Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.     

Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres

We describe the asymptotic behaviour of a cylindrical elasticbody, reinforced along identical -periodically distributed fibresof size r, with 0 < r < , filled in with some differentelastic material, when this small parameter goes to 0. Thecase of small deformations and small strains is considered.We exhibit a critical size of the fibres and a critical linkbetween the radius of the fibres and the size of the Lamécoefficients of the reinforcing elastic material. Epi-convergencearguments are used in order to prove this asymptotic behaviour.The proof is essentially based on the construction of appropriatetest functions.     

Low-energy measurement of the 7Be(p,gamma)8B cross section

We have measured the cross section of the 7Be(p,gamma)8B reaction for E(c.m.) = 185.8, 134.7, and 111.7 keV using a radioactive 7Be target (132 mCi). Single and coincidence spectra of beta+ and alpha particles from 8B and 8Be* decay, respectively, were measured using a large acceptance spectrometer. The zero energy S factor inferred from these data is 18.5+/-2.4 eV b and a weighted mean value of 18.8+/-1.7 eV b (theoretical uncertainty included) is deduced when combining this value with our previous results at higher energies.     

New Total Synthesis of O‐Methyledgeworin and Edgeworin

Upon thermal condensation of diethyl (coumarinyl‐7‐oxy)malonate with O‐methyl‐resorcine, the corresponding bis[coumarinyl]ether, 4‐hydroxy‐O‐methyledgeworin is obtained in good yield. This leads to 4‐bromo‐O‐methyledgeworin, O‐methyledgeworin, and edgeworin.     

Asymptotic analysis and boundary homogenization in linear elasticity

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of ℝ³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ϵ distributed on the boundary of this domain, when the parameter ϵ goes to 0. We use epi‐convergence arguments in order to establish this asymptotic behaviour. We then specialize this general situation to the case of identical strips of size r_ϵ ϵ‐periodically distributed on the lateral surface of an axisymmetric body. We exhibit a critical size of the strips through the limit of the non‐negative quantity −1/(ϵ ln r_ϵ) and we identify the different limit problems according to the values of this limit. Copyright © 2000 John Wiley & Sons, Ltd.     

Asymptotic behaviour of contact problems between two elastic materials through a fractal interface

Two linear elastic materials are brought into contact along a fractal interface Σ. We suppose that the contact is perfect on small zones disposed on Σ. Using Γ-convergence arguments, we establish the possible limit contact laws which appear when letting the common size of these zones tend to 0. We also generalise these results to the case of more general obstacle problems on this fractal interface.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Green's function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.