Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T03:35:41.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Friction factors for a lattice of Brownian particles

Published online by Cambridge University Press:  20 April 2006

Alan J. Hurd ,
Noel A. Clark ,
Richard C. Mockler  and
William J. O'Sullivan
Alan J. Hurd
Affiliation:
Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02254 Permanent address: Division 1152, Sandia National Laboratories, Albuquerque, NM 87185.
Noel A. Clark
Affiliation:
Department of Physics, University of Colorado, Boulder, Colorado 80309
Richard C. Mockler
Affiliation:
Department of Physics, University of Colorado, Boulder, Colorado 80309
William J. O'Sullivan
Affiliation:
Department of Physics, University of Colorado, Boulder, Colorado 80309
Get access Rights & Permissions [Opens in a new window]

Abstract

The resistance to oscillatory motions of arbitrary wavelengths in an infinitely dilute lattice of identical spheres, immersed in a viscous fluid, is calculated from the linearized Navier–Stokes equation to lowest order in fluid inertia and sphere-volume fraction. The application we have in mind is to analyse the hydrodynamic modes in colloidal crystals (a lattice of Brownian particles repelling each other electrically), although other applications are possible. We find that the friction per particle for both compressional and transverse shear modes is close to the Stokes value at short wavelengths, whereas at long wavelengths fluid backflow within the lattice is important and causes the friction to increase for compressional modes. For shear modes, in which backflow is not present, the friction decreases from the Stokes value at short wavelengths to zero at long wavelengths. At sufficiently long wavelengths, when the shear-mode friction becomes small enough, propagating viscoelastic modes are possible in a lattice with elastic forces between spheres. Fluid inertia is most important for long-wavelength transverse motions, since a significant amount of fluid mass gets carried along by each particle. Explicit results for a bcc lattice are presented along with interpolation formulas, and the pertinence of these results to colloidal crystals is discussed. Finally, the effects of constraining walls are explored by considering a one-dimensional lattice near a wall. Backflow imposed by the wall increases the friction factors for the lattice modes, showing that propagating modes are unlikely in colloidal crystals that are confined to a cell thinner than a critical length.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 153 , April 1985 , pp. 401 - 416
Copyright
© 1985 Cambridge University Press

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.)

References

Beenakker, C. W. J. & Mazur, P. 1984 Preprint.
Benzing, D. W. & Russel, W. B. 1981 J. Coll. Interface Sci. 83, 178.
Glasser, M. L. 1974 J. Math. Phys. 15, 188.
Goren, S. L. 1983 J. Fluid Mech. 132, 185.
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.
Hasimoto, H. 1959 J. Fluid Mech. 5, 317.
Hurd, A. J. 1981 Ph.D. thesis, University of Colorado.
Hurd, A. J., Clark, N. A., Mockler, R. C. & O'Sullivan, W. J.1982 Phys. Rev. A26, 2869.
Joanny, J. F. 1979 J. Coll. Interface Sci. 71, 622.
Lindsay, H. M. & Chaikin, P. M. 1982 J. Chem. Phys. 76, 3774.
Mazur, P. 1982 Physica 110A, 128.
Mazur, P. & Bedeaux, D. 1974 Physica 76, 235.
Mazur, P. & van Saarloos, W. 1982 Physica 115A, 21.
Noble, B. 1969 Applied Linear Algebra. Prentice-Hall.
Pieranski, P., Dubois-Violette, E., Rothen, F. & Strzelecki, L. 1981 J. Phys. (Paris) 42, 53.
Russel, W. B. & Benzing, D. W. 1981 J. Coll. Interface Sci. 83, 163.
Saffman, P. G. 1973 Stud. Appl. Math. 52, 115.
Saarloos, W. Van & Mazur, P. 1983 Physica 120A, 77.
Zuzovsky, M., Adler, P. M. & Brenner, H. 1983 Phys. Fluids 26, 1714.