Marcone Corrêa Pereira

In this paper we present the analytical study of the magnetohydrodynamic (MHD) flow through a rectangular duct driven by the pressure gradient and under the action of the transverse magnetic field. Motivated by various MHD applications in which hydrodynamic slip naturally occur, we prescribe the slipping boundary condition on the upper boundary which contains irregularities as well. Depending on the period of the boundary roughness, we derive three different limit problems by using rigorous analysis in the appropriate functional setting. This approach also enables us to determine the relative contribution of the MHD effect and the slip itself in the governing coupled system satisfied by the velocity and induced magnetic field.

In this paper we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard formula is computed allowing domain perturbations given by embeddings of $n$-dimensional Riemannian manifolds (possibly with boundary) of finite volume while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.