Resultados numéricos en el problema de la rejilla de Stokes

En este artículo se presenta un estudio numérico sobre el movimiento de un fluido viscoso incompresible que atraviesa una pared finamente perforada de espesor pequeño (una rejilla). Se muestran resultados numéricos para rejillas simétricas y no simétricas respecto a un plano de sustentación, los cuales confirman numéricamente ciertos resultados teóricos sobre el comportamiento del fluido cerca de la rejilla. En particular, se observa numéricamente que para rejillas muy finamente perforadas, el flujo se organiza cerca de la rejilla de modo de atravesarla con una velocidad constante. En el caso simétrico, esta velocidad es además perpendicular a la rejilla. Se estudian tres problemas tests.ljn this article we present a numerical analysis of a viscous incoinpressible fluid as it moves through a very thinly periodically perforated sieve with a non-zero thickness. Numerical results are obtained for symmetric and non symmetric sieves which confirm certain theorical results on the asymptotic behavior of the fluid flow near the sieve. In particular, we numerically verify that for a very thinly perforated sieve the fluid gets organized near the sieve in such a way that it crosses it with a constant velocity. In the symmetric case this velocity is also perpendicular to the sieve. We study three test problems.Peer Reviewe

Detection of a Moving Rigid Solid in a Perfect Fluid

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid-solid system fills the whole two dimensional space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in [3], that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, like ellipses for instance, for which the problem of detection admits a unique solution. Using complex analysis, we prove that the well-posedness of the inverse problem is equivalent to the solvability of an infinite set of nonlinear equations. This result allows us to show that when the solid enjoys some symmetry properties, it can be partially detected. Besides, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. Finally, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.Comment: 19 pages, 14 figure

Resultados numéricos en el problema de la rejilla de Stokes

En este artículo se presenta un estudio numérico sobre el movimiento de un fluido viscoso incompresible que atraviesa una pared finamente perforada de espesor pequeño (una rejilla). Se muestran resultados numéricos para rejillas simétricas y no simétricas respecto a un plano de sustentación, los cuales confirman numéricamente ciertos resultados teóricos sobre el comportamiento del fluido cerca de la rejilla. En particular, se observa numéricamente que para rejillas muy finamente perforadas, el flujo se organiza cerca de la rejilla de modo de atravesarla con una velocidad constante. En el caso simétrico, esta velocidad es además perpendicular a la rejilla. Se estudian tres problemas tests.ljn this article we present a numerical analysis of a viscous incoinpressible fluid as it moves through a very thinly periodically perforated sieve with a non-zero thickness. Numerical results are obtained for symmetric and non symmetric sieves which confirm certain theorical results on the asymptotic behavior of the fluid flow near the sieve. In particular, we numerically verify that for a very thinly perforated sieve the fluid gets organized near the sieve in such a way that it crosses it with a constant velocity. In the symmetric case this velocity is also perpendicular to the sieve. We study three test problems.Peer Reviewe

On the detection of several obstacles in 2D Stokes flow: topological sensitivity and combination with shape derivatives

International audienceWe consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain Ω from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape

Resultados numéricos en un modelo de lavado de una resina macroporosa

En este artículo se proponen dos métodos numéricos para resolver un sistema de ecuaciones en derivadas parciales formado por una ecuación del tipo parabólica, y otra hiperbólica. Estos métodos están basados en dos aproximaciones teóricas de la solución de la ecuación parabólica del sistema, y en un uso adecuado de un esquema de diferencias finitas para aproximar la solución de la ecuación hiperbólica. El sistema de ecuaciones en estudio tiene su origen en un problema real proveniente de la físico-química, que se explica en detalle en el artículo. En este problema concreto, ambos métodos fueron implementados computacionalmente, presentándose aquí los resultados numéricos obtenidos.Peer Reviewe

Bloch Approximation in Homogenization and Applications

The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in this paper. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions $u^\varepsilon$ of boundary value problems associated with such operators when the period $\varepsilon>0$ of the coefficients is small. In a previous work by C. Conca and M. Vanninathan [SIAM J. Appl. Math., 57 (1997), pp. 1639--1659], a new proof of weak convergence as $\varepsilon\to 0$ towards the homogenized solution was furnished using Bloch wave decomposition. Following the same approach here, we go further and introduce what we call Bloch approximation, which will provide energy norm approximation for the solution $u^\varepsilon$. We develop several of its main features. As a simple application of this new object, we show that it contains both the first and second order correctors. Necessarily, the Bloch approximation will have to capture the oscillations of the solution in a sharper way. The present approach sheds new light and offers an alternative for viewing classical results

Fourier approach to homogenization problems

This article is divided into two chapters. The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in the first chapter. Following a Fourier approach, we discuss some of the basic issues of the subject: main convergence theorem, Bloch approximation, estimates on second order derivatives, correctors for the medium, and so on. The second chapter is devoted to the discussion of some non-classical behaviour of vibration problems of periodic structures

An Inverse Problem in the Mathematical Modelling of our Sense of Smell

Páginas: 8-15The first step in our sensing of smell is the conversion of chemical odorants into electrical signals. This happens when odorants stimulate ion channels along cilia, which are long thin cylindrical structures in our olfactory system. Determining how the ion channels are distributed along the length of a cilium is beyond current experimental methods. Here we describe how this can be approached as a mathematical inverse problem. Precisely, two integral equations based mathematical models are studied for the inverse problem of deter- mining the distribution of ion channels in cilia of olfactory neurons from experimental data. The Mellin transform allows us to write an explicit formula for their solutions. Proving observability and continuity inequalities for the second integral equation is then a question of estimating the Mellin transform of the kernel on vertical lines. For the first integral model, an identifiability and a non observability (in some weighted spaces) results are proven