J. Phys. II France
Volume 2, Numéro 5, May 1992
Page(s) 1023 - 1043
DOI: 10.1051/jp2:1992184
J. Phys. II France 2 (1992) 1023-1043

Hydrodynamic dispersion on self-similar structures : a Laplace space renormalization group approach

Emmanuel Villermaux1 and Daniel Schweich2

1  Institut de mécanique de Grenoble UMR 101, BP 53 X, 38041 Grenoble Cedex, France
2  Laboratoire des Sciences du Génie Chimique, CNRS, ENSIC, 1 rue Grandville, BP 451, 54001 Nancy Cedex, France

(Received 18 March 1991, revised 9 November 1991, accepted 8 January 1992)

A general method for calculating the properties of the Residence Time Distribution (RTD) of a fluid flowing though a self-similar network, with or without stagnation (trapping) effects, in the high Péclet number limit is developed. The renormalization procedure adopted yields the Laplace transform of the Residence Time Distribution and allows one to calculate its time moments. The incidence of the connectivity of the medium on dispersion is discussed. The fractal dimension does not appear explicitly in the dispersion properties of the network. Geometrical dispersion is shown to result from the difference of pathlengths offered to the fluid in the generating pattern of the network. The dispersion front is strongly non-Gaussian and presents several maxima (short circuits) in some extreme cases. An approximate expression for the dispersion front is derived when the distribution of pathlengths in naroow. In the latter situation, a dispersion coefficient can be defined, and it is characterised by two parametres, $\gamma$ and $(1-f)t_{\rm m}$, which represent respectively the intensity of the disorder (or the ability to mixing) related to geometric dispersion, and the characteristic hold-up time of a tracer particle in the stagnation phase supposed to occupy a fraction (1-f) of the volume and to be uniformly distributed in the medium. We also discuss the conditions of moments convergence and the long time aymptotic form of the RTD.

47.55M - 05.60

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