Radial flow of viscous non-Newtonian fluids between disks

doi:10.1016/0020-7462(67)90027-3

International Journal of Non-Linear Mechanics

Volume 2, Issue 3, September 1967, Pages 261-273

https://doi.org/10.1016/0020-7462(67)90027-3 Get rights and content

Abstract

A theoretical solution is given for the laminar flow of a non-Newtonian fluid between two circular parallel disks. The fluid under consideration is assumed to obey a model recently proposed by Sisko [1] which has been checked experimentally to fit accurately the viscosity data of greases over a very wide range of shear rates. By equating to zero one of the constants in the model, the model equation reduces to the power law equation which is known to represent the behavior of a large number of non-Newtonian fluids.

Résumé

On donne une solution théorique pour le flux laminaire d'un fluide non Newtonien entre deux disques circulaires parallèles. Le fluide considéré est supposé obéir à un modèle récemment proposé par Sisko [1] que l'expérience a vérifié conforme avec précision aux données de la viscosité des graisses sur une très grande étendue de contraintes. En égalant à zéro une des constantes du modèle, l'équation se réduit à une fonction puissance qui, on le sait, représente le comportement d'un grand nombre de fluides non Newtoniens.

Zusammenfassung

Eine theoretische Lösung für die laminare Strömung einer nicht-Newton'schen Flüssigkeit zwischen zwei kreisförmigen, parallelen Scheiben wird angegeben. Es wird angenommen, dass die betrachtete Flüssigkeit einem Modell foglt, das kürzlich von Sisko [1] vorgeschlagen wurde und das im Experiment fähig gewesen war, die Viskositätswerke von Schmiermitteln über einen sehr weiten Schubraten berich recht genau wiederzugeben. Wird in dem Modell eine der Konstanten gleich null gesetzt, do wird die Modellgleichung zu der Potenzgesetzgleichung reduziert, welche bekanntlich das verhalten einer grossen Zahl von nicht-Newton 'sehen Flüssigkeiten beschreibt.

Реферат

Пpивoдитcя тeopeтиɥecкoe peшeниe пpoблeмы лaминapнo гo тeɥeния нeньyтoнoвcкoйжидкocти мeждy двyмя кpyглым и пapaллeльными диcкaми. Пpeдлoпoжeнo, ɥтo paccмaтpивaeмaя жидкocть пoдɥиняeтcя зaкoнy cпpaв eдливoмy для мoдeли пpeдлoжeннoи в пocлeднee вpeмя cиcкo [1] и пo экcпepимeнтaм oблaдayщeй тoɥнo дaнными жиpoвыч cмaзoк в oɥeнь шиpoкoм диaпaзoнe cкopocтeи cдвигa. Пpиpaвнeни e нyлy oднoи из кoнcтaнт мoдeли, пpивoдит ypaвнeниe мoдeли к видy ypaвнeния cтeпeннoгo зaкoнa, o кoтopoм извecтнo, ɥтo oнo oпиcывaeт пoвeдeниe мнoгич нe-ньyтoнoвcкич жидкoc тeи.

References (4)

A.W. Sisko
Capillary viscometer for non-Newtonian liquids
J. Colloid Sci.
(1960)
A.W. Sisko
The flow of lubricating greases
Ind. Engng Chem.
(1958)

There are more references available in the full text version of this article.

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Close-contact melting of phase change materials with a non-Newtonian power-law fluid liquid phase—Modeling and analysis
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In the current study, we model and analyze close-contact melting of a vertical cylinder on an isothermal surface, where the molten liquid phase rheological properties correspond to the non-Newtonian power-law fluid model. Under the assumptions of thin layer approximation and quasi-steady conditions, we derive analytical solutions for the initial molten layer thickness and maximum pressure as well as the time-dependent melt fraction, molten layer thickness, and pressure and velocity distributions in the thin molten layer. Dimensional analysis of the various derived analytical expressions reveals the governing dimensionless groups and their fundamental scaling laws. We show that the derived scaling laws generalize the classical and well-established Newtonian close-contact melting scaling laws. Moreover, the new model provides quantitative estimations that are extensively compared against the Newtonian case. It is shown that for typical values of the flow behavior index, the dimensionless pressure and hydraulic resistance in the molten layer decrease monotonically as the flow behavior index decreases. As a result, the dimensionless molten layer thickness decreases. Thus, we demonstrate that non-Newtonian effects can significantly impact the melting dynamics. As such, in comparison with the Newtonian fluid case, the melting rate is enhanced for shear-thinning fluids, whereas the melting rate is reduced for shear-thickening fluids.
Discussion on “Analysis of Bingham fluid radial flow in smooth fractures” [J Rock Mech Geotech Eng 12 (2020) 1112–1118]
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Recently, Zou et al. (2020a) published a theoretical analysis on the radial flow of a Bingham fluid, where they argued that the classical analysis by Dai and Bird (1981) violates the mass conservation. The present discussion aims to clarify this conflict between those two studies. It is noted that Zou et al. (2020a) presumed the gap-wise mass flux is negligible in the mass conservation equation, while Dai and Bird (1981) did not require so in their model, and this is found to be the origin of the conflict. In fact, Dai and Bird (1981)’s model is shown to not violate the mass conservation. Therefore, those two models should be viewed as separate models derived from different perspectives. Details of the major difference between the two models are discussed.
Flow classification of radial and squeeze flows between parallel disks
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Citation Excerpt :
For instance, it can be employed for the construction of a constitutive equation that benefits from being simple while rheological material functions can be independently fitted to experimental data [18]. Particularly in this study, power-law and Bingham fluids are considered, of which lubrication approximation solutions for the radial and squeeze flows are available in the literature [4,5,11,19–21]. Though, it should be mentioned at this point that the lubrication approximation solution possesses some issues which will be described later.
The flow classification developed by Thompson and Souza Mendes [Int. J. Engng. Sci. 43 (2005) 79-105] is applied to radial and squeeze flows between two parallel disks. Based on the lubrication approximation, explicit expressions of the flow classification parameter, namely $ℛ$ , could be obtained for those flows, and the results for power-law and Bingham fluids are presented. It is shown that $0 \leq ℛ \leq 1$ in the radial flow of both power-law and Bingham fluids, which reflects mixed shear and extensional flow. In the squeeze flow of a power-law fluid, the overall distribution of $ℛ$ is similar to the radial flow case. In the squeeze flow of a Bingham fluid, on the other hand, it is noted that there can be a particular region where $ℛ > 1$ , which is due to the rapid rotation of the principal axes of the rate-of-strain tensor.
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Assuming that the flow is laminar and that the fracture aperture is much smaller than its radius, the three-dimensional radial flow can be simplified as two-dimensional flow based on the lubrication approximation that has been widely adopted in the literature. For instance, Na and Hansen1 analyzed the steady-state radial flow of Sisko fluids for which the power-law and Newtonian fluids are special cases. Laurencena and Williams2 theoretically and experimentally studied radial flow of power-law fluids in homogeneous fractures.
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Radial flow of viscous non-Newtonian fluids between disks

Abstract

Résumé

Zusammenfassung

Реферат

J. Colloid Sci.

The flow of lubricating greases

Ind. Engng Chem.