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Swapping trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres

Published online by Cambridge University Press:  14 November 2007

M. ZURITA-GOTOR*
Affiliation:
Department of Mechanical Engineering, P.O. Box 20-8286, Yale University, New Haven, CT 06520-8286, USA
J. BŁAWZDZIEWICZ
Affiliation:
Department of Mechanical Engineering, P.O. Box 20-8286, Yale University, New Haven, CT 06520-8286, USA
E. WAJNRYB
Affiliation:
IPPT, Świetokrzyska 21, Warsaw, Poland
*
Present address: Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Camino de los descubrimientos s/n, Sevilla 41092, Spain.

Abstract

Binary encounters between spherical particles in shear flow are studied for a system bounded by a single planar wall or two parallel planar walls under creeping flow conditions. We show that wall proximity gives rise to a new class of binary trajectories resulting in cross-streamline migration of the particles. The spheres on these new trajectories do not pass each other (as they would in free space) but instead they swap their cross-streamline positions. To determine the significance of the wall-induced particle migration, we have evaluated the hydrodynamic self-diffusion coefficient associated with a sequence of uncorrelated particle displacements due to binary particle encounters. The results of our calculations quantitatively agree with the experimental value obtained by Zarraga & Leighton (Phys. Fluids, vol. 14, 2002, p. 2194) for the self-diffusivity in a dilute suspension of spheres undergoing shear flow in a Couette device. We thus show that the wall-induced cross-streamline particle migration is the source of the anomalously large self-diffusivity revealed by their experiments.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

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, 651657.CrossRefGoogle Scholar
Arp, P. A. & Mason, S. G. 1977 Kinetics of flowing dispersions. VIII. Doublets of rigid spheres (theoretical). J. Colloid Interface Sci. 61, 2143.CrossRefGoogle 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, 375400.CrossRefGoogle Scholar
Bhattacharya, S. & Blawzdziewicz, J. 2002 Image system for Stokes-flow singularity between two parallel planar walls. J. Math. Phys. 43, 5720–31.CrossRefGoogle Scholar
Bhattacharya, S., Blawzdziewicz, J. & Wajnryb, E. 2005 a Hydrodynamic interactions of spherical particles in suspensions confined between two planar walls. J. Fluid Mech. 541, 263292.CrossRefGoogle Scholar
Bhattacharya, S., Blawzdziewicz, J. & Wajnryb, E. 2005 b Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method. Physica A 356, 294340.CrossRefGoogle Scholar
Bhattacharya, S., Blawzdziewicz, J. & Wajnryb, E. 2006 a Far-field approximation for hydrodynamic interactions in parallel-wall geometry. J. Comput. Phys. 212, 718738.CrossRefGoogle Scholar
Bhattacharya, S., Blawzdziewicz, J. & Wajnryb, E. 2006 b Hydrodynamic interactions of spherical particles in Poiseuille flow between two parallel walls. Phys. Fluids 18, 053301.CrossRefGoogle Scholar
Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspensions under shear. J. Chem. Phys. 87, 54375448.CrossRefGoogle Scholar
Cichocki, B., Jones, R. B., Kutteh, R. & Wajnryb, E. 2000 Friction and mobility for colloidal spheres in Stokes flow near a boundary: The multipole method and applications. J. Chem. Phys. 112, 2548–61.CrossRefGoogle Scholar
da Cunha, F. R., & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211223.CrossRefGoogle Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2002 Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307335.CrossRefGoogle Scholar
Eckstein, E., Bailey, D. & Shapiro, A. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191208.CrossRefGoogle Scholar
Hackborn, W. W. 1990 Asymmetric Stokes flow between parallel planes due to a rotlet. J. Fluid Mech. 218, 531–46.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.CrossRefGoogle Scholar
Kulkarni, P. M. & Morris, J. F. 2007 Pair-sphere trajectories in finite Reynolds number shear flow. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid. Mech. 181, 415.CrossRefGoogle Scholar
Lin, C., Lee, K. & Sather, N. 1970 Slow motion of 2 spheres in a shear field. J. Fluid Mech. 43, 3547.CrossRefGoogle Scholar
Lorentz, H. A. 1907 A general theory concerning the motion of a viscous fluid. Abhandl. Theor. Phys. 1, 23.Google Scholar
Mikulencak, D. R. & Morris, J. F. 2004 Stationary shear flow around fixed and free bodies at finite Reynolds number. J. Fluid Mech. 520, 215242.CrossRefGoogle Scholar
Press, W. H., Vetterling, W. T., Flannery, B. P. & Teukolsky, S. A. 1992 Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge University Press.Google Scholar
Subramanian, G. & Koch, D. L. 2006 Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear. Phys. Fluids 18, 073302.CrossRefGoogle Scholar
Wang, Y., Mauri, R. & Acrivos, A. 1996 The transverse shear-induced liquid and particle tracer diffusivites of a dilute suspension of spheres undergoing a simple shear flow. J. Fluid Mech. 327, 255272.CrossRefGoogle Scholar
Wang, Y., Mauri, R. & Acrivos, A. 1998 Transverse shear-induced gradient diffusion in a dilute suspension of spheres. J. Fluid Mech. 357, 279287.CrossRefGoogle Scholar
Zarraga, I. E. & Leighton, D. T. 2001 Shear-induced diffusivity in a dilute bidisperse suspension of hard spheres. J. Colloid Interface Sci. 243, 13.CrossRefGoogle Scholar
Zarraga, I. E. & Leighton, D. T. 2002 Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres. Phys. Fluids 14, 21942201.CrossRefGoogle Scholar
Zinchenko, A. 1984 Effect of hydrodynamic interactions between the particles on the rheological properties of dilute emulsions. Prikl. Matem. Mekhan. 48, 198.Google Scholar
Zurita-Gotor, M., Blawzdziewicz, J. & Wajnryb, E. 2005 Boltzmann Monte-Carlo simulations of a suspension of non-spherical particles in a parallel-wall channel. Bull. Am. Phys. Soc. 50, 229.Google Scholar
Zurita-Gotor, M., Blawzdziewicz, J. & WAjnryb, E. 2007 Motion of a rod-like particle between parallel walls with application to suspension rheology. J. Rheol. 51, 7197.CrossRefGoogle Scholar