Ben Schweizer

Homogenization of Maxwell's Equations in a Split Ring Geometry

We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer

Homogenization of Maxwell’s equations with split rings

\Omega\subset\mathbb{R}^3

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial.

Two-phase flow equations with a dynamic capillary pressure

We analyze the time harmonic Maxwell’s equations in a complex geometry. The scatterer Ω ⊂ R contains a periodic pattern of small wire structures of high conductivity, the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed meta-material.

Free boundary fluid systems in a semigroup approach and oscillatory behavior

We consider the one-dimensional degenerate two-phase flow equations as a model for water drive in oil recovery. The effect of oil trapping is observed in strongly heterogeneous materials with large variations in the permeabilities and in the capillary pressure curves. In such materials, a vanishing oil saturation may appear at interior interfaces and inhibit the oil recovery. We introduce a free boundary problem that separates a critical region with locally vanishing permeabilities from a strictly parabolic region and we give a rigorous derivation of the effective conservation law.

Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations

We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

Instability of gravity wetting fronts for Richards equations with hysteresis

We consider the free boundary problem of a liquid drop with viscosity and surface tension. We study the linearized equations with semigroup methods to get existence results for the nonlinear problem. The spectrum of the generator is computed. Large surface tension creates nonreal eigenvalues, and an exterior force results in a Hopf bifurcation. The methods are used to study wind-generated surface waves.

PERIODIC HOMOGENIZATION OF THE PRANDTL–REUSS MODEL WITH HARDENING

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in

Effective Maxwell Equations in a Geometry with Flat Rings of Arbitrary Shape

\mathbb{R}^n

Uniform estimates in two periodic homogenization problems

,