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We present a new basis of representation for the graphene honeycomb structure that facilitates the solution of the eigenvalue problem by reducing it to one dimension. We define spaces in these geometrical basis that allow us to solve the Hamiltonian in the edges of the lattice. We conclude that it is enough to analyze a one-dimensional problem in a set of coupled ordinary second-order differential equations to obtain the behavior of the solutions in the whole graphene structure.

We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.

International audience; We propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier–Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of weak solution in the two–dimensional problem and the existence of at least one solution in the three–dimensional case, together with regularity properties and an energy estimate. We also propose a fully discrete scheme of our problem using the characteristic method and we present numerical simulations in two physical examples.

International audience; We propose a new model in a fluid-structure system composed by a rigid body and a viscous incompress-ible fluid using a boundary condition based on Coulomb's law. This boundary condition allows the fluid to slip on the boundary if the tangential component of the stress is too large. In the opposite case, we recover the standard Dirichlet boundary condition. The governing equations are the Navier-Stokes system for the fluid and the Newton laws for the body. The corresponding coupled system can be written as a variational inequality. We prove that there exists a weak solution of this system.

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.

Numerical investigations of unsteady laminar flow and heat transfer in a channel of height H with periodically mounted square bars of height d = 0.2H arranged side by side to the approaching flow have been conducted for different transverse separation distances of the bars. Five cases with transverse separation distance of 0, 0.5, 1, 1.5 and 2d for a Reynolds number of 300 in a channel with a periodicity length of 2H were studied. The unsteady Navier–Stokes equations and the energy equation have been solved by a finite volume code with staggered grids combined with the SIMPLEC algorithm and a fine grid resolution. Due to the arrangement of bars detached from the channel walls the flow is unsteady with vortex shedding from the bars. The amplitude and mean values of the drag coefficients, skin friction coefficients, friction factor and Nusselt numbers have a strong dependence of the transverse separation distance of the bars.

This paper considers the periodic spectral problem associated with the Laplace operator written in \({\mathbb{R}^N}\) (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.

Papier accepté pour la publication dans Quarterly of Applied Mathematics; International audience; In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.

In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

In this paper, we consider a Lagrange-Galerkin scheme to approximate a two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.