70 research outputs found
Rigorous derivation of a hyperbolic model for Taylor dispersion
In this paper we upscale the classical convection-diffusion equation in a narrow slit. We suppose that the transport parameters are such that we are in Taylor's regime i.e. we deal with dominant Peclet numbers. In contrast to the classical work of Taylor, we undertake a rigorous derivation of the upscaled hyperbolic dispersion equation. Hyperbolic effective models were proposed by several authors and our goal is to confirm rigorously the effective equations derived by Balakotaiah et al in recent years using a formal Liapounov - Schmidt reduction. Our analysis uses the Laplace transform in time and an anisotropic singular perturbation technique, the small characteristic parameter " being the ratio between the thickness and the longitudinal observation length. The Peclet number is written as CeÂża, with a<2. Hyperbolic effective model corresponds to a high Peclet number close to the threshold value when Taylor's regime turns to turbulent mixing and we characterize it by supposing 4/3 <a <2. We prove that the difference between the dimensionless physical concentration and the effective concentration, calculated using the hyperbolic upscaled model, divided by e2Âża (the local Peclet number) converges strongly to zero in L2-norm. For Peclet numbers considered in this paper, the hyperbolic dispersion equation turns out to give a better approximation than the classical parabolic Taylor model
Asymptotic analysis of pollution filtration through thin random fissures between two porous media
We describe the asymptotic behaviour of a filtration problem from a
contaminated porous medium to a non-contaminated porous medium through thin
vertical fissures of fixed height h>0, of random thinness of order {\epsilon}
and which are -periodically distributed. We compute the limit
velocity of the flow and the limit flux of pollutant at the interfaces between
the two porous media and the intermediate one
Numerical simulation of biofilm formation in a microchannel
The focus of this paper is the numerical solution of a pore-scale model for
the growth of a permeable biofilm. The model includes water flux inside the
biofilm, different biofilm components, and shear stress on the biofilm-water
interface. To solve the resulting highly coupled system of model equations, we
propose a splitting algorithm. The Arbitrary Lagrangian Eulerian (ALE) method
is used to track the biofilm-water interface. Numerical simulations are
performed using physical parameters from the existing literature. Our
computations show the effect of biofilm permeability on the nutrient transport
and on its growth
Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution
Using a Maximum Entropy Production Principle (MEPP), we derive a new type of
relaxation equations for two-dimensional turbulent flows in the case where a
prior vorticity distribution is prescribed instead of the Casimir constraints
[Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a
Gaussian prior is specifically treated in connection to minimum enstrophy
states and Fofonoff flows. These relaxation equations are compared with other
relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776
(1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a
small-scale parametrization of 2D turbulence or serve as numerical algorithms
to compute maximum entropy states with appropriate constraints. We perform
numerical simulations of these relaxation equations in order to illustrate
geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure
Upscaling Flow and Transport Processes
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