Juan Casado-Díaz

Why viscous fluids adhere to rugose walls: A mathematical explanation

Abstract The main purpose of this paper is to justify rigorously the following assertion: A viscous fluid cannot slip on a wall covered by microscopic asperities because, due to the viscous dissipation, the surface irregularities bring to rest the fluid particles in contact with the wall. In mathematical terms, this corresponds to an asymptotic property established in this paper for any family of fields that slip on oscillating boundaries and remain uniformly bounded in the H1-norm.

The two-scale convergence method applied to generalized Besicovitch spaces

The two–scale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems. In the present paper we develop this theory to generalized Besicovitch spaces, which include the almost–periodic functions. The main difficulty comes from the fact that these spaces are not separable. We also show how to apply these results to the homogenization of partial differential problems in this framework.

Two-Dimensional Div-Curl Results: Application to the Lack of Nonlocal Effects in Homogenization

In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L1 combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two. More precisely, in the two-dimensional case we extend the Murat–Tartar H-convergence which holds for uniformly bounded and equi-coercive conductivity sequences, as well as the compactness result which holds for bounded and equi-integrable conductivity sequences in L1. Our homogenization results are based on extensions of the classical div-curl lemma, which are also specific to the dimension two.

ASYMPTOTIC BEHAVIOR OF A VISCOUS FLUID WITH SLIP BOUNDARY CONDITIONS ON A SLIGHTLY ROUGH WALL

For an oscillating boundary of period and amplitude e, it is known that the asymptotic behavior when e tends to zero of a three-dimensional viscous fluid satisfying slip boundary conditions is the same as if we assume no-slip (adherence) boundary conditions. Here we consider the case where the period is still e but the amplitude is δe with δe/e converging to zero. We show that if tends to infinity, the equivalence between the slip and no-slip conditions still holds. If the limit of belongs to (0, +∞) (critical size), then we still have the slip boundary conditions in the limit but with a bigger friction coefficient. In the case where tends to zero the boundary behaves as a plane boundary. Besides the limit equation, we also obtain an approximation (corrector result) of the pressure and the velocity in the strong topology of L2 and H1 respectively.

An adaptation of the multi-scale methods for the analysis of very thin reticulated structures

Abstract The purpose of this Note is to present a new approach to the analysis of thin reticulated structures involving several parameters. The method is related to the two-scale convergence method.

Asymptotic Behavior of the Navier--Stokes System in a Thin Domain with Navier Condition on a Slightly Rough Boundary

We study the asymptotic behavior of the solutions of the Navier--Stokes system in a thin domain

The limit of Dirichlet systems for variable monotone operators in general perforated domains

\Omega_\varepsilon

Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence

of thickness

A derivation theory for generalized Besicovitch spaces and its application for partial differential equations

\varepsilon

Relaxation of a Control Problem in the Coefficients with a Functional of Quadratic Growth in the Gradient

satisfying the Navier boundary condition on a periodic rough set