The converging flow of viscoplastic fluid in a wedge or cone

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When viscous fluid in a corner is disturbed, eddies can form in the absence of inertia. Examples of flow configurations in which this motion occurs include flow through an abrupt contraction and over a cavity. Six decades ago, Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18) calculated the slow viscous flow of Newtonian fluids in sharp corners, detailing his eponymous ‘Moffatt eddies’. In this study we examine corner flows of viscoplastic materials, a class of non-Newtonian fluids which exhibit solid-like behaviour for stresses below a yield stress. Specifically, we consider a Bingham fluid, for which the material is perfectly rigid at stresses below the yield stress. While a static unyielded plug forms at the tip of the corner, eddies analogous to those found by Moffatt can also form. We examine these viscoplastic eddies numerically, by computing finite element solutions using the augmented-Lagrangian method, and analytically, by employing a viscoplastic boundary layer formulation and scaling arguments. We measure the depth of the static plug as a function of the Bingham number (dimensionless yield stress), show that the process of a new eddy forming as the Bingham number is decreased is driven by the pressure in the yielded fluid adjacent to the static plug, and provide a heuristic argument for the critical Bingham number at which this occurs.

Section: Results and Key Scalingsmentioning

confidence: 99%

Viscoplastic corner eddies

Taylor-West

Hogg

2022

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“…We investigate the incompressible flow of a Herschel–Bulkley viscoplastic fluid between two rigid, semi-infinite plates, hinged at the origin and rotating towards one another with angular velocity, (see figure 1 a ), thus extending the classical problem of viscous Newtonian fluid flow in this configuration (see, for example, Moffatt (1964)). Recent studies of viscoplastic fluids in converging and recirculating corner flows have demonstrated how the existence of a yield-stress changes the structure of the Newtonian solutions significantly, leading to the occurrence of rigid unyielded regions of fluid, or ‘plugs’, and the development of viscoplastic boundary layers when the dimensionless yield-stress is large (Taylor-West & Hogg 2021, 2022). In both of these previous studies, the magnitude of the strain rate varies with the distance from the vertex, resulting in viscously and yield-stress dominated behaviour in different regions of the wedge.…”

Section: Introductionmentioning

confidence: 99%

Viscoplastic flow between hinged plates

Taylor-West

Hogg

2023

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The incompressible motion of viscoplastic fluid between two semi-infinite rigid plates, hinged at their ends and rotating towards one another at constant angular velocity, generates self-similar flow fields because there is no externally imposed length scale in the absence of inertia. The magnitude of the strain rate scales with the angular velocity of the plates and the dimensionless deviatoric stresses are functions only of the polar angle and a dimensionless measure of the yield stress; they are independent of the radial distance from the corner. These flows feature unyielded regions adjacent to the boundaries for sufficiently large angles between the plates. Moreover, when the dimensionless yield stress is large, there are viscoplastic boundary layers that are attached to the boundary or the plug, the asymptotic structures of which are constructed.

“…Hewitt & Balmforth 2018). Taylor-West & Hogg (2021) have previously considered converging viscoplastic flow in a wedge; in this volume the same authors tackle the problem of viscoplastic corner flow (Taylor-West & Hogg 2022).…”

Section: Introductionmentioning

confidence: 99%

Clogged corners

Hewitt

2022

Viscoplastic materials cannot flow if the local stress falls below the yield stress of the material. Such materials tend to ‘clog up’ geometrical features like corners or side branches by forming rigid plugs, because the stress becomes insufficient to yield them. In this volume, Taylor-West and Hogg (J. Fluid Mech., vol. 941, 2022, A64) consider the problem of flow of an idealised viscoplastic fluid in a wedge forced by a disturbance far from the corner; this is the viscoplastic analogue of Moffatt's famous corner eddies. They demonstrate and describe the details of how the fluid clogs up the corner of the wedge with a rigid plug bounded by a viscoplastic eddy. More generally, this study provides a way to explain some of the details of cloggings in different geometries that have previously been observed in viscoplastic materials.