Skip to main content

Self-diffusion in sheared suspensions

  • Jeffrey F. Morris (a1) and John F. Brady (a1)

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number $Pe = \dot{\gamma}a^2/2D$, where $\dot{\gamma}$ is the shear rate, a is the particle radius, and D = kBT/6πa is the diffusion coefficient of an isolated particle. Here, kB is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by kBT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction $\phi = \frac{4}{3}\pi a^3n$ and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, Ds, is given by the sum of D0s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D0s and D are anisotropic, in general, with the anisotropy of D0s due solely to that of the steady microstructure. The influence of flow upon Ds is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations.

The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to OPe3/2) for Pe ≤ 1 and ø ≤ 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D0s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with $\dot{\gamma}$. Regardless of whether or not the particles interact hydrodynamically, flow influences Ds at OPe) and OPe3/2). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of OPe3/2), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe3/2DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor.

At high ø a scaling theory based on the approach of Brady (1994) is used to approximate Ds. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of $\bar{P}e = \dot{\gamma}a^2/2D^s_0(\phi)$. At small $\bar{P}e$ the dependence on $\bar{P}e$ is the same as at low ø.

Hide All
Acrivos, A., Batchelor, G. K., Hinch, E. J., Koch, D. L. & Mauri, R. 1992 Longitudinal shear-induced diffusion of spheres in a dilute suspension. J. Fluid Mech. 240, 651.
Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer from single spheres in Stokes flow. Phys. Fluids 5, 387.
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 1.
Batchelor, G. K. 1977 The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech. 83, 97.
Batchelor, G. K. 1979 Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution. J. Fluid Mech. 95, 369.
Batchelor, G. K. 1982 Sedimentation in a polydisperse system of interacting spheres. Part 1. General theory. J. Fluid Mech. 199, 379.
Batchelor, G. K. 1983 Diffusion in a dilute polydisperse system of interacting spheres. J. Fluid Mech. 131, 155.
Batchelor, G. K. & Green, J. T. 1972 The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56, 375.
Berne, B. J. & Pecora, R. 1976 Dynamic Light Scattering. Wiley.
Blawzdziewicz, J. & Szamel, G. 1993 Structure and rheology of semidilute suspension under shear. Phys. Rev. E 48, 4632.
Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspensions under shear. J. Chem. Phys. 87, 5437.
Bossis, G. & Brady, J. F. 1989 The rheology of Brownian suspensions. J. Chem. Phys. 91 1866.
Brady, J. F. 1994 The long-time self diffusivity in concentrated colloidal dispersions. J. Fluid Mech. 272, 109.
Brady, J. F. & Vicic, M. 1995 Normal stresses in colloidal dispersions. J. Rheol. 39, 545.
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous medium. Phil. Trans. R. Soc. Lond. A 297, 81.
Duffy, J. W. 1984 Diffusion in shear flow. Phys. Rev. A 30, 1465.
Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion in shear flow of a suspension. J. Fluid Mech. 79, 191.
Elrick, D. E. 1962 Source functions for diffusion in uniform shear flow. Austral. J. Phys. 15, 283.
Foister, R. T. & Ven, T. G. M. van de 1980 Diffusion of Brownian particles in shear flows. J. Fluid Mech. 96, 105.
Frankel, I. & Brenner, H. 1991 Generalized Taylor dispersion phenomena in unbounded homogeneous shear flows. J. Fluid Mech. 230, 147.
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261.
Jones, R. B. & Burfield, G. S. 1982 Memory effects is the diffusion of an interacting polydisperse suspension. Part 1. Physica A 111, 562.
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Leal, L. G. 1973 On the effective conductivity of a dilute suspension of spherical drops in the limit of low particle Péclet number. Chem. Engng Commun. 1, 21.
Leal, L. G. 1992 Laminar Flow and Convective Transport Processes. Butterworth-Heinemann.
Leighton, D. & Acrivos, A. 1987 Measurement of self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109.
Miguel, M. san & Sancho, J. M. 1979 Brownian motion in shear flow. Physica A 99, 357.
Morris, J. F. & Brady, J. F. 1996 Microstracture of strongly sheared suspensions and its impact on rheology and diffusion. To be submitted to J. Fluid Mech.
Novikov, E. A. 1958 Concerning turbulent diffusion in a stream with transverse gradient of velocity. Prikl. Mat. Mech. 22, 412.In Russian.
Phung, T. N. 1993 Behavior of Concentrated Colloidal Dispersions by Stokesian Dynamics. PhD thesis, California Institute of Technology.
Phung, T. N., Brady, J. F. & Bossis, G. 1996 Stokesian Dynamics simulation of Brownian suspensions J. Fluid Mech. 313, 181.
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237.
Pusey, P. N. 1991 Colloidal suspensions. In Liquids, Freezing, and Glass Transition (ed.J. P. Hansen, D. Levesque & J. Zinn-Justin). Elsevier.
Qiu, X, Ou-Yang, H. D., Pine, D. J. & Chaikin, P. M. 1988 Self-diffusion of interacting colloids far from equilibrium. Phys. Rev. Lett. 61, 2554.
Rallison, J. M. & Hinch, E. J. 1986 The effect of particle interactions on dynamic light scattering from a dilute suspension. J. Fluid Mech. 167, 131.
Russel, W. B. & Glendinning, A. B. 1981 The effective diffusion coefficient detected by dynamic light scattering. J. Chem. Phys. 74, 948.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186.
Wang, Y., Mauri, R. & Acrivos, A. 1996 Transverse shear-induced diffusion of spheres in dilute suspension. J. Fluid Mech. (submitted).
Woodcock, L. V. 1981 Glass transition in the hard sphere model and Kauzman's paradox. Ann. NY Acad. Sci. 37, 274.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 32 *
Loading metrics...

Abstract views

Total abstract views: 122 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th November 2017. This data will be updated every 24 hours.