Boundary conditions for contacts lines in coextrusion flows

A bicomponent coextrusion process is modelled using a 3-D finite element formulation. The layer uniformity problem in coextrusion is addressed by examining the effects of the polymer melt/polymer melt/die wall contact line boundary condition. It has been observed that the less viscous polymer layer will tend to displace the more viscous polymer layer near the die wall. The behaviour of the contact lisle is considered to be either a stick or slip boundary condition. In the stick boundary condition, the contact line does not move from its original position after the two polymer layers meet, A slip boundary condition allows the contact line to move along the die wall. The calculated interfaces which result from different contact line assumptions are determined. Results show that if a stick boundary condition is appropriate for a given fluid/fluid/solid contact line, then a very thin entrained layer of the more viscous polymer melt will be trapped between the less viscous polymer melt and the die wall. Slip boundary conditions would allow complete displacement of the contact line along the die wall. Both slip and stick boundary conditions produce similar interface profiles far away from the die wall for small viscosity ratios. In certain eases, the displacement of the more viscous material by the less viscous material will cease and a static interface structure is produced regardless of die length. Experimental work with polycarbonate melts is compared with the numerical simulations.A. Torres on leave from Investigación y Desarrollo,, C.A. (INDESCA), P.O. Box 10319, Complejo Petroquímico El Tablazo, Maracaibo, 4001, Venezuela.     

Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

Die Viskosität niedermolekularer,unpolarer Flüssigkeiten in Abhängigkeit von Temperatur,Druck und Konzentration

Zusammenfassung Flüssigkeiten mit unpolaren oder schwachpolaren Molekülen folgen gewöhnlich gleichzeitig der Gleichung vonTammann für die Temperaturabhängigkeit und der Gleichung vonArrhenius für die Konzentrationsabhängigkeit ihrer Viskosität. Hieraus ergibt sich eine bestimmte Konzentrationsabhängigkeit der Konstanten der Temperaturgleichung und damit auch eine Beziehung für die Viskositätszahl als Funktion der Temperatur und der Eigenschaften der Lösungspartner. Ferner wird gefunden, daß die Konstanten der Temperaturgleichung zueinander in Beziehung stehen, d. h. daß der Häufigkeitsfaktor und die Aktivierungsenergie von der Einfriertemperatur abhängen.Ändert man die für Moleküle mit isotroper Dichteverteilung entwickelte Theorie vonBorn undGreen so ab, daß das freie Volumen unterhalb der Einfriertemperatur verschwindet, so wird die Viskosität auch anisotroper Moleküle gut als Funktion von Temperatur, Druck und Konzentration wiedergegeben. Die Berücksichtigung von Volumeneffekten beim Mischen solcher Flüssigkeiten erweitert den Gültigkeitsbereich der gefundenen Beziehungen. Oligomere folgen diesen Gesetzmäßigkeiten noch, aber nicht geknäuelte Fadenmoleküle.

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Summary Fluids with unpolar or weakly polar molecules consisting of several atoms usually follow the equation ofTammann for temperature dependence and as well the equation ofArrhenius for concentration dependence of their viscosity. This means that the constants of the temperature-equation have a definite dependence on concentration and that the intrinsic viscosity may be expressed in terms of temperature and physical constants of the pure components. Besides that the constants of the temperature dependence are correlated, i. e. the activation energy and the frequency-factor both depend on the softening temperature.If the theory ofBorn andGreen for molecules with an isotropic density distribution is modified in such a way that the free volume disappears below the softening point a good representation of experimental viscosities of anisotropic molecules as a function of temperature, pressure and concentration is obtained. Taking into account the volume-effect on mixing such fluids, the range of validity of the relations is extended. Oligomers still obey these laws, but no longer coiled thread molecules.

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Herrn Prof. Dr. Dr. h. c. Dr. e. h. Dr. h. c.Otto Bayer zum 60. Geburtstag gewidmet.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

Negative wake generation of FENE-CR fluids in uniform and Poiseuille flows past a cylinder

The effect of flow conditions on the negative wake generation (longitudinal velocity overshoot behind a cylinder in the viscoelastic fluid flow along the centerline) has been investigated. FENE-CR model that predicts constant shear viscosity and controlled extensional viscosity was considered as a constitutive equation. The discrete elastic viscous split stress-G/streamline upwind Petrov–Galerkin (DEVSS-G/SUPG) formulation was employed and the high-resolution solutions were obtained with an efficient iterative solver based on the incomplete LU(0)-type preconditioner and BiCGSTAB. We found that the negative wake generation was more obvious in uniform flow conditions than in Poiseuille flow, which suggests that the experimentally unrevealed negative wake generation of Boger fluids could be partially attributed to the geometrical effect of Poiseuille flow. The negative wake generation was more discernable at low extensibility and high value of viscosity ratio, which agrees well with the previous studies. In addition, we could observe an undershoot phenomenon in Poisseuille flow condition, which has never been reported.     

The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

An extended numerical method of calculating twodimensional or axisymmetric heated flows allowing for dissipation

Summary The analysis of [1] is extended to cover viscous flow for both plane twodimensional and axisymmetric flow. The equations used are the Navier-Stokes equations in streamwise coordinates, except that for simplicity terms of order are neglected, where is a coefficient of viscosity and the flow direction. The supersonic combustion chamber of [1] is used as an example, but the technique of specifying the streamlines in advance has to be modified for a boundary layer with pressure gradient. Thermal conductivity is included but does not have a very important effect. Flow separation is indicated in some circumstances.

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Übersicht Das Rechenverfahren von [1] wird erweitert auf Strömungen mit Reibung und zwar für zweidimensionale ebene und rotationssymmetrische Strömungen. Benutzt werden die Navier-Stokesschen Gleichungen in Stromlinienkoordinaten; zur Vereinfachung werden Glieder der Größenordnung vernachlässigt, wobei der Reibungskoeffizient und die Richtung der Strömung sind. Als Beispiel dient die Überschallbrennkammer von [1]. Das Verfahren, im voraus Stromlinien festzulegen, muß bei Grenzschichten mit Druckgradient abgeändert werden. Wärmeleitfähigkeit wird berücksichtigt, aber ihr Einfluß ist gering. In einigen Fällen wird Strömungsablösung vorausgesagt.

     

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

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

Fully explicit and self-consistent algebraic Reynolds stress model

A fully explicit, self-consistent algebraic expression (for Reynolds stress) which is the exact solution to the Reynolds stress transport equation in the weak-equilibrium limit for two-dimensional mean flows for all linear and some quasi-linear pressure-strain models, is derived. Current explicit algebraic Reynolds stress models derived by employing the weak-equilibrium assumption treat the production-to-dissipation (P/) ratio as a constant, resulting in an effective viscosity that can be singular away from the equilibrium limit. In this paper the set of simultaneous algebraic Reynolds stress equations in the weak-equilibrium limit are solved in the full nonlinear form and the eddy viscosity is found to be nonsingular. Preliminary tests indicate that the model performs adequately, even for three-dimensional mean-flow cases. Due to the explicit and nonsingular nature of the effective viscosity, this model should mitigate many of the difficulties encountered in computing complex turbulent flows with the algebraic Reynolds stress models.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480.     

Solidification behavior of the theta system 2-propanol/poly(n-butyl methacrylate)

Thermoreversible gelation of the system 2-propanol/poly (n-butyl methacrylate) — as detected by D'SC or dielectric experiments — does not manifest itself in a straightforward manner in the dynamic-mechanical properties. Its occurrence can, however, be seen in many ways: i) For constant composition of the system and a reference temperature lower than T _gel, the storage modulus G is larger than the loss modulus G in the glass transition zone of the master curve and both vary in an almost parallel manner with the angular frequency over almost two decades (whereas this feature is normally found for other gelling systems within the rubber plateau or the flow region). ii) The entanglement molecular weight obtained from G_max is markedly less max than the entanglement molecular weight in the melt divided by ₂, the volume fraction of the polymer. iii) The temperature influences change from WLF like to Arrhenius-like behavior as T is lowered in the case of highly concentrated polymer solutions; analogous considerations hold true as ₂ is increased at constant T. iv) For sufficiently low temperatures, the activation energy of flow exhibits a maximum in the concentration range where the gelation is — according to DSC experiments — most pronounced. Like with ordinary non-gelling systems it is possible to construct master curves. On the basis of Graessley's theory identical dependencies are obtained for the variation of the entanglement parts of the stationary viscosity with shear rate and for the dependence of the entanglement part of the complex viscosity on the frequency of oscillation. Zero shear viscosity and limiting value of the complex viscosity for vanishing as a function of ₂ match smoothly and exhibit two points of inflection.     

Der Spannungs- und Verschiebungszustand drehsymmetrischer Membrane mit beliebig gekrümmter Meridiankurve

Zusammenfassung Die Einführung von Zylinderkoordinaten (x, r, ) in die Gleichgewichtsbedingungen der Schnittkräfte bzw. in die Beziehungen zwischen Verzerrung und Verschiebungen am differentialen Schalenabschnitt ermöglicht die Berechnung des Spannungs- und Verschiebungszustandes von drehsymmetrischen Membranen mit beliebig gekrümmter Meridiankurve auf die Integration einer einfachen, linearen partiellen Differentialgleichung zweiter Ordnung für eine charakteristische FunktionF bzw. zurückzuführen. Eine geschlossene Lösung und damit eine Darstellung der Schnittkräfte und Verschiebungen durch explizite Formeln ist bei harmonischer Belastung cosn für zwei Funktionsgruppen=x ² und=x ^–3 möglich. Im Sonderfall der drehsymmetrischen und der antimetrischen Belastung mitn=0 undn=1 gelten die Gleichungen der Schnitt- und Verschiebungsgrößen für eine beliebige Meridianfunktion=(). Die Betrachtungen der Randbedingungen offener Schalen bei harmonischer Belastung geben über die infinitesimalen Deformationen einer drehsymmetrischen Membran mit überall negativer Krümmung Aufschluß.     

Effects of temperature and pressure on the rheological behavior of montmorillonite sols

Summary The effects of high temperatures up to 180 C and high pressures up to 560 kg/cm² on the rheological properties of pure montmorillonite suspensions with well-defined base-exchange cations have been investigated. The suspensions behave very much asBingham plastics according to the equation=+, in which is the shear stress,D the shear rate, the plastic viscosity, and the yield stress which is largely a measure of residual flocculation.The observed effects depend strongly on the interparticle forces that govern the colloidal stability and the rheological behavior of the suspensions. One can distinguish between two categories of suspensions:P-type sols in which the clay particles are associated through Coulombic attraction between positive edges and negative faces and are located in a primary potential energy minimum, andS-type sols in which the particles are associated edge-to-edge and are located in a weaker secondary potential minimum obtained by the summation ofvan der Waals attraction and double layer repulsion.Both theBingham yield stress and the plastic viscosity of theP-type sols decrease with increasing temperature. The temperature dependence of follows theArrhenius equation. TheP-type suspensions are either weakly or not at all thixotropic at room temperature and are definitely non-thixotropic at higher temperatures. Pressure slightly increases both the plastic viscosity and the yield stress.TheS-type sols, on the other hand, display an increase in yield stress and degree of thixotropy with increasing temperature and generally a decrease in the plastic viscosity. This behavior is modified in the case of Ca-montmorillonite suspensions, in which both and pass through a maximum at 150C, followed by a decline. The maximum can be explained by disaggregation of face-to-face aggregated clay packets.Pressure causes a decrease both in and the degree of thixotropy in theS-type suspensions, while it causes a slight increase in the plastic viscosity. This behavior is a consequence of the destruction of the hydration shell caused by high pressure.

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Zusammenfassung Der Einfluß von hohen Temperaturen (bis 180 C) und hohen Drucken (bis zu 560 kp/cm²) auf die rheologischen Eigenschaften reiner Suspensionen von Montmorillonit mit definierten Austauschionen wurde untersucht. Die Suspensionen folgen dem Binghamschen Flie\gesetz gemÄ\ der Gleichung=+, worin die Schubspannung,D das GeschwindigkeitsgefÄlle, dieBingham- ViskositÄt und die Flie\spannung darstellen. Letztere zeigt in erster Linie das Ausma\ der Koagulation an.Die beobachteten Effekte hÄngen stark von den KrÄften zwischen den Teilchen ab, welche die StabilitÄt der Kolloide und das rheologische Verhalten der Suspensionen bestimmen. Man kann zwei Kategorien von Suspensionen unterscheiden: Sole vom sog.P-Typ, in denen die Tonteilchen durch Coulombsche AnziehungskrÄfte zwischen positiven Kanten und negativen FlÄchen assoziiert sind und sich in einem primÄren Potentialminimum befinden, und Sole vomS-Typ, in denen die Teilchen Kante-zu-Kante assoziiert sind und sich in einem flacheren sekundÄren Potentialminimum befinden, welches durch das Zusammenwirken vonvan der Waalsschen AnziehungskrÄften und von Absto\ungskrÄften infolge der Wechselwirkung der elektrischen Doppelschichten entsteht.Sowohl die Flie\grenze als auch dieBingham-ViskositÄt der Sole vomP-Typ nehmen mit wachsender Temperatur ab. Die TemperaturabhÄngigkeit von folgt derArrhenius-Gleichung. Die Suspensionen vomP-Typ sind bei Zimmertemperatur entweder gar nicht oder nur schwach thixotrop, wÄhrend sie bei hoher Temperatur in keinem Fall thixotrop sind. Mit wachsendem Druck erhöht sich sowohl dieBingham-ViskositÄt als auch die Flie\grenze ein wenig.Bei den Solen vomS-Typ andererseits steigt die Flie\spannung und der Grad der Thixotropie mit steigender Temperatur, wÄhrend dieBingham-ViskositÄt im allgemeinen abnimmt. Bei Ca-Montmorillonit-Suspensionen ist das Verhalten etwas anders: Sowohl als auch erreichen bei 150 C ein Maximum und fallen dann wieder ab. Das Maximum kann durch Desaggregation flÄchenhaft aggregierter Tonpartikel erklÄrt werden. Bei steigendem Druck fÄllt sowohl als auch der Grad der Thixotropie in den Suspensionen vomS-Typ ab, wÄhrend dieBingham-YiskositÄt leicht ansteigt. Dieses Verhalten ist eine Folge der dann eintretenden Zerstörung der Solvathüllen.

     

An acoustic/viscous splitting technique for computational aeroacoustics

A new technique for the numerical analysis of aerodynamic noise generation is developed. The approach involves first solving for the time-dependent incompressible flow for the given geometry. A hydrodynamic density correction to the constant incompressible density is then calculated from knowledge of the incompressible pressure fluctuations. The compressible flow solution is finally obtained by considering perturbations about the corrected incompressible flow. This fully nonlinear technique, which is tailored to extract the relevant acoustic fluctuations, appears to be an efficient approach to the numerical analysis of aerodynamic noise generation, particularly in viscous flows. Applications of this technique to some classical acoustic problems of interest, including some with moderately high subsonic Mach numbers, are presented to validate the approach. The technique is then applied to a fully viscous problem where sound is generated by the flow dynamics.     

Prediction of primary normal stress difference from shear viscosity data using a single integral constitutive equation

Summary Based on a single integral constitutive equation with a strain-dependent memory function, a relation between the primary normal stress function and the shear viscosity function is proposed. According to this theory, the primary normal stress function can be obtained from viscosity data by simple differentiation of the viscosity function with respect to the shear rate , and multiplication by a factor (–1/n). The material parametern is thereby associated with the strain dependence of the memory function.This relation was compared with the viscosity and primary normal stress data of six polymer melts, three polymer solutions, and an aluminium-soap solution, which were measured by several research groups and are available in the literature. In spite of the vast differences in physical constitution and chemical structure of the melts and solutions considered, agreement between predicted and measured values was encouraging.

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Zusammenfassung Ausgehend von einer Zustandsgleichung vom Integraltyp mit deformationsabhängiger Gedächtnisfunktion wird eine einfache Beziehung zwischen der ersten Normalspannungsfunktion und der Scherviskositätsfunktion vorgeschlagen. Nach dieser Theorie kann man die erste Normalspannungsfunktion aus Viskositätsdaten erhalten, indem man die Viskositätsfunktion nach der Schergeschwindigkeit ableitet und den entstehenden Ausdruck mit einem Faktor (–1/n) multipliziert. Dabei hängt die Materialgrößen mit der Deformationsabhängigkeit der Gedächtnisfunktion zusammen.Diese Beziehung wurde mit den Viskositäts- und Normalspannungsdaten von sechs Polymerschmelzen, drei Polymerlösungen und einer Aluminiumseifenlösung verglichen, die von verschiedenen Forschergruppen gemessen wurden und die in der Literatur verfügbar sind. Trotz der großen Unterschiede im physikalischen Zustand und in der chemischen Struktur der betrachteten Schmelzen und Lösungen wurde eine gute Übereinstimmung zwischen der Theorie und den experimentellen Daten gefunden.

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With 9 figures and 2 tables     

Calculation of the deceleration of a viscous supersonic gas flow in a channel

The deceleration of nonuniform viscous supersonic gas flows in planar and axisymmetric channels is investigated. A modification of Prandtl's formula for the turbulent viscosity is proposed in order to take into account the dependence of the mixing length on the value of the axial Mach number. The results of the calculations are compared with known experimental data on the deceleration of a supersonic flow in a subsonic pseudoshock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1982.We thank A. N. Sekundov for discussing the work.     

Dissipative instability of a nonisothermal electrically conducting flow between parallel plates in a transverse magnetic field

Problems of dissipative instability (in particular, overheating) in magnetohydrodynamies has been studied in [1–6]. The Leontovieh mechanism of overheating instability is explained in [I] by the example of a stationary homogeneous plasma in a strong magnetic field along which current flows. The rate of buildup of perttbations is estimated in [2] to explain the effect of overheating instability on the operation of an MHD generator. The effect of inhomogeneity in the temperature field and in the boundaries of the region on the formarion of this instability has been studied by the example of discharge in a stationary medium in the absence of a magnetic field [3], Certain cases of overheating instability in magnetohydrodynamies are considered in [4, 6], where it is shown that it can be aperiodic as well as oseillatery (Alfven and acoustic waves). Finally, the hydro-dynamic and overheating branches of instability in the ease of non-isothermal plasma flow in a plane MHD channel was investigated in [6]. But the overheating instability was examined without allowance for the dependence of the viscosity and thermal-conductivity coefficients on temperature in the limiting case S R_m 1 and only for small perturbation wavelengths. The development of shortwave perturbations is studied below with allowance for viscosity and thermal conductivity and for a wider range of conditions A 1. Overheating instability over the entire range of wavelengths for the ease considered in [6] is also studied.The author thanks Yu. M. Zolotaikin for programming and performing the calculations.     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

Applications of maximum entropy method in capillary viscometry

Maximum entropy method (MEM) is presented as a technique for processing data obtained from capillary viscometers. The performance of MEM is assessed by comparing the viscosity versus shear rate curves generated by MEM against that obtained by the standard method based on the Weissenberg-Rabinowitsch-Mooney equation. In all the cases in vestigated, MEM proved to be a reliable technique in coping with the experimental noise in the capillary data.