Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity

This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.     

Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems

This study extends our previous analysis on pressure-drops for strain-hardening Boger-type fluids in contraction flow settings, into those fluids that manifest mild shear-thinning properties. Numerical simulations are compared and contrasted for a variety of constitutive equations, categorised through their differences in viscometric functional response, considering application on 4:1:4 contraction-expansion flow and 2:1 flow past a sphere. Here, prior results on pressure-drop enhancement for constant shear-viscosity fluids have revealed the counter-influences of first normal stress differences and extensional viscosity. The present comparative work advances this study by selectively including the effects of shear-thinning. Suitable models to accomplish this are chosen from the class of Phan-Thien/Tanner (PTT) models, with cross-reference to FENE-models and Oldroyd-B. Furthermore, the work explores the falling sphere problem with comparison of the drag coefficient factor for various implementations. The numerical computations are performed by appealing to a well-founded hybrid finite element/finite volume algorithm, using structured triangular meshing, semi-implicit time-stepping and subcell technology. The cell-vertex finite volume scheme is particularly suited to the solution of the stress subsystem, and invokes fluctuation-distribution for upwinding and median-dual-cells for source-term representation.     

Heat transfer and pressure drop for purely viscous non-Newtonian fluids in turbulent flow through rectangular passages

Experimental measurements of friction factor and heat transfer for the turbulent flow of purely viscous non-Newtonian fluids in a 21 rectangular channel are compared with results previously reported for the circular tube geometry. Comparisons are also made with available analytical and empirical predictions.It is found that the rectangular duct fully established friction factor measurements are within ± 5% of the Dodge-Metzner prediction if the Kozicki generalized Reynolds number is used. A modified form of the simpler explicit equation proposed by Yoo, [i.e.f=0.079n ^0.675(Re ^*)^–0.25], is found to yield predictions for both the rectangular duct and the circular tube geometries with approximately the same accuracy as the Dodge-Metzner equation.Fully developed Stanton numbers for the rectangular duct are in good agreement with the circular tube data over a range ofn from 0.37 to 0.88 for a given Prandtl number,Pr _a , when compared at a fixed value of the Reynolds number based on the apparent viscosity evaluated at the wall shear stress. In general, the experimental data are within ± 20% of Yoo's equation,St=0.0152Re _a ^–0.155 Pr _a ^–2/3 . A new equation is proposed to bring the prediction for circular pipes as well as rectangular channels into better agreement with generally accepted Newtonian heat transfer results.

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Wärmeübergang und Druckverlust für viskose nicht-Newtonsche Fluide in turbulenter Strömung durch rechteckige Kanäle

Zusammenfassung Es werden Messungen des Reibungsfaktors und des Wärmeübergangs bei turbulenter Strömung viskoser nicht-Newtonscher Fluide in einem rechteckigen Kanal mit dem Seitenverhältnis 21 verglichen mit früheren Ergebnissen, die an runden Rohren gewonnen wurden. Weiterhin werden Vergleiche mit aus der Literatur verfügbaren analytischen und empirischen Beziehungen gemacht.Es zeigte sich, daß die Messungen des Reibungsfaktors im rechteckigen Kanal bei vollausgebildeter Strömung auf ± 5% mit der Vorhersage von Dodge-Metzner übereinstimmen, wenn die von Kozicki verallgemeinerte Reynolds-Zahl verwendet wird. Eine modifizierte Form der einfachen von Yoo vorgeschlagenen einfachen Gleichung in explizierter Form (f=0,079n ^0,675(Re ^*)^–0,25) bewies, daß sie sowohl für den rechteckigen Kanal als auch das runde Rohr die Werte mit fast der gleichen Genauigkeit wie die Methode von Dodge-Metzner vorhersagen kann.Die Stanton-Zahlen für den rechteckigen Kanal bei vollausgebildeter Strömung sind in guter Übereinstimmung mit den Werten für das runde Rohr in einem Bereich vonn= 0,37 – 0,88 für eine gegebene Prandtl-Zahl, wenn man den Vergleich bei einem vorgegebenen Wert der Reynolds-Zahl anstellt, die auf die scheinbare Viskosität — abgeleitet aus der Wandschubspannungbezogen ist. Generell läßt sich sagen, daß die Werte auf ± 20% mit der Gleichung von Yoo (St=0,0152Re _a ^–0,155 )Pr _a ^–2/3 ) übereinstimmen. Es wird eine neue Gleichung vorgeschlagen, welche sowohl die Werte für runde Rohre als auch die für rechteckige Kanäle in bessere Übereinstimmung bringt mit den in der Literatur üblichen Ergebnissen für den Wärmeübergang an Newtonsche Fluide.

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Nomenclature a constant in Eq. (8) - A area of cross-section of channel [m²] - b constant in Eq. (8) - c _p specific heat of test fluid [J kg^–1 K^–1] - d capillary tube diameter [m] - D _h hydraulic diameter, 4A/P[m] - f Fanning friction factor, _w/(g9 V²/2) - h axially local (spanwise averaged) heat transfer coefficient,q _w /(T_wi-T_b) [Wm^–2 K^–1] - k _f thermal conductivity of test fluid [Wm^–1K^–1] - K consistency index of power law fluid - n power law index - Nu fully established, local (spanwise averaged) Nusselt numberh D _h /k _f - P perimeter of channel [m] - Pr _a Prandtl number based on apparent viscosjity, c _p /k _f - Pr ^* defined as (Re _a Pr _a )/Re ^* - q _w wall heat flux [Wm^–2] - Re _a Reynolds number based on apparent viscosity, VD _h/ - Re Metzner's generalized Reynolds number in Eq. (2) - Re ^* Reynolds number defined in Eq. (8) - St Stanton number,h/( V c_p) - T _b local bulk temperature of the fluid [K] - T _wi local inside wall temperature [K] - T _wo local outside wall temperature [K] - V bulk flow velocity [m s^–1] - x distance from the inlet of channel along flow direction [m] Greek symbols shear rate [s^–1] - apparent viscosity [Pa s] - density of test fluid [kg m^–3] - shear stress [Pa] - _w shear stress at the wall [Pa] Dedicated to Prof. Dr.-Ing. U. Grigull's 75th birthday     

Die-swell,splashing drop and a numerical technique for solving the Oldroyd B model for axisymmetric free surface flows

This work deals with the development of a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis. The full free surface stress conditions are employed. The resulting governing equations are solved by finite differences on a Marker-and-cell (MAC) type grid. Validation is provided by simulating a pipe flow problem. The classical die-swell problem is solved and swelling ratios are provided. The height of the splash caused by a falling liquid drop for various Reynolds and Weissenberg numbers is then studied, and the height of the splash is shown to diminish with increasing viscoelasticity.     

Wet steam flows in industrial large-diameter pipes: flowrate, moisture and pressure drop measurements

The flowrate, moisture and pressure drop were measured in a water-steam mixture flowing at a pressure of I MPa in pipes with 0.3 and 1.2 m dia. The results are presented here, along with an original method for assessing pressure drops in a water-steam mixture flowing in such industrial pipes.     

Experiments on elongational flow of dilute polymer solutions Part I: Jet reaction and excess pressure drop for the flow through small apertures

Experiments have been made with dilute polymer solutions on the reaction of jets issuing from small orifices and the excess pressure drop for orifice and capillary flows.Under the flow conditions with vortices occurring upstream of the aperture, the jet reaction is nearly zero below some mean velocity for PEO solutions and similarly zero below some generalized Reynolds number for Separan solutions. The normalized jet reactions, when they possess positive values, are correlated with the generalized Reynolds number irrespective of the aperture diameters for both kinds of solution.In most cases, the pressure is higher than in the corresponding water flow, but for some flows with no vortex it is lower. For the vortex flow of PEO solutions the normalized excess pressure drop is inversely proportional to the Reynolds number for both orifices and capillaries, while for Separan solutions this quantity is not correlated with the generalized Reynolds number for orifice flow but is correlated with it for capillary flow.     

Parametric study of a model for determining the liquid flow-rates from the pressure drop and water hold-up in oil–water flows

A flow-pattern-dependent model, traditionally used for calculation of pressure drop and water hold-up, is accustomed for calculation of the liquid production rates in oil–water horizontal flow, based on the known pressure drop and water hold-up. The area-averaged steady-state one-dimensional two-fluid model is used for stratified flow, while the homogeneous model is employed for dispersed flow. The prediction errors appear to be larger when the production rates are calculated instead of pressure drop and water hold-up. The difference in the calculation accuracies between the direct and inverse calculation is most probably caused by the different uncertainties in the measured values of the input variables and a high sensitivity of the calculated phase flow-rates on even small change of the water hold-up for certain flow regimes. In order to locate the source of error in the standard two-fluid model formulation, several parametric studies are performed. In the first parametric study, we investigate under which conditions the momentum equations are satisfied when the measured pressure drop and water hold-up are imposed. The second and third parametric studies address the influence of the interfacial waves and drop entrainment on the model accuracy, respectively. These studies show that both interfacial waves and drop entrainment can be responsible for the augmentation of the wall-shear stress in oil–water flow. In addition, consideration of the interfacial waves offers an explanation for some important phenomena of the oil–water flow, such as the wall-shear stress reduction.     

On the pressure and stress singularities induced by steady flows of a pair of nonmiscible, incompressible, viscous fluids in contact with a wall

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, only of late has there been much in the way of corresponding identifications of flow-induced stress singularities in fluid mechanics. These recent asymptotic identifications are for a single incompressible viscous fluid： Here the asymptotic approach is extended to apply to a configuration entailing two such fluids, For this configuration, various specifications leading to power or log singularities are determined. These results demonstrate that flow-induced stress singularities can occur in a structural container at a location where no singularities are identified within solid mechanics alone.     

Flow-induced nucleation in polymer melts: a study of the GO model for pure and bimodal blends, under shear and extensional flow

We present a rapid rescaling algorithm that enables a systematic comparison between the Graham and Olmsted (GO) model for flow-induced nucleation of polymer melts (Graham and Olmsted, Phys Rev Lett 103(11): 115702, 2009) and direct nucleation rate measurements from a flowing polymer melt. We consider polymer melts consisting of pure long chains and bimodal blends of long and short chains. We simulate the nucleation rate for a wide range of free energy barriers under a wide range of applied shear and extensional flows by using an accelerated nucleation algorithm. We then develop a semi-analytical technique to compute efficiently the nucleation rate under flow for monodisperse melts. We extend our approach to bimodal blends using a method to rescale reference data. This allows us to compare the GO model to experimentally measured nucleation rates at several different temperatures. The GO model is able to consistently account for the effect of temperature on flow-induced nucleation. Our modelling will also contribute to the derivation of computationally inexpensive molecular models of flow-induced nucleation in polymers.     

Similarity solutions of unsteady laminar incompressible boundary layer equations for flow,heat and mass transfer in non-Newtonian fluids around axisymmetric bodies

Summary The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter. In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis. The main analysis shows that the similarity solutions are possible only for the bodies for which where is a cross-sectional radius; and is the longitudinal distance from the nose point to the cross section. In case of rotating bodies, similarity solutions exist only for cones and disks. The analysis, as an example, has successfully been applied to the Powell-Eyring model. It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE. µ, B, andE are the material functions of the Powell-Eyring fluid.

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Zusammenfassung Es wird die mögliche Existenz von Ähnlichkeitslösungen für instationäre drei-dimensionale Grenzschichtströmungen rein-viskoser nicht-newtonscher Flüssigkeiten mit Wärme- und Stoffübertragung um voll eingetauchte stationäre achsensymmetrische Körper in allgemeiner Weise untersucht. Hierbei werden Transformationen verwendet, die einen einzigen linearen Parameter enthalten. Als Spezialfälle der allgemeinen Analyse werden Rotationsströmungen um ruhende und rotierende Körper diskutiert. Die Hauptanalyse ergibt, daß Ähnlichkeitslösungen nur existieren für Körper mit , wo den Abstand von der Achse und den longitudinalen Abstand auf der Oberfläche von der Nase des Körpers aus bedeuten. Im Falle rotierender Körper existieren solche Lösungen nur für Kegel und Kreisscheiben. Die Analyse läßt sich erfolgreich auf das Beispiel einer Powell-Eyring-Flüssigkeit anwenden. Man findet, daß bei gleicher Schergeschwindigkeit der Energieverbrauch zur Aufrechterhaltung der Körperrotation mit wachsender Eyring-Zahl Ey= 1/µBE ansteigt, wobeiµ, B undE Materialfunktionen der Powell-Eyring-Flüssigkeit bedeuten.

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