326 research outputs found
A Bloch wave numerical scheme for scattering problems in periodic wave-guides
We present a new numerical scheme to solve the Helmholtz equation in a
wave-guide. We consider a medium that is bounded in the -direction,
unbounded in the -direction and -periodic for large ,
allowing different media on the left and on the right. We suggest a new
numerical method that is based on a truncation of the domain and the use of
Bloch wave ansatz functions in radiation boxes. We prove the existence and a
stability estimate for the infinite dimensional version of the proposed
problem. The scheme is tested on several interfaces of homogeneous and periodic
media and it is used to investigate the effect of negative refraction at the
interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure
Outgoing wave conditions in photonic crystals and transmission properties at interfaces
We analyze the propagation of waves in unbounded photonic crystals, the waves
are described by a Helmholtz equation with -dependent coefficients. The
scattering problem must be completed with a radiation condition at infinity,
which was not available for -dependent coefficients. We develop an outgoing
wave condition with the help of a Bloch wave expansion. Our radiation condition
admits a (weak) uniqueness result, formulated in terms of the Bloch measure of
solutions. We use the new radiation condition to analyze the transmission
problem where, at fixed frequency, a wave hits the interface between free space
and a photonic crystal. We derive that the vertical wave number of the incident
wave is a conserved quantity. Together with the frequency condition for the
transmitted wave, this condition leads (for appropriate photonic crystals) to
the effect of negative refraction at the interface
Dispersive homogenized models and coefficient formulas for waves in general periodic media
We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix that is periodic with
characteristic length scale ; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
well by a non-dispersive wave equation on fixed time intervals
. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions
describe well on time intervals . More
precisely, we provide a norm and uniform error estimates of the form for . They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
-independent equation of third order that describes dispersion
along rays and we present numerical examples.Comment: 28 pages, 7 figure
Averaging of flows with capillary hysteresis in stochastic porous media
Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form p pc(u,∂t u). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical syste
On the three-dimensional Euler equations with a free boundary subject to surface tension
We study an incompressible ideal fluid with a free surface that is subject to surface tension; it is not assumed that the fluid is
irrotational. We derive a priori estimates for smooth solutions and prove a short-time existence result. The bounds are obtained
by combining estimates of energy type with estimates of vorticity type and rely on a careful study of the regularity properties of
the pressure function. An adequate artificial coordinate system is used instead of the standard Lagrangian coordinates. Under an
assumption on the vorticity, a solution to the Euler equations is obtained as a vanishing viscosity limit of solutions to appropriate
Navier–Stokes systems
Free Boundary Fluid Systems in a Semigroup Approach and Oscillatory Behavior
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
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Darcy's law and groundwater flow modelling
Formulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, the French hydraulic engineer Henry Darcy performed experiments, measuring water flow through a column of sand. He discovered and described a fundamental law: the linear relation between pressure difference and flow rate – known today as Darcy’s law. We describe the law and the evolution of its modern formulation. We furthermore sketch some current mathematical research related to Darcy’s law
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