Stokes’ third problem for Herschel–Bulkley fluids

Ancey, Christophe; Bates, Belinda Margaret

doi:10.1016/j.jnnfm.2017.03.005

Cited by 7 publications

(5 citation statements)

References 40 publications

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“…In that case, Figure 10 of [2] shows that the static/flowing positions computed with the two models differ. A similar conclusion is obtained in [1] when comparing the interface dynamics in an Herschel-Bulkley fluid and in an averaged model.…”

Section: Properties Of the Simplified Modelsupporting

confidence: 83%

Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

et al. 2021

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This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. (2016) from a free surface incompressible model with viscoplastic rheology including a Drucker--Prager yield stress. This asymptotic model is based on the thin-layer approximation (the flow is thin in the direction normal to the topography compared to its down-slope extension), but the equations are not depth-averaged. In addition to the velocity, the model includes a free surface at the top of the flow and a free time-dependent static/flowing interface at the bottom. In the present work, we simplify this asymptotic model by decoupling the space coordinates, and keeping only the dependence on time and on the normal space coordinate $Z$. We introduce a time- and $Z$-dependent source term, assumed here to be given, which represents the opposite of the net force acting on the flowing material, including gravity, pressure gradient, and internal friction. We prove several properties of the resulting simplified model that has a time- and $Z$-dependent velocity and a time-dependent static/flowing interface as unknowns. The crucial advantage of this simplified model is that it can provide explicit solutions in the inviscid case, for different shapes of the source term. These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows.

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Section: Properties Of the Simplified Modelsupporting

confidence: 83%

Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

et al. 2021

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“…For a thixoviscoplastic Houška fluid, there are up to three flow regions: a fully structured core region; a partly structured but unyielded plug region; and an annular yielded region. The boundary between the plug and yielded regions moves in and out over the course of an oscillation; this behavior recalls that predicted in Stokes' third problem for a fluid with distinct static and dynamic yield stresses [26], which can be thought of as a limiting case of thixotropy.…”

Section: Discussionsupporting

confidence: 70%

Thixotropic pumping in a cylindrical pipe

2020

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We consider the flow of a thixotropic fluid in a uniform cylindrical pipe, driven by an oscillating pressure gradient or a body force. For a variety of rheological models, solutions can be obtained by integrating ordinary rather than partial differential equations: We illustrate this approach for the thixoviscoplastic Houška model and the thixoviscous simplified Moore-Mewis-Wagner model. We present asymptotic results in the limits of small and large Deborah numbers and numerical results for intermediate Deborah numbers. Under asymmetrical "sawtooth" forcing, thixotropy leads to the net transport of fluid along the pipe, even when there would be no net transport of the corresponding generalized-Newtonian fluid. We propose the name "thixotropic pumping" for this transport mechanism.

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“…Following start-up, the yield surface moves away from the plate before returning to the plate in finite time, after which the entire material is rigid; see figure 2 (cf. Ancey & Bates 2017).…”

Section: Physical Mechanismsmentioning

confidence: 99%

A layer of yield-stress material on a flat plate that moves suddenly

2022

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The flow of an unbounded Newtonian fluid above a flat solid plate, set into motion suddenly with uniform velocity, is a canonical problem investigated by Stokes (On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc., vol. 9, 1851) and Rayleigh (Lond. Edinb. Dubl. Philos. Mag. J. Sci., vol. 21, issue 126, 1911, pp. 697–711). We tackle the same problem but with the fluid replaced by a yield-stress material of finite height with a free surface; the latter ensures both yielded (fluid-like) and rigid behaviour. The interface between the yielded and rigid regions – the so-called ‘yield surface’ – always emerges from the plate at a rate faster than that for momentum diffusion. The rigid region, adjacent to the free surface, is accelerated by the constant yield stress acting at this yield surface. As the velocity of this region increases following start-up, its acceleration climaxes and subsequently diminishes. In this latter period, the thickness of the rigid region increases owing to relaxation of the velocity gradients and the stress. The yield surface eventually collides with the plate in finite time, after which the entire material moves in concert with the plate as a rigid body. Analytical and numerical results are presented that can form the basis for practical applications. This includes the development of a ‘rheological microscope’ to directly detect and measure the yield stress. The analysis focuses on exploring the flow physics of a Bingham material. This is shown to be similar to that of a general Herschel–Bulkley material with shear-thinning or shear-thickening rheology.

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Stokes’ third problem for Herschel–Bulkley fluids

Cited by 7 publications

References 40 publications

Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

Thixotropic pumping in a cylindrical pipe

A layer of yield-stress material on a flat plate that moves suddenly

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