Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T14:29:17.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear drift interactions between fluctuating colloidal particles: oscillatory and stochastic motions

Published online by Cambridge University Press:  26 April 2006

E. J. Hinch  and
Ludwig C. Nitsche
E. J. Hinch
Affiliation:
Department of Applied Mathematics and Theoretical Physics, The University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Ludwig C. Nitsche
Affiliation:
Department of Chemical Engineering, The University of Illinois at Chicago, 810 South Clinton Street, Chicago, IL 60607, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work we consider how nonlinear hydrodynamic effects can lead to a mean force of interaction between two spheres of equal radius a undergoing translational fluctuations parallel or perpendicular to their line of centres. Motivated by amplitudes and Reynolds numbers characteristic of Brownian motion in colloidal systems, nonlinearities due to motion of the boundaries and to inertia throughout the fluid are treated as regular perturbations of the time-dependent Stokes equations. This formulation ultimately leads to a prescription for computing, at leading order, the time-average nonlinear force for the case of pure oscillatory modes – which represents the Fourier decomposition of more general motions. The associated hydrodynamic problems are solved numerically using a least-squares boundary singularity method. Frequency-dependent results over the whole spectrum are presented for a sphere-sphere gap equal to one radius; illustrative calculations are also carried out at other separations. Subsequently we extend the analysis of nonlinear drift to a Langevin equation formulation of the more complex problem of stochastic motion due to thermal fluctuations in the suspending fluid, i.e. Brownian motion. By integrating (numerically) over the spectrum of frequencies, we quantify how the mutual interactions of all translational disturbance modes give rise, on ensemble average, to a stochastic nonlinear force of interaction between the particles. It is particularly interesting that this net interaction – arising from a zero-mean random force – is of O(1) on the Brownian scale kT/a, even though it represents a small O(Re) correction at each frequency of pure oscillations. Finally, we discuss how the presence of stochastic nonlinear drift would lead to non-uniform equilibrium distributions of dilute colloidal suspensions, unless one adds to the random force in the Langevin equation a cancelling non-zero mean component.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 256 , November 1993 , pp. 343 - 401
Copyright
© 1993 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

Altenberger, A. R. 1979 The role of inertial effects in macroparticle diffusion. J. Polymer Sci: Polymer Phys. Ed. 17, 1317–1324.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1977 Developments in microhydrodynamics. In Theoretical and Applied Mechanics (ed. W.T. Koiter), pp. 33–55. North-Holland.
Birkhoff, G. 1950 Hydrodynamics. A Study in Logic, Fact, and Similtude. Princeton University Press.
Bogomolny, A. 1985 Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22, 644–669.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.Google Scholar
Case, K.M. 1971 Velocity fluctuations of a body in a fluid. Phys. Fluids 14, 2091–2095.Google Scholar
Chow, T.S. & Hermans, J.J. 1974 Limitations of the Langevin equation based on hydrodynamic models. K. Nederl. Akad. Van Wetenschappen — Amst. B 77, 18–25.Google Scholar
Dabroś, T. 1985 A singularity method for calculating hydrodynamic forces and particle velocities in low-Reynolds-number flows. J. Fluid Mech. 156, 1–21.Google Scholar
Dahlquist, G. & Björck, Å. 1974 Numerical Methods. (Transl. by N. Anderson) Prentice Hall.
Dongarra, J.J., Bunch, J.R., Moler, C.B. & Stewart, G.W. 1979 LINPACK Users’ Guide. Philadelphia: SIAM.
El-Kareh, A. W. & Leal, L.G. 1993 Brownian motion of a slightly deformable drop. II Higher-order fluctuation-dissipation theorems. J. Statist. Phys. (submitted.)Google Scholar
Gluckman, M. J., Pfeffer, R. & Weinbaum, S. 1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705–740.Google Scholar
Goldstein, H. 1951 Classical Mechanics. Addison-Wesley.
Grotberg, J. B. 1984 Volume-cycled oscillatory flow in a tapered channel. J. Fluid Mech. 141, 249–264.Google Scholar
Hall, P. 1974 Unsteady viscous flow in a pipe of slowly varying cross section. J. Fluid Mech. 64, 209–226.Google Scholar
Han, P.S. & Olson, M. D. 1987 An adaptive boundary element method. Intl J. Numer. Meth. Engng 24, 1187–1202.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hauge, E. H. & Martin-Löf, A. 1973 Fluctuating hydrodynamics and Brownian motion. J. Statist. Phys. 7, 259–281.Google Scholar
Hinch, E. J. 1975 Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72, 499–511.Google Scholar
Howells, I. D. 1974 Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475.Google Scholar
Kampen, N. G. van 1981 Stochastic Processes in Physics and Chemistry. North-Holland.
Kanwal, R. P. 1964 Drag on an axially symmetric body vibrating slowly along its axis in a viscous fluid. J. Fluid Mech. 19, 631–636.Google Scholar
Karrila, S. J. & Kim, S. 1989 Integral equations of the second kind for Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations. Chem. Engng Commun. 82, 123–161.Google Scholar
Kim, S. & Karrila, S.J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.
Kim, S. & Mifflin, R. T. 1985 The resistance and mobility functions of two equal spheres in low-Reynolds-number flow. Phys. Fluids 28, 2033–2045.Google Scholar
Kim, S. & Russel, W. B. 1985 The hydrodynamic interactions between two spheres in a Brinkman medium. J. Fluid Mech. 154, 253–268.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press (Dover edition 1945.)
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. (Transl. from the Russian by J. B. Sykes & W. H. Reid; Vol. 6 of Course of Theoretical Physics.) Pergamon.
Lawrence, C. J. & Weinbaum, S. 1988 The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489.Google Scholar
Mathon, R. & Johnston, R. L. 1977 The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14, 638–650.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Macmillan.
Nitsche, L. C. & Brenner, H. 1990 Hydrodynamics of particulate motion in sinusoidal pores via a singularity method. AIChE J. 36, 1403–1419.Google Scholar
Pozrikidis, C. 1989a A study of linearized oscillatory flow past particles by the boundary-integral method. J. Fluid Mech. 202, 17–41.Google Scholar
Pozrikidis, C. 1989b A singularity method for unsteady linearized flow. Phys. Fluids A 1, 1508–1520.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Rallison, J. M. 1979 The role of rigidity constraints in the rheology of dilute polymer solutions. J. Fluid Mech. 93, 251–279.Google Scholar
Riley, N. 1966 On a sphere oscillating in a viscous fluid. Q. J. Mech. Appl. Maths 19, 461–472.Google Scholar
Riley, N. 1987 Streaming from a cylinder due to an acoustic source. J. Fluid Mech. 180, 319–326.Google Scholar
Russel, W. B. 1981 Brownian motion of small particles suspended in liquids. Ann. Rev. Fluid Mech. 13, 425–455.Google Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.
Sangani, A. S. 1991 A pairwise interaction theory for determining the linear acoustic properties of dilute bubbly liquids. J. Fluid Mech. 232, 221–284.Google Scholar
Sangani, A. S., Zhang, D. Z. & Prosperetti, A. 1991 The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys. Fluids A 3, 2955–2970.Google Scholar
Sobey, I.J. 1980 On flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Stimson, M. & Jeffrey, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111, 110.Google Scholar
Tabakova, S. S. & Zapryanov, Z. D. 1982a On the hydrodynamic interaction of two spheres oscillating in a viscous fluid. I. Axisymmetric case. Z. Angew. Math. Phys. 33, 344–357.Google Scholar
Tabakova, S. S. & Zapryanov, Z. D. 1982b On the hydrodynamic interaction of two spheres oscillating in a viscous fluid. II. Three dimensional case. Z. Angew. Math. Phys. 33, 487–502.Google Scholar
Wax, N. 1954 Selected Papers on Noise and Stochastic Processes. Dover.
Weinbaum, S., Ganatos, P. & Yan, Z.-Y. 1990 Numerical multipole and boundary boundary integral equation techniques in Stokes flow. Ann. Rev. Fluid Mech. 22, 275–316.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377–403.Google Scholar
Supplementary material: PDF

Hinch and Nitsche supplementary material

Supplementary Material

Download Hinch and Nitsche supplementary material(PDF)
PDF 3.3 MB