230 research outputs found
Effective macroscopic dynamics of stochastic partial differential equations in perforated domains
An effective macroscopic model for a stochastic microscopic system is
derived. The original microscopic system is modeled by a stochastic partial
differential equation defined on a domain perforated with small holes or
heterogeneities. The homogenized effective model is still a stochastic partial
differential equation but defined on a unified domain without holes. The
solutions of the microscopic model is shown to converge to those of the
effective macroscopic model in probability distribution, as the size of holes
diminishes to zero. Moreover, the long time effectivity of the macroscopic
system in the sense of \emph{convergence in probability distribution}, and the
effectivity of the macroscopic system in the sense of \emph{convergence in
energy} are also proved
Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors
We show that any pair of real symmetric tensors \BGve and \BGm can be
realized as the effective electric permittivity and effective magnetic
permeability of a metamaterial at a given fixed frequency. The construction
starts with two extremely low loss metamaterials, with arbitrarily small
microstructure, whose existence is ensured by the work of Bouchitt{\'e} and
Bourel and Bouchitt\'e and Schweizer, one having at the given frequency a
permittivity tensor with exactly one negative eigenvalue, and a positive
permeability tensor, and the other having a positive permittivity tensor, and a
permeability tensor having exactly one negative eigenvalue. To achieve the
desired effective properties these materials are laminated together in a
hierarchical multiple rank laminate structure, with widely separated length
scales, and varying directions of lamination, but with the largest length scale
still much shorter than the wavelengths and attenuation lengths in the
macroscopic effective medium.Comment: 12 pages, no figure
The Effect of Tumbling, Sodium Chloride and Polyphosphates on the Microstructure and Appearance of Whole-Muscle Processed Meats
The properties of a whole-muscle processed meat were determined. The complex action of socium chloride, polyphosphates and mechanical agitation caused extraction of myofibrillar protein, swelling of fibers and loss of cross-strations. A new functional ability was found for the extracted proteins to form a fine cover or membrane on the surface of the whole muscle during cooking. These changes produced a product with improved cooking yield and color appearance
Tunable Double Negative Band Structure from Non-Magnetic Coated Rods
A system of periodic poly-disperse coated nano-rods is considered. Both the
coated nano-rods and host material are non-magnetic. The exterior nano-coating
has a frequency dependent dielectric constant and the rod has a high dielectric
constant. A negative effective magnetic permeability is generated near the Mie
resonances of the rods while the coating generates a negative permittivity
through a field resonance controlled by the plasma frequency of the coating and
the geometry of the crystal. The explicit band structure for the system is
calculated in the sub-wavelength limit. Tunable pass bands exhibiting negative
group velocity are generated and correspond to simultaneously negative
effective dielectric permittivity and magnetic permeability. These can be
explicitly controlled by adjusting the distance between rods, the coating
thickness, and rod diameters
Dynamo effect in parity-invariant flow with large and moderate separation of scales
It is shown that non-helical (more precisely, parity-invariant) flows capable
of sustaining a large-scale dynamo by the negative magnetic eddy diffusivity
effect are quite common. This conclusion is based on numerical examination of a
large number of randomly selected flows. Few outliers with strongly negative
eddy diffusivities are also found, and they are interpreted in terms of the
closeness of the control parameter to a critical value for generation of a
small-scale magnetic field. Furthermore, it is shown that, for parity-invariant
flows, a moderate separation of scales between the basic flow and the magnetic
field often significantly reduces the critical magnetic Reynolds number for the
onset of dynamo action.Comment: 44 pages,11 figures, significantly revised versio
Non-local homogenised limits for composite media with highly anisotropic periodic fibres
We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved
Finite element approximation of the -Laplacian
We study a~priori estimates for the Dirichlet problem of the
-Laplacian,
We show that the gradients of the finite element approximation with zero
boundary data converges with rate if the exponent is
-H\"{o}lder continuous. The error of the gradients is measured in the
so-called quasi-norm, i.e. we measure the -error of
Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains
We derive a new effective macroscopic Cahn-Hilliard equation whose
homogeneous free energy is represented by 4-th order polynomials, which form
the frequently applied double-well potential. This upscaling is done for
perforated/strongly het- erogeneous domains. To the best knowledge of the
authors, this seems to be the first attempt of upscaling the Cahn-Hilliard
equation in such domains. The new homog- enized equation should have a broad
range of applicability due to the well-known versatility of phase-field models.
The additionally introduced feature of systemati- cally and reliably accounting
for confined geometries by homogenization allows for new modeling and numerical
perspectives in both, science and engineering. Our results are applied to
wetting dynamics in porous media and to a single channel with strongly
heterogeneous walls
Homogenization of the planar waveguide with frequently alternating boundary conditions
We consider Laplacian in a planar strip with Dirichlet boundary condition on
the upper boundary and with frequent alternation boundary condition on the
lower boundary. The alternation is introduced by the periodic partition of the
boundary into small segments on which Dirichlet and Neumann conditions are
imposed in turns. We show that under the certain condition the homogenized
operator is the Dirichlet Laplacian and prove the uniform resolvent
convergence. The spectrum of the perturbed operator consists of its essential
part only and has a band structure. We construct the leading terms of the
asymptotic expansions for the first band functions. We also construct the
complete asymptotic expansion for the bottom of the spectrum
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