Die freie Oberfl�che einer Fl�ssigkeit �ber einer rotierenden Scheibe

Abstract

Die freie Oberfl~iche einer Fliissigkeit fiber einer rotierenden ScheibeG. B6hme, R. Vof5 und W. Warnecke Institut ftir Str6mungslehre und Str6mungsmaschinen, Hochschule der Bundeswehr Hamburg Zusammenfassung: Es wird eine modifizierte Form des Weissenberg-Effekts untersucht, wobei sich die viskoelastische Fltissigkeit in einem kreiszylindrischen Gef~iB befindet, an dessen Boden eine Scheibe rotiert. Normalspannungsdifferenzen rufen in der Fliissigkeit eine Str6mung hervor, die auf der Drehachse von unten nach oben gerichtet ist, und die freie Oberfl~iche wrlbt sich nahe der Achse nach auBen. Unter der Voraussetzung hinreichend langsamer Str6mung wird eine Theorie zweiter Ordnung entwickelt. Sic fiihrt auf elliptische Randwertaufgaben zweiter bzw. vierter Ordnung fiir das Geschwindigkeitsfeld der Prim~irstrrmung in Umfangsrichtung und ftir die Stromfunktion der Sekund~irstrrmung in der Meridianebene. Ihnen werden gquivalente Variationsaufgaben zugeordnet und mit der Methode der Finiten Elemente numerisch gelrst. Die Gestalt der freien Oberfl~iche setzt sich bei geeigneter Normierung aus drei universellen Formfunktionen zusammen, die ftir verschiedene Fiillhrhen berechnet werden. Im experimentellen Tell wird nachgewiesen, dab durch entsprechende Messungen der Auslenkung des Fltissigkeitsspiegels die unteren Grenzwerte der beiden Normalspannungskoeffizienten bestimmt werden krnnen. Das Rheometer besitzt den Vorzug, dab die Oberfl~ichenspannung der Fltissigkeit die MeBgrrBe nur unwesentlich beeinfluBt.Abstract: Some kind of Weissenberg effect is considered where the viscoelastic fluid, being within a cylindrical vessel, is set in motion by a rotating disc near the tank bottom. Because of normal-stress differences within the fluid a secondary flow arises which is directed upwards near the axis of symmetry, and thus the free surface is deformed. Under the assumption of sufficiently slow flow a second-order theory is developed. It leads to second-order and fourth-order elliptic boundary value problems for the velocity field in azimuthal direction and for the stream function of the secondary flow, respectively. Equivalent variational problems are formulated and solved by the method of finite elements. When normalized appropriately, the shape of the free surface consists of three shape functions, which are independent of any material constants. It is shown by corresponding experiments, that the zero-shear-rate normal-stress coefficients can be determined by measuring the displacement of the free surface. In this rheometer, the surface tension of the fluid causes only insignificant influence on the quantity to be measured.

Swirling flow of viscoelastic fluids. Part 1. Interaction between inertia and elasticity

“…The surface bulge was found to be larger using the Oldroyd-B model, indicating that the primary normal stress difference caused the free surface to rise while the second normal stress difference acted against the first normal stress difference as far as the bulge shape was concerned. Good quantitative agreement on surface displacement was found between the predictions of Debbaut & Hocq (1992) and the experiments by Böhme et al (1985). However, Siginer (1991) found that surface tension was important when measuring surface deformation to yield normal stress coefficients.…”

“…The open cylinder with rotating bottom lid was used experimentally for a shearthinning elastic liquid (25% polyacrylamide (PAA) in water) and a constant-viscosity elastic liquid (silicon oil) by Böhme, Voss & Warnecke (1985). 'Reverse' flow was observed at low Reynolds number (Re < 0.013 for 2.5% PAA) and a bulge in the free surface was produced which depended on the primary normal stress difference.…”

“…The effect was termed the Quelleffekt because the fluid flowed upwards along the axis of symmetry as a source or Quell. Böhme et al (1985) developed a second-order theory assuming a sufficiently slow flow and solved it using a numerical finite-element method. The results for the surface bulge size agreed well between experiment and the numerical analysis.…”

“…It was also found that the axial bulge deformation was quadratic in the angular velocity of the rotating disk for low angular velocities. Debbaut & Hocq (1992) used the Oldroyd-B and Johnson-Segalman constitutive models to predict the bulge shape observed by Böhme et al (1985). Both models assume a constant-viscosity for the test fluid and a quadratic dependence of the primary normal stress with shear rate.…”