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The International Conference on Multiphase Systems, ICMS'2000
Ufa, RUSSIA, June 15-17, 2000

On the Caflish-Luke Paradox in a Dilute Sedimenting Suspension of Rigid Spheres

Daniel Lhuillier

Université Pierre et Marie Curie et CNRS,
Laboratoire de Modélisation en Mécanique, Case courrier 162,
4 place Jussieu, 75252 Paris Cedex 05, France

Non-brownian sedimenting particles, and the fluid surrounding them, display velocity fluctuations around their mean motion. These fluctuations are non-isotropic and their main features can be represented by two symmetric tensors (somewhat similar to the Reynolds stress in a turbulent fluid), Rp for the particles and <v'v'> for the whole suspension. In the creeping flow limit and for low concentrations of rigid and spherical particles, Rp and <v'v'> are known to diverge in a unbounded suspension (the Caflish-Luke paradox). Here we show that a) the difference Rp - <v'v'> is always finite even for a random and unbounded suspension, and b) <v'v'> obeys an equation which must be solved with the true boundary condition <v'v'> = 0. As a result, <v'v'> generally display steep variations over thin boundary layers and then a much smoother variation in the bulk part of the flow. This behaviour is similar to the one displayed by the suspension velocity close to the boundaries. As a counterpart to the effective boundary conditions proposed by Nozieres (1987) for the suspension velocity, we derive effective boundary conditions for <v'v'>, together with the simplified equation describing its behaviour in the bulk.