dynamiques, mesurées à la température ambiante, d'un système composite bitume-silice, n'évoluent pas simplement avec la fraction volumique de charges comme dans le cas des enrobés usuels. En effet, le module de conservationE varie linéairement avec l'aire de l'interface liant-charge par unité de volume (). Ainsi la relation suivante a pu être mise en évidence:E = E ₀ +a(), la constantea étant fonction d'un module indépendant du taux de vide et d'un terme relié à ce dernier. Si augmente, les modules de perte passent par une valeur maximale. Ces variations sont semblables à celles que l'on aurait si l'on portait ces modules en fonction de la température.

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With 9 figures and 2 tables     

Viscous properties of calcium carbonate filled polymer melts

Summary The viscous properties of calcium carbonate filled polyethylene and polystyrene melts were examined. The relative vircosity _r defined in the previous paper gave an asymtptotic value( _r)_l in the range of the shear stress below 10⁵ dyne/cm².( _r)_l of the calcium carbonate filled system was higher than that of the glass beads or glass balloons filled system at the same volume fraction of the filler. Maron-Pierce equation with ₀ = 0.44 was able to approximate the( _r)_l — relationship. However, it was deduced here that the high value of( _r)_l of calcium carbonyl filled system was due to the apparent increase of and this increase was attributed to the fixed polymer layer formed on the powder particle. By assuming the particle as a sphere with a diameter of 2 µm, the thickness of the fixed polymer layer was estimated as about 0.17 µm. The yield stress estimated from the Casson's plots increased exponentially with.

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Zusammenfassung Es wurden die viskosen Eigenschaften von Polyäthylen-und Polystyrol-Schmelzen untersucht, die mit Kalziumkarbonat-Teilchen gefüllt waren. Für die relative Viskosität _r, wie sie in einer vorangegangenen Veröffentlichung definiert worden war, ergab sich bei Schubspannungen unterhalb 10⁵ dyn/cm² ein asymptotischer Wert( _r)_l. Dieser war bei den mit Kalziumkarbonat gefüllten Schmelzen höher als bei Schmelzen, die bis zur gleichen Volumenkonzentration mit Glaskugeln oder Glasballons gefüllt waren. Die ( _r) _l —-Abhängigkeit ließ sich durch eine Gleichung nachMaron und Pierce mit ₀ = 0,44 beschreiben. Es wurde jedoch geschlossen, daß der hohe( _r)_l-Wert der mit Kalziumkarbonat gefüllten Schmelzen auf eine scheinbare Zunahme von zurückzuführen ist, verursacht durch eine feste Polymerschicht auf der Teilchenoberfläche. Unter Annahme kugelförmiger Teilchen mit einem Durchmesser von 2 µm ließ sich die zugeordnete Schichtdicke zu 0,17 µm abschätzen. Die mittels der Casson-Beziehung geschätzte Fließspannung ergab eine exponentielle-Abhängigkeit.

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With 7 figures and 1 table     

A distributed electrical analog for transient diffusion-techniques of model fabrication and calibration

Simulation of transient two-dimensional diffusion by means of a distributed electrical analog is discussed. After present techniques of analog model construction and calibration are reviewed, an improved calibration technique is presented and a convenient method of analog fabrication, not previously reported, is described. The proposed new method allows complete access to any point on the analog model during a test. Frequency response and step response measurements indicate that an adequate simulation is provided by this particular type of analog model.

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Zusammenfassung Die Simulierung eines nichtstationären, zweidimensionalen Diffusionsvorganges mittels eines kontinuierlichen, elektro-thermischen Analogapparates wird besprochen. Eine Übersicht der gegenwärtigen Methoden für die Konstruktion und Kalibrierung von elektro-thermischen Analogmodellen wird gegeben. Ein verbessertes Verfahren für die Kalibrierung und eine handliche Fertigungsmethode von Analogmodellen, die noch nicht in der Literatur beschrieben wurden, werden dargestellt. Das vorgeschlagene neue Verfahren gestattet vollständigen Zugang zu jedem Punkt im Analogmodell während des Experiments. Meßbeobachtungen des periodischen Frequenzverhaltens und des nichtstationären Verhaltens zeigen, daß dieses spezielle Analogmodell eine ausreichende Simulierung des Diffusionsvorganges gestattet.

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Nomenclature A ₀ Potential amplitude atx=0 (see Fig. 5) - A _x Potential amplitude at locationx (see Fig. 5) - A _L Potential amplitude atx=L (see Fig. 5) - C Capacitance per unit area - j Frequency - L Characteristic length - R Resistance per square - t Time - x Coordinate - X Dimensionless distance,x/L - y Coordinate - Y Dimensionless distance,y/L - Diffusivity - _x Phase angle at locationx (see Fig. 5) - _L Phase angle atx=L (see Fig. 5) - _d Thickness of dielectric sheet - _r Thickness of resistance sheet - Dielectric constant - Resistivity - Dimensionless time,t/L ² - Potential - ₁ Reference potential - ₂ Reference potential - Dimensionless potential, ( – ₁)/( ₂ – ₁) - Angular frequency, 2f - Dimensionless frequency,L ²/ The investigation was performed while the first author was Visiting Associate Professor at Purdue University during 1967/68.     

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f₀ and the diffusion vector fields f₁,...,f_m, and investigate the existence of global random attractors for the associated flows . For this purpose is decomposed into a stationary diffeomorphism given by the stochastic differential equation on the space of smooth flows on R^d driven by m independent stationary Ornstein Uhlenbeck processes z¹,...,z^m and the vector fields f₁,...,f_m, and a flow generated by the nonautonomous ordinary differential equation given by the vector field (_t/x)^–1[f₀(_t)+ _i=1 ¹ f_i(_t)z _t ⁱ ]. In this setting, attractors of are canonically related with attractors of . For , the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f₀ is still a Lyapunov function for , yielding an attractor this way. The criterion is finally tested in various prominent examples.     

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.     

Existence of L 1-Connections Between Equilibria of a Semilinear Parabolic Equation

We give a necessary and sufficient condition for the existence of L ¹-connections between equilibria of a semilinear parabolic equation. By an L ¹-connection from an equilibrium ^– to an equilibrium ⁺ we mean a function u(, t) which is a classical solution on the interval (–, T) for some T and blows up at t = T but continues to exist in the space L ¹ for t [T, ) and satisfies u(, t) ^± (in a suitable sense) as t ±. The main tool in our analysis is the zero number.     

Die kompressible Grenzschichtströmung am dreidimensionalen Staupunkt bei starkem Absaugen oder Ausblasen

Zusammenfassung Es wird die kompressible, laminare Grenzschichtströmung am dreidimensionalen Staupunkt mit Absaugen oder Ausblasen an der Wand untersucht und daraus Wandschubspannung, Wärmeübergang und Verdrängungsdicke in Abhängigkeit von der Normalgeschwindigkeit an der Wand bestimmt. Besonders ausführlich wkd auf die Grenzfälle sehr starken Absaugens bzw. Ausblasens eingegangen, die auf singuläre Störungsprobleme führen, deren Lösung mit der Methode der angepaßten asymptotischen Entwicklungen erfolgt. – Die Unsymmetrie am Staupunkt wird durch den Parameter c gekennzeichnet mit den Spezialfällen c=0 (ebener Staupunkt) und c=1 (rotationssymmetrischer Staupunkt). Im Grenzfall starken Absaugens sind die beiden Wandschubspannungskomponenten und der Wärmeübergang unabhängig von c, im Grenzfall starken Ausblasens ist nur eine der beiden Wandschubspannungskomponenten von c unbeeinflußt.

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The compressible boundary layer flow at a threedimensional stagnation point with intensive suction or injection

The compressible laminar boundary layer flow at a general three-dimensional stagnation point including large rates of injection or suction on the porous surface is considered. The wall shear stress, heat flux and displacement thickness as function of the mass transfer parameter are determined. The two limiting cases of intensive suction and intensive blowing lead to singular perturbations problems, which are solved by the method of matched asymptotic expansions.—The asymmetry of the stagnation point flow is characterized by the ratio c of the two velocity gradients including the special cases of two-dimensional (c=0) and axisymmetric (c=1) stagnation point flow.-For intensive suction the wall shear stresses and the heat flux become independent of c, whereas for intensive blowing only one of the two wall shear stress components is independent of c.

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Bezeichnungen x, y, z kartesische Koordinaten, siehe Bild 1 - u, v, w Geschwindigkeitskomponenten in x-, y- und z-Richtung innerhalb der Grenzschicht - U, V Geschwindigkeitskomponenten in x- und y-Richtung am Außenrand der Grenzschicht - a =(dU/dx)_x=0 Geschwindigkeitsgradient in x-Richtung der Außenströmung im Staupunkt - b=(dV/dy)_y=0 Geschwindigkeitsgradient in y-Richtung der Außenströmung im Staupunkt - c=b/a Staupunkt-Parameter (c=0: ebene Strömung, c=1: axialsymmetrische Strömung) - Dichte - p Druck - h spezifische Enthalpie - Viskosität - Pr Prandtl-Zahl - _x, _y Wandschubspannungskomponenten in x- bzw. y-Richtung - c_m=–ga Ausblaseparameter, siehe Gl. (21) - t_w= h_w/h_e bezogene Wandenthalpie - Ähnlichkeitsvariable, siehe Gl. (11) - F () G () dimensionslose Funktionen nach den Gln. (12), - (),() (13), (15) und (32) - ¯ Ähnlichkeitsvariable, siehe Gl. (29) - F (¯), G (¯) dimensionlose Funktionen nach Gl. (29) - ₁=1/c_m Störparameter für starkes Absaugen - ₂=1/c _m ² Störparameter für starkes Ausblasen - ^*, ^* Verdrängungsdicken nach Gl. (26) - R, R rechte Seiten der Differentialgln. (40) bzw. (41) - z=/ in Abschnitt 4: bezogener Wandabstand nach Gl. (43) - =t_w·z² Ähnlichkeitsvariable in Abschnitt 4 - ^* (c) Lösung der Gl. (58) - A(c) Definition nach Gl. (63) Indizes w an der Wand - e am Außenrand der Grenzschicht - a äußere Lösung - i innere Lösung     

Time-resolved, 3D, laser-induced fluorescence measurements of fine-structure passive scalar mixing in a tubular reactor

A three-dimensional, time-resolved, laser-induced fluorescence (3D-LIF) technique was developed to measure the turbulent (liquid-liquid) mixing of a conserved passive scalar in the wake of an injector inserted perpendicularly into a tubular reactor with Re=4,000. In this technique, a horizontal laser sheet was traversed in its normal direction through the measurement section. Three-dimensional scalar fields were reconstructed from the 2D images captured at consecutive, closely spaced levels by means of a high-speed CCD camera. The ultimate goal of the measurements was to assess the downstream development of the 3D scalar fields (in terms of the full scalar gradient vector field and its associated scalar energy dissipation rate) in an industrial flow with significant advection velocity. As a result of this advection velocity, the measured 3D scalar field is artificially skewed during a scan period. A method to correct for this skewing was developed, tested and applied. Analysis of the results show consistent physical behaviour.List of symbols  A  Deformation tensor - D_t, D_f  Reactor and injector diameter - L_x, L_y, L_z  Dimensions of the 3D-LIF measurement volume - N_x, N_y, N_z  Number of data samples per measurement volume - Re_m  Reynolds number based on mean velocity - Sc  Schmidt number - f  Focal length - f_c,lens, f_c,array  Cut-off frequency for camera lens and sensor array - f, f  Marginal probability density function for and - f  Joint probability density function of and -  Temporal separation of the 2D data planes -  Temporal resolution of the measurement volume -  Spatial resolution of the measurement volume - ,  Deformation angle and deformation, where =tan -  Fluid energy dissipation rate - ,  Strain limited vorticity and scalar diffusion layers -  Scalar concentration - , _B Kolmogorov and Batchelor length scale - ,  Spherical angles of the scalar gradient vector, -  Kinematic viscosity - _e^–2 Half-thickness (1/e²) of the laser sheet - , _a Kolmogorov and Kolmogorov advection time scales -  Scalar energy dissipation rate -  Scalar diffusivity - 2D, 3D Two- and three-dimensional - DNS Direct numerical simulation - LIF Laser-induced fluorescence - SED Scalar energy dissipation rate - TR Tubular reactor

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Utility of parallel-plate plastometer for rheological study of filled polymer melts

Summary In order to test the applicability of the parallel-plate plastometer to the rheological study of filled polymer melts, viscous properties of polyethylene and polystyrene filled with particles and short fibers were measured by this apparatus. Flow curves in the low range of shear rate were obtained by approximating the materials as Ostwald-de-Waele liquids. Up to about 0.5 of the volume fraction of the filler,, there was no problem and the utility of the apparatus was clearly shown. When increased more, the results were not necessarily self-consistent because the flow became unstable. Experiments on samples with high showed that the unstable flow resulted in an apparent yield stress which agreed with that estimated from the Casson's plots.

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Zusammenfassung Um die Anwendbarkeit des Parallel-Platten-Plastometers für die rheologische Untersuchung von Schmelzen gefüllter Polymerer zu prüfen, wurden mit einem solchen Gerät die viskosen Eigenschaften von Polyäthylen und Polystyrol-Proben gemessen, die mit Teilchen oder kurzen Fasern gefüllt waren. Im Bereich niedriger Schergeschwindigkeiten ließen sich die Fließkurven mit Hilfe der Ostwald-de Waele-Gleichung approximieren. Bis zu einer Füllstoff-Volumenkonzentration von etwa 50% traten keine Komplikationen auf, so daß die Eignung des Geräts für diese Fälle klar erwiesen werden konnte. Wenn jedoch noch weiter gesteigert wurde, waren die Ergebnisse durchweg nicht mehr widerspruchsfrei, da dann der Fließvorgang instabil wurde. Versuche mit hohen Volumenkonzentrationen zeigten, daß die Fließinstabilität eine scheinbare Fließgrenze zur Folge hat, deren Größe mit dem aus einer Casson-Darstellung abgeschätzten Wert in etwa übereinstimmt.

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With 8 figures and 1 table     

Nonlinear axisymmetric static and transient analysis of orthotropic thick annular plates

Summary This paper deals with the geometrically nonlinear axisymmetric static and transient response of moderately thick cylindrically orthotropic circular plates subjected to uniformly distributed and ring loads. Immovable clamped and simply supported annular plates with and without a rigid plug subjected to static and step function loads have been considered. Shear deformation and rotary inertia have been included. Orthogonal point collocation method and Newmark- scheme have been employed to solve the differential equations expressed in terms of transverse displacement , shear rotation and stress function . The effect of transverse shear has been investigated for isotropic and orthotropic plates. A simple approximate method has also been used to predict the maximum dynamic response to step loads from the results for static loads.

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Nichtlineare, axialsymmetrische, statische und dynamische Analyse orthotroper, dicker Kreisringplatten

Übersicht Es wird das geometrisch nichtlineare, axialsymmetrische Verhalten orthotroper Kreisring-platten unter gleichmäßig bzw. ringförmig verteilten Lasten untersucht. Eingespannte und gelenkig gelagerte Platten mit oder ohne starrem Pfropfen unter statischer oder dynamischer Belastung werden behandelt, wobei Schubdeformation und Drehträgheit berücksichtigt werden. Eine Kollokationsmethode und das Newmarksche -Schema werden angewendet, um die Differentialgleichungen für die Durchbiegung , die Verdrehung und die Spannungsfunktion zu lösen.

     

Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations

We characterize the occurrence of complete or incomplete blow-up (and extinction) for a general quasilinear heat equation of the form in terms of the constitutive functions and f We assume that (u)>0 for u>0 and that f(u)0. For the positive sign + before f(u) in (HE), with f(u) superlinear as u, blow-up occurs in finite time: sup _x u(x, t) as tT<. For the negative sign, we consider the case of singular absorption: f(u) as u0. Then initially positive solutions vanish at some point in finite time (extinction), and a singularity in the equation occurs there. An important aspect of blow-up or extinction problems is the possibility of having a nontrivial extension of the solution for t>T, i.e., after the singularity occurs. If such continuation exists, we say that the blow-up (extinction) is incomplete; otherwise it is called complete. Our characterization is based on the qualitative behaviour of the family of travelling-wave solutions and a proper use of the Intersection-Comparison argument. The analysis applies to other nonlinear models, like the equations with gradient-dependent diffusivity.     

Rheodynamic lubrication of a journal bearing

Summary The problem of a journal bearing lubricated by a Bingham material has been solved. It has been found that the load capacity, and the moment of friction of the bearing are larger than in a journal bearing, lubricated with a Newtonian material.Nomenclature r radius of the journal - c radial clearance - r + c radius of the bearing - e eccentricity - w angular velocity of the journal - h thickness of the lubricant film at any point - thickness of the core - ø angular distance of a point, from the point, where film thickness is maximum - eccentricity ratio (e/c) - x distance along the bearing surface - y distance normal to the bearing surface - T shear stress in the lubricant - T ₀ yield value of a Bingham solid - viscosity of a Newtonian fluid - plastic viscosity of a Bingham solid - p fluid pressure in the lubricant film - Q volume flow of the lubricant - W ₀ load capacity of the bearing for ordinary lubricants - W load capacity of the bearing - M moment of friction - F coefficient of friction - ₁ maximum thickness of the inlet core - ₂ maximum thickness of the outlet core - ₁ circumferential extent of the inlet core in the journal bearing - ₂ circumferential extent of the outlet core in the journal bearing - h ₀ minimum hieght of core formation in the slider bearing - h _p maximum height of core formation in the slider bearing - u velocity of the fluid in the direction of x in the slider bearing - V velocity in the y direction - h ₁ height of the inlet core at the circumferential extent ₁ - h ₂ height of the outlet core at the circumferential extent ₂ - h ₃ height of the outlet core in the region ₂ - q Q/(cwr) - q ₀ value of q for Newtonian lubricants - p ₀ pressure at =0 - H h/c - H ₁ h ₁/c - H ₂ h ₂/c - B T ₀ C/wr = Bingham number     

An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers

Summary Based on Maron-Pierce's equation, an empirical equation was suggested, which relates the relative viscosity ( _r ) of the polymer melt filled with various inorganic filler, such as glass fiber, carbon fiber, talc, precipitated- and natural-calcium carbonate powder, and glassy small sphere, to the volume fraction () of the filler. The equation is _r = (1 –/A)^–2, whereA is a parameter relating to the packing geometry of the filler, which is similar to the parameter ₀ in Maron-Pierce's equation. In the equation _r is defined as the ratio of the viscosity of the filledsystem to that of the medium at the same shear stress not the shear rate. The applicability of the equation is above the shear stress about 10⁴ dyne/cm². The equation has a simple form and is considered to have a practical utility for filled-polymer melt systems.With 2 figures and 1 table     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

The Two-Dimensional Attractor of a Differential Equation with State-Dependent Delay

The delay differential equation