Accessibility setting

Section: New Results

Asymptotic Analysis

Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling

G. Allaire, O. Bernard, J.-F. Dufrêche and A. Mikelic

We study the homogenization (or upscaling) of the transport of a multicomponent electrolyte in a dilute Newtonian solvent through a deformable porous medium. The pore scale interaction between the flow and the structure deformation is taken into account. After a careful adimensionalization process, we first consider so-called equilibrium solutions, in the absence of external forces, for which the velocity and diffusive fluxes vanish and the electrostatic potential is the solution of a Poisson-Boltzmann equation. When the motion is governed by a small static electric field and small hydrodynamic and elastic forces, we use O'Brien's argument to deduce a linearized model. Then we perform the homogenization of these linearized equations for a suitable choice of time scale. It turns out that the deformation of the porous medium is weakly coupled to the electrokinetics system in the sense that it does not influence electrokinetics although the latter one yields an osmotic pressure term in the mechanical equations. As a consequence, the effective tensor satisfies Onsager properties, namely is symmetric positive definite.

On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder

G. Allaire and A. Piatnitski

This work studies the asymptotic behaviour of the principal eigenfunction of the adjoint Neumann problem for a convection diffusion operator defined in a long cylinder. The operator coefficients are 1-periodic in the longitudinal variable. Depending on the sign of the so-called longitudinal drift (a weighted average of the coefficients), we prove that this principal eigenfunction is equal to the product of a specified periodic function and of an exponential, up to the addition of fast decaying boundary layer terms.

A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

G. Allaire, M. Briane and M. Vanninathan

In this work we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation where the homogenized equation can be improved by adding small additionnal higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects. Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved by Conca et al. that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case.

Influence of the geometry on plasmonic waves

L. Chesnel, X. Claeys and S.A. Nazarov

In the modeling of plasmonic technologies in time harmonic regime, one is led to study the eigenvalue problem $-\mathrm{div}\left(\sigma \nabla u\right)=\lambda u\left(P\right)$, where $\sigma$ is a physical coefficient positive in some region ${\Omega }_{+}$ and negative in some other region ${\Omega }_{-}$. We highlight an unusual instability phenomenon for the source term problem associated with $\left(P\right)$: for certain configurations, when the interface between ${\Omega }_{+}$ and ${\Omega }_{-}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. We explain this property studying the eigenvalue problem $\left(P\right)$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. These theoretical results are illustrated by numerical experiments.

Effective boundary conditions for thin periodic coatings Participants

Mathieu Chamaillard, Houssem Haddar and Patrick Joly

We study the derivation of asymptotic model (generalized impedance boundary conditions) for periodic coating in 3-D configurations. The definition of periodicity for $3D$ surfaces cannot be done in an intrinsic way in general. We propose a definition based on the use of local parametrisations of the surface. This parametrization-dependent definition is somehow inspired from practical considerations in the manufacturing of periodic coatings. The asymptoic of the problem is constructed for the scalar problem and also for the electormagnetic problem. Approximate models of order 1 and 2 are derived for the scalar problem and are validated numerically. In the electromagnetic case, only conditions or order 1 are exhibited in the genral case.