Hostname: page-component-7f64f4797f-kjzhn Total loading time: 0 Render date: 2025-11-03T15:45:07.074Z Has data issue: false hasContentIssue false
  • English
  • Français

The analysis of self-diffusion and migration of rough spheres in nonlinear shear flow using a traction-corrected boundary element method

Published online by Cambridge University Press:  25 February 2008

MARC S. INGBER ,
SHIHAI FENG ,
ALAN L. GRAHAM  and
HOWARD BRENNER
MARC S. INGBER
Affiliation:
Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA
SHIHAI FENG
Affiliation:
Institutes Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
ALAN L. GRAHAM
Affiliation:
Institutes Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
HOWARD BRENNER
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The phenomena of self-diffusion and migration of rough spheres in nonlinear shear flows are investigated using a new traction-corrected boundary element method (TC-BEM) in which the near-field asymptotics for the traction solution in the interstitial region between two nearly touching spheres is seamlessly coupled with a traditional direct boundary element method. The TC-BEM is extremely accurate in predicting particle trajectories, and hence can be used to calculate both the particle self-diffusivity and a newly defined migration diffusivity for dilute suspensions. The migration diffusivity is a function of a nonlinearity parameter characterizing the shear flow and arises from the net displacement of the centre of gravity of particle pairs. This net displacement of the centre of gravity of particle pairs does not occur for smooth particles, nor for rough particles in a linear shear flow. An explanation is provided for why two-particle interactions of rough spheres in a nonlinear shear flow result in particle migration.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 598 , 10 March 2008 , pp. 267 - 292
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Abbott, J.R., Tetlow, N., Graham, A. L., Altobelli, S. A., Fukushima, E., Mondy, A. L. & Stephens, T. S. 1991 Experimental observation of particle migration in concentrated suspensions: Couette flow. J. Rheol 35, 773–795.CrossRefGoogle Scholar
Ascoli, E. P., Dandy, D. S. & Leal, L. G. 1989 Low Reynolds number hydrodynamic interaction of solid particle with planar wall. Intl J. Numer. Meth. Fluids 9, 651–688.Google Scholar
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, 375–400.Google Scholar
Bossis, G. & Brady, J. F. 1984 Dynamic simulation of sheared suspensions. PartI. General method. J. Chem. Phys 80, 5141–5154.Google Scholar
Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspension under shear. J. Chem. Phys 87, 5437–5448.Google Scholar
Brady, J. F. & Bossis, G. 1985 The Rheology of concentrated suspensions of spheres in simple shear-flow by numerical simulation. J. Fluid Mech 155, 105–129.CrossRefGoogle Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly shear suspensions and its impact on rheology and diffusion. J. Fluid Mech 348, 103–139.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a planar surface. Chem. Engng Sci 16, 242–251.Google Scholar
Buyevich, I. A. 1995 Particle distribution in suspension shear flow. Chem. Engng Sci 51, 635–647.Google Scholar
Chan, C.Y., Beris, A.N. & Advani, S. G. 1992 Second order boundary element method calculations of hydrodynamic interactions between particles in close proximity. Intl J. Numer. Meth. Fluids 14, 1063–1086.Google Scholar
Chang, C. & Powell, R. L. 1994 The rheology of bimodal hard-sphere dispersions. Phys. Fluids 6, 1628–1636.Google Scholar
Cooley, M. D. & O'Neill, M. E. 1969 On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or a stationary sphere. Mathematika 16, 37–49.Google Scholar
Corless, R. M. & Jeffrey, D. J. 1988 Stress moments of nearly touching spheres in low Reynolds number flow. Z. Angew. Math. Phys 39, 874–884.Google Scholar
Cox, R.G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface. PartII. Small gap widths, including inertial effects. Chem. Engng Sci 22, 1753–1777.Google Scholar
da Cunha, F. R. & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech 309, 211–223.Google Scholar
Dance, S. L. & Maxey, M. R. 2003 Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comput. Phys 189, 212–238.CrossRefGoogle Scholar
Eckstein, E. C., Bailey, G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech 79, 191–208.CrossRefGoogle Scholar
Fang, Z., Mammoli, A. A., Brady, J. F., Ingber, M. S., Mondy, L. A. & Graham, A. L. 2002 Flow-aligned tensor models for suspension flows. Intl J. Mult. Flow 28, 137–166.Google Scholar
Graham, A. L. & Bird, R. B. 1984 Particle clusters in concentrated suspensions. Part1. experimental-observations of particle clusters. Ind. Engng Chem Fund 23, 406–410.Google Scholar
Hampton, R. E., Mammoli, A. A. & Ingber, M. S. 2003 A corrected BEM for the inclusion of near-field effects associated with suspension flows at low Reynolds numbers. In Boundary Element Technology XV (ed. Brebbia, C. A. & Dippery, R. E.), pp. 249–258. WIT Press.Google Scholar
Hsiao, S. -C., Christensen, D., Ingber, M. S., Mondy, L. A. & Altobelli, S. A. 2003 Particle migration rates in a Couette apparatus. In Computational Methods in Multiphase Flow II (ed. Brebbia, C. A. & Mammoli, A. A.), pp. 241–251. WIT Press.Google Scholar
Ingber, M. S. 1989 Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle. Intl J. Num. Meth. Fluids 9, 263–273.CrossRefGoogle Scholar
Ingber, M. S., Mammoli, A. A., Vorobieff, P., McCollum, T. & Graham, A. L. 2006 Experimental and numerical analysis of irreversibilities particles suspended in a Couette device. J. Rheol 50, 99–114.Google Scholar
Ingber, M. S., Subia, S. R. & Mondy, L. A. 2000 Massively parallel boundary element method modeling of particles in low-Reynolds number Newtonian fluid flows. In Applications of High-Performance Computing in Engineering (ed. Ingber, M. S., Power, H. & Brebbia, C. A.), pp. 313–324. WIT Press.Google Scholar
Jánosi, I. M., Tel, T., Wolf, D. E. & Garcia, J. A. C. 1997 Chaotic particle dynamics in viscous flows: The three particle Stokeslet problem. Phys. Rev. E 56, 2858–2868.Google Scholar
Jayaweera, K. O. L. F., Mason, B. F. & Slack, G. W. 1964 The behaviour of clusters of spheres falling in a viscous fluid. J. Fluid Mech 20, 121–128.Google Scholar
Jeffrey, D. J. & Onishi, T. 1984 The forces and couples acting on two nearly touching spheres in low-Reynolds number flow. Z. Angew. Math. Phys 35, 634–641.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
Leighton, D. & Acrivos, A. 1987a Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech 177, 109–131.Google Scholar
Leighton, D. & Acrivos, A. 1987b The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech 181, 415–439.Google Scholar
Mammoli, A. A. 2002 Towards a reliable method for predicting the rheological properties of multiphase fluids. Engng Anal. Bound. Elem 26, 747–756.Google Scholar
Mammoli, A. A. 2005 The treatment of lubrication forces in boundary integral equations. Proc. R. Soc. Lond. A 462, 855–881.Google Scholar
Mammoli, A. A. & Ingber, M. S. 2000 Parallel multipole BEM simulation of two-dimensional suspension flows. Engng Anal. Bound. Elem 24, 65–73.Google Scholar
Marchioro, M. & Acrivos, A. 2001 Shear-induced particle diffusivities from numerical simulation. J. Fluid Mech 443, 101–128.Google Scholar
Morris, J. F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: The role of normal stresses. J. Rheol 43, 1213.Google Scholar
Nasseri, S., Phan-Thien, N. & Fan, X. J. 2000 Lubrication approximation in completed double layer boundary element method. Comput. Mech 26, 388–397.Google Scholar
Nott, P. & Brady, J. F. 1994 Pressure-driven flow of suspensions: Principles and computational methods. J. Fluid Mech 275, 157–199.Google Scholar
O'Neill, M. E. & Majumdar, S. R. 1970a Asymmetrical slow viscous fluid motion caused by the translation or rotation of two spheres. Part II. asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero. Z. Angew. Math. Phys 21, 180–187.Google Scholar
O'Neill, M. E. & Majumdar, S. R. 1970b Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I. The determination of exact solutions for any values of the ratio of radii and separation parameters. Z. Angew. Math. Phys 21, 164–179.Google Scholar
Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. R. 1992 A constitutive equation for concentrated suspension that accounts for shear-induced particle migration. Phys. Fluids A 4, 30–40.Google Scholar
Pozarnik, M. & Skerget, L. 2003 Boundary element method numerical model based on mixture theory of two-phase flow. In Computational Methods in Multiphase Flow II (ed. Mammoli, A. A. & Brebbia, C. A.), pp. 3–12. WIT Press.Google Scholar
Rampall, I., Smart, J. R. & Leighton, D. T. 1997 The influence of surface roughness on the particle-pair distribution function of dilute suspensions of non-colloidal spheres in simple shear flow. J. Fluid Mech 339, 1–24.Google Scholar
Sangani, A. S. & Mo, G. 1994 Inclusion of lubrication forces in dynamic simulations. Phys. Fluids 6, 1653–1662.CrossRefGoogle Scholar
Sangani, S. & Mo, G. 1996 An O(n) algorithm for Stokes and Laplace interactions of particles. Phys. Fluids 8, 1990–2010.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech 448, 115–146.Google Scholar
Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol 46, 1031–1056.Google Scholar
Telles, J. C. F. & Oliveira, R. F. 1994 Third degree polynomial transformation for for boundary element integrals: Further improvements. Engng Anal. Bound. Elem 13, 135–142.Google Scholar
Tetlow, N., Graham, A. L., Ingber, M. S., Subia, S. R., Mondy, A. L. & Altobelli, S. A. 1998 Particle migration in a Couette apparatus: Experiment and modeling. J. Rheol 42, 307–327.Google Scholar
Youngren, G. K. & Acrivos, A. 1973 Stokes flow past a particle of arbitrary shape: A numerical method of solution. J. Fluid Mech 60, 377–403.Google Scholar
Zinchenko, A. Z. 1998 Effective conductivity of loaded granular materials by numerical simulation. Phil. Trans. R. Soc. Lond. A 356, 2953–2998.Google Scholar