Dispersion Relation of a Ferrofluid Layer of Any Thickness and Viscosity in a Normal Magnetic Field; Asymptotic Regimes

We have calculated the general dispersion relationship for surface waves on a ferrofluid layer of any thickness and viscosity, under the influence of a uniform vertical magnetic field. The amplification of these waves can induce an instability called peaks instability (Rosensweig instability). The expression of the dispersion relationship requires that the critical magnetic field and the critical wavenumber of the instability depend on the thickness of the ferrofluid layer. The dispersion relationship has been simplified into four asymptotic regimes: thick or thin layer and viscous or inertial behaviour. The corresponding critical values are presented. We show that a typical parameter of the ferrofluid enables one to know in which regime, viscous or inertial, the ferrofluid will be near the onset of instability.Comment: 21 pages, 6 eps figures, Latex, to be published in Journal de Physique I

Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation

We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.Comment: 6 pages, 2 figure

Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave and the spatial decay. When the motility is unbounded we show that the position of the front scales as $t^{3/2}$. When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with source term.Comment: 7 page

High frequency wave packets for the Schr\"odinger equation and its numerical approximations

We build Gaussian wave packets for the linear Schr\"odinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat, Zuazua, Numerical dispersive schemes for the nonlinear Schr\"odinger equation, SIAM. J. Numer. Anal., 47(2) (2009), 1366-1390. It is by now well known that bigrid algorithms provide filtering mechanisms allowing to recover the uniformity of the dispersive properties as the mesh size goes to zero. We analyze and illustrate numerically how these high frequency wave packets split and propagate under these bigrid filtering mechanisms, depending on how the fine grid/coarse grid filtering is implemented.Comment: 8 pages, 3 figure

Spectral analysis of surface waves for the characterization of the EDZ in circular galleries

Lors du creusement de galeries profondes ou de tunnels, les propriétés hydromécaniques de la roche encaissante autour de l'ouvrage sont altérées sur une certaine distance qui dépend de la nature de la roche et du type d'excavation. Une telle zone est appelée Excavation Damaged Zone (EDZ). Cette altération de l'encaissant se caractérise par une densification de la fracturation intrinsèque de la roche. La connaissance des caractéristiques mécaniques de l'EDZ ainsi que son extension est actuellement un axe majeur de recherche notamment pour la conception de centres de stockage souterrains des déchets nucléaires. En effet, l'EDZ, par son réseau de fractures, est considéré comme un chemin potentiel pour les radionucléides et donc comme un facteur de possible contamination du milieu. Les méthodes géophysiques initialement utilisées à des échelles kilométriques pour analyser les événements géologiques, sont dorénavant transposées à des échelles métriques voire centimétriques et appliquées en génie civil ou dans tout autre domaine de l'ingénierie. L'intérêt de telles méthodes est leur caractère non destructif qui les rend faciles d'utilisation et généralement moins coûteuses que d'autres méthodes destructives. Elles permettent aussi un suivi dans le temps de l'évolution des propriétés des matériaux auscultés. La MASW (Multiple Acquisition of Surface Wave) est une méthode géophysique utilisant le principe de dispersion des ondes de surface (Park et al, 1999). Cette méthode a été transposée dans ce contexte afin d'obtenir un profil 1D des vitesses des ondes de cisaillement (S) autour d'un ouvrage souterrain et ainsi de déterminer l'extension et les caractéristiques en terme de vitesse des ondes S de l'EDZ. L'intérêt de cette méthode est sa facilité de mise en oeuvre et la possibilité de l'utiliser sans restriction majeure

Relations de dispersion pour des chaînes linéaire comportant des interactions harmoniques auto-similaires

Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, etc. and look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity? How can we describe the mechanics of self-similar structures in the static and dynamic framework? Physical systems with self-similarity as a symmetry property require the introduction of non-local particle-particle interactions and a (quasi-) continuous distribution of mass. We construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D'Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals and which yields in the low-frequency regime a power law frequency dependence for the oscillator density. For details of the present model we refer to our recent paper (Michelitsch et al., Phys. Rev. E 80, 011135 (2009))