72 research outputs found
Rigorous derivation of the thin film approximation with roughness-induced correctors
50 pagesWe derive the thin film approximation including roughness-induced correctors. This corresponds to the description of a confined Stokes flow whose thickness is of order~\eps (designed to be small)~; but we also take into account the roughness patterns of the boundary that are described at order~\eps^2, leading to a perturbation of the classical Reynolds approximation. The asymptotic expansion leading to the description of the scale effects is rigorously derived, through a sequence of Reynolds-type problems and Stokes-type (boundary layer) problems. Well-posedness of the related problems and estimates in suitable functional spaces are proved, at any order of the expansion. In particular, we show that the micro-/macro-scale coupling effects may be analysed as the consequence of two features: the interaction between the macroscopic scale (order~1) of the flow and the microscopic scale (order~\eps of the thin film) is perturbed by the interaction with a microscopic scale of order~\eps^2 related to the roughness patterns (as expected through the classical Reynolds approximation)~; moreover, the converging-diverging profile of the confined flow, which is typical in lubrication theory (note that the case of a constant cross-section channel has no interest) provides additional micro-macro-scales coupling effects
Effects of rough boundary and nonzero boundary conditions on the lubrication process with micropolar fluid
The lubrication theory is mostly concerned with the behavior of a lubricant flowing through a narrow gap. Motivated by the experimental findings from the tribology literature, we take the lubricant to be micropolar fluid and study its behavior in a thin domain with rough boundary. Instead of considering (commonly used) simple zero boundary condition, we impose physically
relevant (nonzero) boundary condition for microrotation and perform asymptotic analysis of the corresponding 3D boundary value problem. We formally derive a simplified mathematical model acknowledging the roughness-induced effects and the effects of the nonzero boundary conditions on the macroscopic flow. Using the obtained asymptotic model, we study numerically the influence of the specific rugosity profile on the performance of a linear slider bearing. The numerical results clearly indicate that the use of the rough surfaces may contribute to enhance the mechanical performance of such device.Croatian Science FoundationUniversity of ZagrebMinisterio de EconomÃa y CompetitividadJunta de AndalucÃ
Roughness effect on the Neumann boundary condition
36 pagesInternational audienceWe study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written using an energy developed in the roughness boundary layer. The first part deals with the case of a Laplace operator in a simple domain but many more general results are next given: when the domain or the operator are more complex, or with Robin-Fourier boundary conditions. Some numerical illustrations are used to obtain magnitudes for the coefficients appearing in the new wall laws. Finally, these wall laws can be interpreted using a fictive boundary without rugosity. That allows to give an application to the water waves equation
Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary
We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media
Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries
In this paper we address the large-scale regularity theory for the stationary
Navier-Stokes equations in highly oscillating bumpy John domains. These domains
are very rough, possibly with fractals or cusps, at the microscopic scale, but
are amenable to the mathematical analysis of the Navier-Stokes equations. We
prove: (i) a large-scale Calder\'on-Zygmund estimate, (ii) a large-scale
Lipschitz estimate, (iii) large-scale higher-order regularity estimates,
namely, and estimates. These nice regularity
results are inherited only at mesoscopic scales, and clearly fail in general at
the microscopic scales. We emphasize that the large-scale
regularity is obtained by using first-order boundary layers constructed via a
new argument. The large-scale regularity relies on the
construction of second-order boundary layers, which allows for certain boundary
data with linear growth at spatial infinity. To the best of our knowledge, our
work is the first to carry out such an analysis. In the wake of many works in
quantitative homogenization, our results strongly advocate in favor of
considering the boundary regularity of the solutions to fluid equations as a
multiscale problem, with improved regularity at or above a certain scale.Comment: 72 page
Uniform Lipschitz Estimates in Bumpy Half-Spaces
This paper is devoted to the proof of uniform H\"older and Lipschitz
estimates close to oscillating boundaries, for divergence form elliptic systems
with periodically oscillating coefficients. Our main point is that no structure
is assumed on the oscillations of the boundary. In particular, those are
neither periodic, nor quasiperiodic, nor stationary ergodic. We investigate the
consequences of our estimates on the large scales of Green and Poisson kernels.
Our work opens the door to the use of potential theoretic methods in problems
concerned with oscillating boundaries, which is an area of active research.Comment: 54 page
Conference on Binary Optics: An Opportunity for Technical Exchange
The papers herein were presented at the Conference on Binary Optics held in Huntsville, AL, February 23-25, 1993. The papers were presented according to subject as follows: modeling and design, fabrication, and applications. Invited papers and tutorial viewgraphs presented on these subjects are included
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