Homogenization of the Schrodinger equation with a time oscillating potential

We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory

Diffraction of Bloch Wave Packets for Maxwell's Equations

We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schr\"odinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations

A bound on the group velocity for Bloch wave packets

We give a direct proof that the group velocities of Bloch wave packet solutions of periodic second order wave equations cannot exceed the maximal speed of propagation of the periodic wave equation

Instability thresholds for flexible rotors in hydrodynamic bearings

Two types of fixed pad hydrodynamic bearings (multilobe and pressure dam) were considered. Optimum and nonoptimum geometric configurations were tested. The optimum geometric configurations were determined by using a theoretical analysis and then the bearings were constructed for a flexible rotor test rig. It was found that optimizing bearings using this technique produces a 100% or greater increase in rotor stability. It is shown that this increase in rotor stability is carried out in the absence of certain types of instability mechanisms such as aerodynamic crosscoupling. However, the increase in rotor stability should greatly improve rotating machinery performance in the presence of such forces as well

Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media

We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of $N$ chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behavior of its solution depending on a parameter $\beta$ which is the square of the ratio between a characteristic pore length and the Debye length. For small $\beta$ we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the {\it Donnan effect}, observing that the ions for which the charge is the one of the solid phase are expelled from the pores. For large $\beta$ we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid, so that the simplified additive theories, such as the one of the popular Derjaguin, Landau, Verwey and Overbeek (DLVO) approach, can be used. We show that the asymptotic behavior is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential)

Aerodynamic stiffness of an unbound eccentric whirling centrifugal impeller with an infinite number of blades

An unbounded eccentric centrifugal impeller with an infinite number of log spiral blades undergoing synchronous whirling in an incompressible fluid is considered. The forces acting on it due to coriolis forces, centripetal forces, changes in linear momentum, changes in pressure due to rotating and changes in pressure due to changes in linear momentum are evaluated

Oil seal effects and subsynchronous vibrations in high-speed compressors

Oil seals are commonly used in high speed multistage compressors. If the oil seal ring becomes locked up against the fixed portion of the seal, high oil film crosscoupled stiffnesses can result. A method of analysis for determining if the oil seals are locked up or not is discussed. The method is then applied to an oil seal in a compressor with subsynchronous vibration problems