A consistent thin-layer theory for Bingham plastics

Balmforth, N. J.; Craster, Richard V.

doi:10.1016/s0377-0257(98)00133-5

Cited by 181 publications

(261 citation statements)

References 29 publications

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“…Since to leading order, the governing equations are similar to those describing the steady uniform regime, it is expected that the plug structure subsists here, but it cannot be a true plug because this would conflict with the flow structure (which depends on x). To avoid inconsistencies in the perturbation analysis, we follow the treatment suggested by Balmforth and Craster [38], which consists in considering two asymptotic expansions (one for the sheared layer and the other one for the pseudo-plug layer) and matching them through a 'fake' yield surface, i.e., the interface at which the second stress invariant is at the yield-stress value to leading order. We refer the reader to Ref.…”

Section: Slope-dominated Regimementioning

confidence: 99%

“…An alternative approach is lubrication theory, which takes its roots in Reynolds' pioneering work. The theory is based on an approximation to the governing equations for shallow slopes and thin low-inertia flows through an asymptotic expansion in the aspect ratio ε = H * /L * , with 0377 H * and L * being the flow-depth and length scales, respectively [13,24,[33][34][35][36][37][38][39][40][41]. As pointed out by Balmforth et al [40], this theory can be extended to steep slopes by changing the scaling that underpins the asymptotic reduction of the local equations.…”

Section: Introductionmentioning

confidence: 99%

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The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

Ancey

Cochard

2009

Journal of Non-Newtonian Fluid Mechanics

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a b s t r a c tIn this paper we investigate the dam-break problem for viscoplastic (Herschel-Bulkley) fluids down a sloping flume: a fixed volume of fluid initially contained in a reservoir is released onto a slope and flows driven by gravitational forces until these forces are unable to overcome the fluid's yield stress. Like in many earlier investigations, we use lubrication theory and matched asymptotic expansions to derive the evolution equation of the flow depth, but with a different scaling for the flow variables, which makes it possible to study the flow behavior on steep slopes. The evolution equation takes on the form of a nonlinear diffusion-convection equation. To leading order, this equation simplifies into a convection equation and reflects the balance between gravitational forces and viscous forces. After presenting analytical and numerical results, we compare theory with experimental data obtained with a long flume. We explore a fairly wide range of flume inclinations from 6 • to 24 • , while the initial Bingham number lies in the 0.07-0.26 range. Good agreement is found at the highest slopes, where both the front position and flowdepth profiles are properly described by theory. In contrast, at the lowest slopes, theoretical predictions substantially deviate from experimental data. Discrepancies may arise from the formation of unsheared zones or lateral levees that cause slight flow acceleration.

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Section: Slope-dominated Regimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

Ancey

Cochard

2009

Journal of Non-Newtonian Fluid Mechanics

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“…At the yield surface between the two, the velocity field and shear stress are continuous. As demonstrated by Balmforth and Craster [4], this interface is a 'fake' yield surface above which there is 'pseudo-plug flow', with the deviation from a truly rigid, plug-like region being represented at higher asymptotic orders.…”

Section: Formulationmentioning

confidence: 88%

“…(This balance yields the velocity scale, U, introduced above.) In this regime the shear stress is then given to leading order by 4) and motion occurs provided the shear stress exceeds the yield stress at some point within the flow, a condition that requires…”

Section: Formulationmentioning

confidence: 99%

“…Free surface flows of viscoplastic materials have wide application to geophysical and industrial situations [1][2][3] and in recent years considerable progress has been made in developing consistent mathematical models of their motion when the flows are relatively shallow and predominantly parallel to the underlying boundary [4]. Viscoplasticity affects these flows in important ways: first the shear stresses developed within the flow diminish as the free surface is approached and so fluid close to the surface must exhibit 'unyielded' behaviour.…”

Section: Introductionmentioning

confidence: 99%

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Slumps of viscoplastic fluids on slopes

Hogg

Matson

2009

Journal of Non-Newtonian Fluid Mechanics

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a b s t r a c tFree-surface slumps of viscoplastic fluid, modelled using a Herschel-Bulkley constitutive law, are studied as they flow from rest, behind a rapidly removed dam, along an inclined two-dimensional channel. These dam-break flows are eventually arrested and attain a final, static state in which the streamwise pressure gradient and the along-slope component of gravitational acceleration are balanced by the yield stress. The shapes of the arrested free surfaces are compactly represented as Lambert-W functions and are characterised by two dimensionless parameters that measure the magnitude of the slope relative to the initial aspect ratio of the release and the magnitude of the yield stress relative to the weight of fluid layer. These states are only attained asymptotically for long times after the release and perturbations to the final profiles are shown to decay as 1/t n , where n is the power index in the Herschel-Bulkley model. This analysis requires careful formulation, because, formally, within a diminishing boundary layer close to the front of the motion, the size of the perturbation exceeds the arrested state. Thus, while straightforward linearisation of the governing equation is possible within the bulk of the flow outside of the boundary layer, this must be matched asymptotically to the solutions within the boundary layer close to the front. The presence of this region does not obviate computation of the rate of approach to the arrested state, but is required to permit complete calculation of the evolving shape of the free surface.

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Three‐dimensional surface deformation derived from airborne interferometric UAVSAR: Application to the Slumgullion Landslide

et al. 2016

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In order to provide surface geodetic measurements with “landslide‐wide” spatial coverage, we develop and validate a method for the characterization of 3‐D surface deformation using the unique capabilities of the Uninhabited Aerial Vehicle Synthetic Aperture Radar (UAVSAR) airborne repeat‐pass radar interferometry system. We apply our method at the well‐studied Slumgullion Landslide, which is 3.9 km long and moves persistently at rates up to ∼2 cm/day. A comparison with concurrent GPS measurements validates this method and shows that it provides reliable and accurate 3‐D surface deformation measurements. The UAVSAR‐derived vector velocity field measurements accurately capture the sharp boundaries defining previously identified kinematic units and geomorphic domains within the landslide. We acquired data across the landslide during spring and summer and identify that the landslide moves more slowly during summer except at its head, presumably in response to spatiotemporal variations in snowmelt infiltration. In order to constrain the mechanics controlling landslide motion from surface velocity measurements, we present an inversion framework for the extraction of slide thickness and basal geometry from dense 3‐D surface velocity fields. We find that the average depth of the Slumgullion Landslide is 7.5 m, several meters less than previous depth estimates. We show that by considering a viscoplastic rheology, we can derive tighter theoretical bounds on the rheological parameter relating mean horizontal flow rate to surface velocity. Using inclinometer data for slow‐moving, clay‐rich landslides across the globe, we find a consistent value for the rheological parameter of 0.85 ± 0.08.

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A consistent thin-layer theory for Bingham plastics

Cited by 181 publications

References 29 publications

The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

Slumps of viscoplastic fluids on slopes

Three‐dimensional surface deformation derived from airborne interferometric UAVSAR: Application to the Slumgullion Landslide

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