Abstract
This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.
Highlights
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As observed in [18], since the location of the center of gravity is determined by timedependent integrals, (1) is a non-local partial integro-differential equation
The main goal of this paper is to introduce a simple mathematical equation based on physically intuitive heuristics that may help explain the limiting oval shapes of stones wearing down solely by frictional abrasion by waves on a flat sandy beach
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Berger (center); and beach stones collected by the author on several continents (right; the largest is about 30 cm long and weighs about 13 kg). In studying the evolving shapes of beach stones, Aristotle conjectured that spherical shapes dominate (see [18]). In support of his theory, he proposed that the inward rate of abrasion in a given direction is an increasing function of the distance from the center of mass of the stone to the tangent plane (the beach) in that direction, the intuition being that the further from the center of mass a point is, the more likely incremental pieces are to be worn off, since the moment arm is larger. For many natural choices of f , the shape of a convex stone eroding under (1) apparently becomes spherical in the limit; see numerical experiments below
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