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An effective gravitational temperature for sedimentation

Nature volume 409, pages 594–597 (2001)Cite this article

Abstract

The slow sedimentation of suspensions of solid particles in a fluid results in complex phenomena that are poorly understood. For a low volume fraction (φ) of particles, long-range hydrodynamic interactions result in surprising spatial correlations1 in the velocity fluctuations; these are reminiscent of turbulence, even though the Reynolds number is very low2,3,4. At higher values of φ, the behaviour of sedimentation remains unclear; the upward back-flow of fluid becomes increasingly important, while collisions and crowding further complicate inter-particle interactions5,6,7,8. Concepts from equilibrium statistical mechanics could in principle be used to describe the fluctuations and thereby provide a unified picture of sedimentation, but one essential ingredient—an effective temperature that provides a mechanism for thermalization—is missing. Here we show that the gravitational energy of fluctuations in particle number can act as an effective temperature. Moreover, we demonstrate that the high-φ behaviour is in fact identical to that at low φ, provided that the suspension viscosity and sedimentation velocity are scaled appropriately, and that the effects of particle packing are included.

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Figure 1: PIV velocity fluctuation (δV = V - V(φ)ẑ) vector maps of a typical sample at φ = 0.30.
Figure 2: Normalized velocity fluctuations ΔV/V(φ) as a function of φ.
Figure 3: Correlation lengths, ξ, as a function of φ.
Figure 4: Normalized diffusion coefficients D(φ) ≡ ΔV2τd from this work and Ds(φ) from ref.

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Acknowledgements

We thank M. Brenner, S. Tee, P. Tong, A. J. C. Ladd, B. J. Ackerson, P. Mucha and A. Levine for discussions. This work was supported by NASA, NSF and the donors of the Petroleum Research Fund, administered by the ACS. Current address of P.N.S. is NASA MSFC, Huntsville, AL 35802 (Phil.Segre@msfc.nasa.gov).

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Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, 02138, Massachusetts, USA

    P. N. Segrè & D. A. Weitz

  2. Department of Physics, University of Pennsylvania, Philadelphia, 19104, Pennsylvania, USA

    F. Liu

  3. Department of Physics and Astronomy, Northwestern University, Evanston, 60208, Illinois, USA

    P. Umbanhowar

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  1. P. N. Segrè
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  4. D. A. Weitz
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Correspondence to D. A. Weitz.

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Segrè, P., Liu, F., Umbanhowar, P. et al. An effective gravitational temperature for sedimentation. Nature 409, 594–597 (2001). https://doi.org/10.1038/35054518

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