Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-06T09:13:45.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-axisymmetric motion of rigid closely fitting particles in fluid-filled tubes

Published online by Cambridge University Press:  26 April 2006

T. W. Secomb  and
R. Hsu
T. W. Secomb
Affiliation:
Department of Physiology, University of Arizona, Tucson, AZ 85724, USA
R. Hsu
Affiliation:
Department of Physiology, University of Arizona, Tucson, AZ 85724, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider non-axisymmetric motion of a rigid particle in a cylindrical fluid-filled tube, with negligible inertial effects. The particle is assumed to fit closely in the tube, and lubrication theory is used to describe the fluid flow in the narrow gap between the particle and the tube wall. The solution to the Reynolds lubrication equation and the components of the resistance matrix are expressed in terms of a Green's function. For the case in which the gap is almost uniform, the Green's function is expanded as a power series in a small parameter δ, characteristic of the variations in gap width, and the first two terms are obtained.

The velocity of a freely suspended axisymmetric particle driven by a pressure difference along the tube is deduced from the resistance matrix. According to the results at first order in δ, in general the particle moves transversely with a constant velocity. In the absence of higher-order effects, it would eventually collide with the wall. Motion along the tube axis is a neutrally stable solution to the equations of motion at first order. However, if effects at second order in δ are included, motion of an axisymmetric particle along the tube axis is stable or unstable depending on its shape. Generally, if the particle is narrower near the front than near the rear, and the width near the middle is at least as large as the mean of the widths near the front and rear, then its motion is stable. Numerical calculations (not restricted to small δ) confirm these results for axisymmetric particles, and show that a non-axisymmetric shape similar to a red blood cell has a stable equilibrium position in the tube.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 257 , December 1993 , pp. 403 - 420
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

Barnard, A. C. L., Lopez, L. & Hellums, J. D. 1968 Basic theory of blood flow in capillaries. Microvasc. Res. 1, 23–34.Google Scholar
Boyd, J. 1964 The influence of fluid forces on the sticking and the lateral vibration of pistons. Trans. ASME E: J. Appl. Mech. 31, 397–401.Google Scholar
Bungay, P. M. & Brenner, H. 1973 The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1, 25–56.Google Scholar
Cameron, A. 1966 The Principles of Lubrication. Wiley.
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
Christopherson, D. G. & Dowson, D. 1959 An example of minimum energy dissipation in viscous flow. Proc. R. Soc. Lond. A 251, 550–564.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 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
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics 2nd revised edn. Noordhoff.
Hochmuth, R. M. & Sutera, S. P. 1970 Spherical caps in low Reynolds-number flow. Chem. Engng Sci. 25, 593–604.Google Scholar
Hsu, R. & Secomb, T. W. 1989 Motion of non-axisymmetric red blood cells in cylindrical capillaries. J. Biomech. Engng 111, 147–151.Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Ann. Rev. Fluid Mech. 12, 435–476.Google Scholar
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34, 113–143.Google Scholar
Secomb, T. W. & Skalak, R. 1982 A two-dimensional model for capillary flow of an asymmetric cell. Microvasc. Res. 24, 194–203.Google Scholar
Secomb, T. W., Skalak, R., Özkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.Google Scholar
Sugihara-Seki, M., Secomb, T. W. & Skalak, R. 1990 Two-dimensional analysis of two-file flow of red cells along capillaries. Microvasc. Res. 40, 379–393.Google Scholar
Tözeren, H. & Skalak, R. 1978 The steady flow of closely fitting incompressible elastic spheres in a tube. J. Fluid Mech. 87, 1–16.Google Scholar
Wakiya, S. 1957 Viscous flows past a spheroid. J. Phys. Soc. Japan 12, 1130–1141.Google Scholar
Zarda, P. R., Chien, S. & Skalak, R. 1977 Interaction of viscous incompressible fluid with an elastic body. In Computational Methods for Fluid-Solid Interaction Problems (ed. T. Belytschko & T. L. Geers), pp. 65–82. ASME.
Supplementary material: PDF

Secomb and Hsu supplementary material

Appendix

Download Secomb and Hsu supplementary material(PDF)
PDF 576.1 KB