Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2025-01-03T22:40:10.460Z Has data issue: false hasContentIssue false
  • English
  • Français

The motion of a rigid body in viscous fluid bounded by a plane wall

Published online by Cambridge University Press:  26 April 2006

Richard Hsu  and
Peter Ganatos
Richard Hsu
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, New York, NY 10031, USA
Peter Ganatos
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, New York, NY 10031, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The boundary-integral method is used to calculate the hydrodynamic force and torque on an arbitrary body of revolution whose axis of symmetry is oriented at an arbitrary angle relative to a planar wall in the zero-Reynolds-number limit. The singular solution of the Stokes equations in the presence of a planar wall is used to formulate the integral equations, which are then reduced to a system of linear algebraic equations by satisfying the no-slip boundary conditions on the body surface using the boundary collocation method or weighted residual technique.

Numerical tests for the special case of a sphere moving parallel or perpendicular to a planar wall show that the present theory is accurate to at least three significant figures when compared with the exact solutions for gap widths as small as only one-tenth of the particle radius. Higher accuracy can be achieved and solutions can be obtained for smaller gap widths at the expense of more computation time and larger storage requirements.

The hydrodynamic force and torque on a spheroid with varying aspect ratio and orientation angle relative to the planar wall are obtained. The theory is also applied to study the motion of a toroidal particle or biconcave shaped disc adjacent to a planar wall. The coincidence of the drag and torque of a biconcave-shaped body and a torus having an aspect ratio b/a = 2 with the same surface area shows that in this case the hole of a torus has little influence on the flow field. On the other hand, for an aspect ratio b/a = 10, the effect of the hole is significant. It is also shown that when the body is not very close to the wall, an oblate spheroid can be used as a good approximation of a biconcave-shaped disc.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 207 , October 1989 , pp. 29 - 72
Copyright
© 1989 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

Blake, J. R. 1971 A note on the image system for a stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.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
Brenner, H. 1964 The Stokes resistance of an arbitrary particle – II. An extension. Chem. Engng Sci. 19, 599–629.Google Scholar
Brenner, H. & Gaydos, L. J. 1977 The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius. Models of the diffusive and convective transport of solute molecules in membranes and porous media. J. Colloid Interface Sci. 58, 312–356.Google Scholar
Chen, T. C. & Skalak, R. 1970 Stokes flow in a cylindrical tube containing a line of spheroidal particles. Appl. Sci. Res. 22, 403–441.Google Scholar
Dabros, 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
Fung, Y. C. 1981 Biomechanics: Mechanical Properties of Living Tissues. Springer.
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
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Engng Sci. 22, 653–660.Google Scholar
Goren, L. & O'Neill, M. E. 1980 Asymmetric creeping motion of an open torus. J. Fluid Mech. 101, 97–110.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics, 2nd rev. edn. Noordhoff.
Hsu, R. 1985 Hydrodynamic interaction of an arbitrary particle with a planar-wall at low Reynolds number. Ph.D. dissertation, City University of New York.
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
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
Kucaba-Pietal, A. 1986 Nonaxisymmetric Stokes flow past a torus in the presence of a wall. Arch. Mech. 38, 647–663.Google Scholar
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
Leal, L. G. & Lee, S. H. 1982a Particle motion near a deformable fluid interface. Adv. Colloid Interface Sci. 17, 61–81.Google Scholar
Lee, S. H. & Leal, L. H. 1982b The motion of a sphere in the presence of a deformable interface. II. A numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87, 81–106.Google Scholar
Leichtberg, S., Weinbaum, S. & Pfeffer, R. 1976 A theory for the coaxial slow viscous motion of finite clusters of spheres in unbounded Poiseuille flow and its application to rouleaux formation. Biorheol. 13, 165–179.Google Scholar
Lewellen, P. 1982 Hydrodynamic analysis of microporous mass transport. Ph.D. dissertation. University of Wisconsin-Madison.
Lightfoot, E. N. 1974 Transport Phenomena and Living Systems. Wiley.
Liron, N. 1978 Fluid transport by cilia between parallel plates. J. Fluid Mech. 86, 705–726.Google Scholar
Liron, N. & Mochon, S. 1976 The discrete-cilia approach to propulsion of ciliated microorganisms. J. Fluid Mech. 75, 593–607.Google Scholar
Odqvist, F. K. G. 1930 Uber die Randwertaufgaben der Hydrodynamik Zaher Flussigkeiten. Math. Z. 32, 329–375.Google Scholar
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
Skalak, R., Chen, P. H. & Chien, S. 1972 Effect of hematocrit and rouleaux on apparent viscosity in capillaries. Biorheol. 9, 67–89.Google Scholar
Tozeren, H. 1984 Boundary integral equation method for some Stokes problems. Intl J. Num. Meth. Fluids 4, 159–170.Google Scholar
Wakiya, S. 1959 Effect of a submerged object on a slow viscous flow (Report V). Spheroid at an arbitrary angle of attack. Res. Rep. Fac. Engng Niigata Univ. Japan 8, 17–30 (in Japanese).Google Scholar
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75–96.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 (and corrigendum 69, 813).Google Scholar
Yuan, F. & Wu, W. Y. 1987 The Stokes flow of an arbitrary prolate axisymmetric body towards an infinite plane wall. Appl. Math. Mech. SUT, Shanghai, China, 8, 17–30.Google Scholar