Asymptotic Analysis

Using the periodic unfolding method we derive the homogenized equations for the quasi-static initial boundary value problem with internal variables, which model the deformation behavior of viscoplastic materials with a periodic microstructure. The free energy associated with the problem is allowed to be positive semi-definite.

We consider the two-dimensional Euler equation with periodic boundary conditions. We construct time quasi-periodic solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction of propagation is orthogonal to this variable, and the support is concentrated on flat strips of the stationary state. The frequencies of the solution are given by the locally constant velocities associated with the stationary state.

In this paper we obtain the Reynolds equation for the case of a Newtonian incompressible fluid with a stationary isothermal motion in a tribological system of radial face seals when one of the contact surfaces (the rotor) has an oscillatory motion around its equilibrium point. Moreover, it is supposed that this surface is rough. We also obtain the homogenized equation whose coefficients are given explicitly. Finally, we present numerical results for the homogenized equation.

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former.

First we give a Dobrushin-type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from the paper “On the motion of a small body immersed in a two-dimensional incompressible perfect fluid”, accepted for publication in Bulletin de la SMF where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation.

Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.