Abrasion of flat rotating shapes

Roth, Aaron; Marques, Carlos M.; Durian, Douglas J.

doi:10.1103/physreve.83.031303

Cited by 5 publications

(7 citation statements)

References 10 publications

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“…Clearly, shape must be explicitly considered when assessing the contribution of abrasion to downstream fining of sediments [17]. Several recent experiments that examined shape evolution under abrasion [18,19] provided qualitative confirmation of geometric models [20][21][22], which predict that regions of high curvature are preferentially eroded. Building on this work, we present the first quantitative test of the curvature-driven abrasion model originally proposed by Firey [20], using laboratory experiments and a discrete chopping model.…”

Section: Introductionmentioning

confidence: 90%

How River Rocks Round: Resolving the Shape-Size Paradox

et al. 2014

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River-bed sediments display two universal downstream trends: fining, in which particle size decreases; and rounding, where pebble shapes evolve toward ellipsoids. Rounding is known to result from transport-induced abrasion; however many researchers argue that the contribution of abrasion to downstream fining is negligible. This presents a paradox: downstream shape change indicates substantial abrasion, while size change apparently rules it out. Here we use laboratory experiments and numerical modeling to show quantitatively that pebble abrasion is a curvature-driven flow problem. As a consequence, abrasion occurs in two well-separated phases: first, pebble edges rapidly round without any change in axis dimensions until the shape becomes entirely convex; and second, axis dimensions are then slowly reduced while the particle remains convex. Explicit study of pebble shape evolution helps resolve the shape-size paradox by reconciling discrepancies between laboratory and field studies, and enhances our ability to decipher the transport history of a river rock.

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Section: Introductionmentioning

confidence: 90%

How River Rocks Round: Resolving the Shape-Size Paradox

et al. 2014

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“…However, a choice like this would also seem physically unmotivated. The function f allows for more general radius-speed relations, and the tree picture makes it clear that all that changes is the relative rate of growth or shrinkage of each edge, and not the overall qualitative picture; in particular, this may explain the observation in [6] that the model was robust to changing r to r α with α = 1/2, 1, 2, 3. We speculate next on some possible choices for f .…”

Section: Discussionmentioning

confidence: 99%

“…We did not carry out an extensive comparison here with the numerical solution of [6], but the plotted curves appeared indistinguishable in a few checks. Indeed, it would be interesting to place this work on firmer mathematical ground along the lines of [8] by analyzing how the corners generated by the amputation and interpolation processes are smoothed by the addition of higher derivative terms, and how this happens in the finite difference scheme of Roth, Marques and Durian.…”

Section: Discussionmentioning

confidence: 99%

“…More recently, deterministic erosion processes have been studied by Roth, Marques and Durian. They performed an experiment to measure the contours of linoleum tiles of fixed thickness and varying shape that they had rotated for differing amounts of time in a slurry of grit [6]. This paper describes the solution to their rotational abrasion model.…”

mentioning

confidence: 99%

“…It is obvious that circles (r(θ, t) = constant) are stationary solutions to this equation, and the evolution of experimental contours computed in [6] by a finite differencing scheme also evolved towards circles unerringly. These solutions also matched quantitatively the evolution of several geometrical quantities extracted from their experimental data, such as area, perimeter, and the width of the curvature distribution.…”

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confidence: 99%

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Solution of the Roth-Marques-Durian rotational abrasion model

Chen

2011

Phys. Rev. E

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We solve the rotational abrasion model of Roth, Marques and Durian [Phys. Rev. E (2010)], a one-dimensional quasilinear partial differential equation resembling the inviscid Burgers equation with the unusual feature of a step function factor as a coefficient. The complexity of the solution is primarily in keeping track of the cases in the piecewise function that results from certain amputation and interpolation processes, so we also extract from it a model of an evolving planar tree graph that tracks the evolution of the coarse features of the contour.What determines the shapes of pebbles is an intriguing physical question with interest not just to beachcombers out for walks but also geologists, who are interested in the history of erosion at a site [1], as well as mechanical engineers [2], who wish to understand wear processes. Recently several models have been proposed to explain these shapes. Two stochastic models are of note, a "cutting model" [3,4] which accompanied an experimental measurement of pebbles rotating in a tray and an analytically tractable "chipping model" [5]. These models lead to distributions of non-circular shapes. More recently, deterministic erosion processes have been studied by Roth, Marques and Durian. They performed an experiment to measure the contours of linoleum tiles of fixed thickness and varying shape that they had rotated for differing amounts of time in a slurry of grit [6]. This paper describes the solution to their rotational abrasion model. Supposing r(θ, t) describes the radial distance of the contour from the rotational axis as a function of angle θ and time t, they proposedHere C is a positive proportionality factor with dimensions of Angle/(Time×Length) and H(x) denotes a Heaviside step function defined so that H(x) = 0 for x ≤ 0 and H(x) = 1 for x > 0. It is obvious that circles (r(θ, t) = constant) are stationary solutions to this equation, and the evolution of experimental contours computed in [6] by a finite differencing scheme also evolved towards circles unerringly. These solutions also matched quantitatively the evolution of several geometrical quantities extracted from their experimental data, such as area, perimeter, and the width of the curvature distribution. Summarized here are the key ideas behind our exact solution of this equation. First, we exploit a connection to the Burgers equation at zero viscosity, a well-studied equation from gas dynamics [7]. Second, the solution r(θ, t) can be written in a piecewise fashion as a union of r = constant (circular) arcs and certain "stretched" segments of the initial contour. More precisely, these segments are curves of the form r 0 (θ(θ 0 , t)) where r 0 (θ 0 ) is the initial contour and θ(θ 0 , t) at fixed θ 0 is a linear function in t. The solution is constructed to be continuous, but will admit corners with discontinuous slope generically. Finally, the organization of the solution has a strong combinatorial flavor, and the evolution of the pattern of critical points in the contour is captured by a model of an evo...

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Downstream fining in a megaclast-dominated fluvial system: The Sabeto River of western Viti Levu, Fiji

et al. 2019

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Megaclast-dominated fluvial systems have been largely ignored by researchers. Yet megaclasts exert an important control on channel geomorphology, flow behaviour and sediment transport, and such systems are probably far more common than this neglect would suggest. Despite the explicit identification of megaclasts as the products of cataclysmic flow, observation of modern fluvial systems confirms that megaclasts can be entrained by flows that lie within the normal flow regime. The pattern of fining along the megaclast-dominated Sabeto River of western Viti Levu, Fiji reveals a strong negative exponential relationship between particle size and distance downstream, and a rate of downstream fining comparable with that of other gravel-bed rivers. The downstream fining pattern is straightforward and step-like discontinuities in the fining gradient are absent. This is suggestive of a sedimentary system only minimally disrupted by the supply of sediment from tributaries or the reworking of material from sources such as hillslopes and alluvial terraces. There is evidence, however, of anomalously coarse sub-populations at two sites. These coarser components must predate the other deposits at the sites. We speculate that they represent the products of earlier and higher-magnitude events. The bed sediments may thus be composite features, made up of individual components each transported by a specific event of particular magnitude. Our work confirms the relationship between the rate of downstream fining and particle size. This may arise because the distance over which individual particles are transported is likely to be size dependent. It is also possible that abrasion may be more effective in the case of larger particles than smaller ones, with the result that the coarser fraction of the river's bed sediment fines downstream more rapidly than the finer fraction. There is a long-standing debate about the relative roles of abrasion and sorting in downstream fining. Our data reveal little evidence for a downstream increase in the roundness of the river's gravels, implying that abrasion is likely to play a minor role in the fining process in this system.

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Abrasion of flat rotating shapes

Cited by 5 publications

References 10 publications

How River Rocks Round: Resolving the Shape-Size Paradox

How River Rocks Round: Resolving the Shape-Size Paradox

Solution of the Roth-Marques-Durian rotational abrasion model

Downstream fining in a megaclast-dominated fluvial system: The Sabeto River of western Viti Levu, Fiji

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