15,149 research outputs found
Existence and homogenization of the Rayleigh-B\'enard problem
The Navier-Stokes equation driven by heat conduction is studied. As a
prototype we consider Rayleigh-B\'enard convection, in the Boussinesq
approximation. Under a large aspect ratio assumption, which is the case in
Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the
existence of a global strong solution to the 3D Navier-Stokes equation coupled
with a heat equation, and the existence of a maximal B-attractor. A rigorous
two-scale limit is obtained by homogenization theory. The mean velocity field
is obtained by averaging the two-scale limit over the unit torus in the local
variable
The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms
An asymptotic theory is developed to generate equations that model the global
behaviour of electromagnetic waves in periodic photonic structures when the
wavelength is not necessarily long relative to the periodic cell dimensions;
potentially highly-oscillatory short-scale detail is encapsulated through
integrated quantities.
The theory we develop is then applied to two topical examples, the first
being the case of aligned dielectric cylinders, which has great importance in
the modelling of photonic crystal fibres. We then consider the propagation of
waves in a structured metafilm, here chosen to be a planar array of dielectric
spheres. At certain frequencies strongly directional dynamic anisotropy is
observed, and the asymptotic theory is shown to capture the effect, giving
highly accurate qualitative and quantitative results as well as providing
interpretation for the underlying change from elliptic to hyperbolic behaviour
Extension of Lorenz Unpredictability
It is found that Lorenz systems can be unidirectionally coupled such that the
chaos expands from the drive system. This is true if the response system is not
chaotic, but admits a global attractor, an equilibrium or a cycle. The
extension of sensitivity and period-doubling cascade are theoretically proved,
and the appearance of cyclic chaos as well as intermittency in interconnected
Lorenz systems are demonstrated. A possible connection of our results with the
global weather unpredictability is provided.Comment: 32 pages, 13 figure
Complex dynamics in double-diffusive convection
The dynamics of a small Prandtl number binary mixture in a laterally heated
cavity is studied numerically. By combining temporal integration, steady state
solving and linear stability analysis of the full PDE equations, we have been
able to locate and characterize a codimension-three degenerate Takens-Bogdanov
point whose unfolding describes the dynamics of the system for a certain range
of Rayleigh numbers and separation ratios near S=-1.Comment: 8 pages, 5 figure
Radiation from structured-ring resonators
We investigate the scalar-wave resonances of systems composed of identical Neumann-type inclusions arranged periodically around a circular ring. Drawing on natural similarities with the undamped Rayleigh-Bloch waves supported by infinite linear arrays, we deduce asymptotically the exponentially small radiative damping in the limit where the ring radius is large relative to the periodicity. In our asymptotic approach, locally linear Rayleigh-Bloch waves that attenuate exponentially away from the ring are matched to a ring-scale WKB-type wave field. The latter provides a descriptive physical picture of how the mode energy is transferred via tunnelling to a circular evanescent-to-propagating transition region a finite distance away from the ring, from where radiative grazing rays emanate to the far field. Excluding the zeroth-order standing-wave modes, the position of the transition circle bifurcates with respect to clockwise and anti-clockwise contributions, resulting in striking spiral wavefronts
- …