Abstract
The steady-state Couette flow of a yield-stress material obeying the Bingham-plastic constitutive equation is analyzed assuming that slip occurs when the wall shear stress exceeds a threshold value, the slip (or sliding) yield stress. The case of Navier slip (zero slip yield stress) is studied first in order to facilitate the analysis and the discussion of the results. The different flow regimes that arise depending on the relative values of the yield stress and the slip yield stress are identified and the various critical angular velocities defining those regimes are determined. Analytical solutions for all the regimes are presented and the implications for this important rheometric flow are discussed.
Highlights
Yield-stress, or viscoplastic, materials constitute a very important class, which includes foams, emulsions, colloids, gels, pastes, and suspensions that are of great interest in pharmaceutics, cosmetics, food, oil, and construction industries [1]
If τ∗ is the viscous stress tensor, γ ≡ ∇u∗ + (∇u∗ )T is the rate-of-strain tensor, where u∗ is the velocity q vector and the superscript T denotes the transpose, and τ∗ ≡ τ∗ : τ∗ /2 and γ ≡ γ : γ /2 are the magnitudes of τ∗ and γ, respectively, the Bingham-plastic model can be written as follows:
Philippou et al [41] solved analytically both the steady-state and time-dependent Couette flows of a Newtonian fluid with wall slip following Equation (3), i.e., with non-zero slip yield stress, showing the existence of three steady-state regimes, defined by the critical values of the angular velocity at which slip starts at the two cylinders
Summary
Yield-stress, or viscoplastic, materials constitute a very important class, which includes foams, emulsions, colloids, gels, pastes, and suspensions that are of great interest in pharmaceutics, cosmetics, food, oil, and construction industries [1]. Philippou et al [41] solved analytically both the steady-state and time-dependent Couette flows of a Newtonian fluid with wall slip following Equation (3), i.e., with non-zero slip yield stress, showing the existence of three steady-state regimes, defined by the critical values of the angular velocity at which slip starts at the two cylinders. The objective of the present work is to investigate the effect of wall slip with nonzero slip yield stress on the circular Couette flow of an ideal Bingham plastic and to identify the various flow regimes that arise depending on the relative values τ∗0 and τ∗c Besides rheometry, this flow is encountered in industrial applications, such as electrorheological clutches [42], catalytic chemical reactors, filtration devices, liquid-liquid extractors, journal bearings, and oil and gas drilling [43].
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