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The effect of Brownian motion on the bulk stress in a suspension of spherical particles

  • G. K. Batchelor (a1)
Abstract

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

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References
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Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545.
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 1.
Batchelor, G. K. & Green, J. T. 1972a The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56, 375.
Batchelor, G. K. & Green, J. T. 1972b The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401.
Einstein, A. 1905 On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. Annln Phys. 17, 549.
Einstein, A. 1906 A new determination of molecular dimensions. Annln Phys. 19, 289 (and 34, 591).
Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683.
Krieger, I. M. 1972 Rheology of monodisperse latices. Advan. Colloid Interface Sci. 2, 111.
Leal, L. G. & Hinch, E. J. 1973 Theoretical studies of a suspension of rigid particles affected by Brownian couples. Rheol. Acta 12, 127.
Lin, D. C. 1973 Rheological properties of moderately concentrated suspensions. Ph.D. dissertation, University of Washington, Seattle.
Lorentz, H. A. 1906 A general theorem concerning the motion of a viscous fluid. Abhandl. theoret. Phys. 1, 23.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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