Lévy flights, which are Markovian continuous time random walks possibly accounting for extreme events, serve frequently as small-scale models for the spreading of matter in heterogeneous media. Among them, Brownian motion is a particular case where Fick's law holds: for a cloud of walkers, the flux is proportional to the gradient of the probability density of finding a particle at some place. Lévy flights resemble Brownian motion, except that jump lengths are distributed according to an α-stable Lévy law, possibly showing heavy tails and skewness. For α between 1 and 2, a fractional form of Fick's law is known to hold in infinite media: that the flux is proportional to a combination of fractional derivatives or the order of α − 1 of the density of walkers was obtained as a consequence of a fractional dispersion equation. We present a direct and natural proof of this result, based upon a novel definition of usual fractional derivatives, involving a convolution and a limiting process. Taking account of the thus obtained fractional Fick's law yields fractional dispersion equation for smooth densities. The method adapts to domains, limited by boundaries possibly implying non-trivial modifications to this equation.

Nous étudions numériquement l'influence d'un champ magnétique vertical, du nombre de Reynolds et d'une stratification de température sur la stabilité de l'écoulement de Hartmann chauffé par le bas. Pour Pr = 0,001 et Ha 2,5, nos résultats montrent que le champ magnétique stabilise aussi bien les modes transverses oscillatoires progressifs (T ) que les modes longitudinaux stationnaires (L). Quant à la stratification de température, elle est à l'origine d'une déstabilisation de l'écoulement de Hartmann par rapport au cas isotherme et de l'apparition des modes (L) inexistants dans le cas Ra = 0. On note aussi que l'étendue du domaine des valeurs de Re où les modes transverses (T ) sont les plus dangereux se rétrécit lorsque Ha augmente et qu'elle s'élargit quand Ra augmente à Ha donné. Quant à l'étendue du domaine des valeurs de Re où les modes (L) prévalent, elle augmente quand Ha croît.

We illustrate in the present paper how steady-state and unsteady-steady three-dimensional transport phenomena problems can be solved using the Chebyshev orthogonal collocation technique. All problems are elucidated using the ubiquitous software: MATLAB 1 . The treated case studies include: (1) unsteadystate heat conduction in a parallelepiped body, (2) quenching of a brick, (3) transient diffusion of a colorant in a liquid, and (4) steady-state natural convection in an enclosure. Whenever possible, the results obtained with the spectral method are bench-marked against the analytical solution in order to validate the proposed numerical technique. ß 2016 Wiley Periodicals, Inc. Comput Appl Eng Educ 24:866-875, 2016; View this article online at wileyonlinelibrary.com/journal/cae; 1 , POLYMATH, MATHCAD, JAVA, ASPEN-HYSYS or EXCEL is common in chemical engineering pedagogical literature [1 -12]. A literature survey over last three decades shows that only two papers report on the application of the Chebyshev orthogonal collocation technique, implemented in Mathematica © , for educational purposes [12,13], restricted however to one-and two-dimensional problems.Our main goal in this paper is to illustrate the utilization of this numerical method, commonly applied in research [14][15][16][17], in graduate-level classrooms in order to solve miscellaneous problems Correspondence to H. Binous (binoushousam@yahoo.com).

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This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 17918Open Archive Toulouse Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract. The authors studied the effect of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a porous cavity. The horizontal bottom plate was submitted to constant uniform heat flux and the top one was maintained at constant temperature while no mass flux was imposed. The influence of vertical vibrations on the onset of convection and on the stability of the unicellular flow was investigated for positive separation ratio ψ. The case of high-frequency and small-amplitude vibrations was considered so that a formulation using time averaged equations could be used. For an infinite horizontal porous layer, the equilibrium solution was found to lose its stability via a stationary bifurcation leading to unicellular flow or multicellular one depending on the value of ψ. The analytical solution of the unicellular flow was obtained and separation was expressed in terms of Lewis number, separation ratio and thermal Rayleigh number. The direct numerical simulations using the averaged governing equations and analytical stability analysis showed that the unicellular flow loses its stability via oscillatory bifurcation. The vibrations decrease the value of ψ uni, which allows species separation for a wide variety of binary mixtures. The vibrations can be used to maintain the unicellular flow and allow species separation over a wider range of Rayleigh numbers. The results of direct numerical simulations and analytical model are in good agreement. To cite this version: