87 research outputs found
On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow
AbstractWe consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem, and the tangential drag force and the effective mass flow are determined up to order O(ε3/2). The value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of O(ε)
Sigma-convergence for thin heterogeneous domains and application to the upscaling of Darcy-Lapwood-Brinkmann flow
The sigma-convergence concept has been up to now used to derive macroscopic
models in full space dimensions. In this work, we generalize it to thin
heterogeneous domains given rise to phenomena in lower space dimensions. More
precisely, we provide a new approach of the sigma-convergence method that is
suitable for the study of phenomena occurring in thin heterogeneous media. This
is made through a systematic study of the sigma-convergence method for thin
heterogeneous domains. Assuming that the thin heterogeneous layer is made of
microstructures that are distributed inside in a deterministic way including as
special cases the periodic and the almost periodic distributions, we make use
of the concept of algebras with mean value to state and prove the main
compactness results. As an illustration, we upscale a Darcy-Lapwood-Brinkmann
micro-model for thin flow. We prove that, according to the magnitude of the
permeability of the porous domain, we obtain as effective models, the Darcy law
in lower dimensions. The effective models are derived through the solvability
of either the local Darcy-Brinkmann problems or the local Hele-Shaw problems.Comment: 32 page
Approximation of homogenized coefficients in deterministic homogenization and convergence rates in the asymptotic almost periodic setting
For a homogenization problem associated to a linear elliptic operator, we
prove the existence of a distributional corrector and we find an approximation
scheme for the homogenized coefficients. We also study the convergence rates in
the asymptotic almost periodic setting, and we show that the rates of
convergence for the zero order approximation, are near optimal. The results
obtained constitute a step towards the numerical implementation of results from
the deterministic homogenization theory beyond the periodic setting. To
illustrate this, numerical simulations based on finite volume method are
provided to sustain our theoretical results.Comment: 49 pages, 10 figure
About loss of regularity and "blow up" of solutions for quasilinear parabolic systems.
Starting from sufficient conditions for regularity of weak solutions
to quasilinear parabolic systems, uecessary conditions for loss of regularity are
formulated. It is shown numerically that in some situations loss of regularity
("blow up") really happens accordingly to these conditions
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