The Theory of Distributions

“…It should be noted that this is different from the regularization described in some other texts (e.g. [16]), but that the 'finite part regularization' used here ensures that the consistency property holds, whereas the regularization used in [16] does not.…”

Section: (424)mentioning

confidence: 84%

“…where the dashed integral sign is used to indicate a finite part integral (see, for example, [16]). Using this result, we find that (3.23) becomes…”

Section: Continuum Formulationmentioning

confidence: 99%

The Mechanics of a Chain or Ring of Spherical Magnets

Hall¹,

Vella²,

Goriely³

2013

SIAM J. Appl. Math.

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Abstract. Strong magnets, such as neodymium-iron-boron magnets, are increasingly being manufactured as spheres. Because of their dipolar characters, these spheres can easily be arranged into long chains that exhibit mechanical properties reminiscent of elastic strings or rods. While simple formulations exist for the energy of a deformed elastic rod, it is not clear whether or not they are also appropriate for a chain of spherical magnets. In this paper, we use discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres. We find that the mechanical properties of a chain of magnets differ significantly from those of an elastic rod: while both magnetic chains and elastic rods support bending by change of local curvature, nonlocal interaction terms also appear in the energy formulation for a magnetic chain. This continuum model for the energy of a chain of magnets is used to analyse small deformations of a circular ring of magnets and hence obtain theoretical predictions for the vibrational modes of a circular ring of magnets. Surprisingly, despite the contribution of nonlocal energy terms, we find that the vibrations of a circular ring of magnets are governed by the same equation that governs the vibrations of a circular elastic ring.

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“…(12) can be explicitly solved for δv ξ using distribution Fourier calculus [12] to yield the result Eq. (10).…”

mentioning

confidence: 99%

Kinetic Roughening in Two-Phase Fluid Flow through a Random Hele-Shaw Cell

Pauné¹,

Casademunt²

2003

Phys. Rev. Lett.

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A nonlocal interface equation is derived for two-phase fluid flow, with arbitrary wettability and viscosity contrast c = (µ1 − µ2)/(µ1 + µ2), in a model porous medium defined as a Hele-Shaw cell with random gap b0 + δb. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, non-conserved and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which scale with capillary number as ℓ1 ∼ b0(cCa) −1/2 and ℓ2 ∼ b0Ca −1 . Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments.PACS numbers: 47.55. Mh, 68.35.Ct The displacement of a fluid by another in a porous medium is a problem of fundamental interest in nonequilibrium physics as a paradigm of interface dynamics in disordered media [1,2]. Experiments on bead packs in Hele-Shaw cells [3] in particular, have stimulated considerable theoretical efforts, but the problem has consistently revealed itself rather elusive [1,2]. More recently a new surge of interest has arisen with the recognition of the inherently non-local character of the problem as a key ingredient [4,5], and the realization of a new series of experiments in Hele-Shaw cells with random gap [6,7,8]. Roughening exponents of the proposed nonlocal equations have been explored by means of Flory-type scaling arguments [4] and phase field simulations [5,6]. While the specific properties of noise are known to be crucial to determine the universal aspects of interface roughening, fluctuations are usually modeled at a phenomenological level, and including only local capillary effects. Noise related to the non-Laplacian viscous pressure due to quenched disorder in the permeability has been so far neglected. While this may be justified for imbibition experiments [5], other situations, such as forced fluid invasion, do require a quantitative assessment of this point. In addition, it would be desirable to have a unified formulation for general conditions of viscosity contrast c = (µ 1 −µ 2 )/(µ 1 +µ 2 ) and wettability given the rich variety of phenomena that the experimental evidence has unveiled as a function of those parameters [2].Here we address the general problem of fluid displacement in a Hele-Shaw cell with random gap, as a simple model of a porous medium. This model system has the great advantage that no coarse-graining procedure must be invoked in the theoretical formulation, thus allowing us to derive ab initio a general and complete interface equation, quantitatively accurate, with explicit dependence on 'bare' parameters, and including all noise sources. On the experimental side, the system is also appealing since a direct control of the disorder is locally possible on the microscopic scale [6,7].A complete description of interface fluctuations must contain three basic physical effects of a porous matrix on its motion, namely local variations of (i) capillary pressure, (ii) permeability, and (iii) available vol...

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The Theory of Distributions

Cited by 52 publications

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Uniform central limit theorems for kernel density estimators

Uniform central limit theorems for kernel density estimators

The Mechanics of a Chain or Ring of Spherical Magnets

Kinetic Roughening in Two-Phase Fluid Flow through a Random Hele-Shaw Cell

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