Effective viscosity and Reynolds number of non-Newtonian fluids using Meter model
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ORIGINAL CONTRIBUTION
Effective viscosity and Reynolds number of non-Newtonian fluids using Meter model Takshak Shende1 · Vahid J. Niasar1 · Masoud Babaei1 Received: 22 May 2020 / Revised: 30 September 2020 / Accepted: 3 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The Meter model (a four-parameter model) captures shear viscosity–shear stress relationship (S-shaped type) of polymeric non-Newtonian fluids. We devise an analytical solution for radial velocity profile, average velocity, and volumetric flow rate of steady-state laminar flow of non-Newtonian Meter model fluids through a circular geometry. The analytical solution converts to the Hagen–Posseuille equation for the Newtonian fluid case. We also develop the formulations to determine effective viscosity, Reynolds number, and Darcy’s friction factor using measurable parameters as available rheological models do not correctly define these parameters for a given set of flow condition in circular geometry. The analytical solution and formulations are validated against experimental data. The results suggest that the effective Reynolds number and effective friction factor estimated using the proposed formulation help characterize non-Newtonian fluid flow through a circular geometry in laminar and turbulent flows. Keywords Non-Newtonian fluid · Shear stress · Viscosity · Analytical solution · Shear-thinning fluid · Reynolds number · Micro-capillary fluid flow
Introduction The laminar flow of a non-Newtonian fluid (described using generalized Newtonian fluid model) through a circular capillary/tube has broader engineering application (e.g., polymer fluid flow through pipes in industrial settings (Bird et al. 1987), capillary bundle model of porous media (Savins 1969), pore-network model (Sochi and Blunt 2008)). Among generalized Newtonian fluid models (Yilmaz and Gundogdu 2008), Cross (Cross 1965), Carreau (Yasuda 1979), Carreau–Yasuda (Yasuda 1979), Meter (Meter and Bird 1964; Meter 1964; Savins 1969; Tsakiroglou 2002; Tsakiroglou et al. 2003; Tsakiroglou et al. 2003), and Steller–Ivako (Steller and Iwko 2018) models can predict
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00397-020-01248-y) contains supplementary material, which is available to authorized users. Masoud Babaei
[email protected] 1
Department of Chemical Engineering and Analytical Science, The University of Manchester, Manchester, UK
S-shaped rheological properties (i.e., constant viscosity at low and high shear values and decreasing viscosity at intermediate shear values) of many shear-thinning fluids. Attempts have been made by many investigators to obtain an analytical solution for flow of non-Newtonian fluid through a circular tube. Matsuhisa and Bird derived an analytical solution for the laminar flow of a fluid obeying shear stress–dependent Ellis model (Matsuhisa and Bird 1965). Meter and Bird proposed the analytical solution for the flow of shear stress–dependent Meter model fluid in a c
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