Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-04T08:40:08.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Axisymmetric creeping motion of drops through circular tubes

Published online by Cambridge University Press:  26 April 2006

M. J. Martinez  and
K. S. Udell
M. J. Martinez
Affiliation:
Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA Current address: Fluid and Thermal Sciences Department, Sandia National Laboratories, Albuquerque, NM 87185, USA.
K. S. Udell
Affiliation:
Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The axisymmetric creeping motion of a neutrally buoyant deformable drop flowing through a circular tube is analysed with a boundary integral equation method. The fluids are immiscible, incompressible, and the bulk flow rate is constant. The drop to suspending fluid viscosity ratio is arbitrary and the drop radius varies from 0.5 to 1.15 tube radii. The effects of the capillary number, viscosity ratio, and drop size on the deformation, the drop speed, and the additional pressure loss are examined.

Drops with radius ratios less than 0.7 are insensitive to substantial variation in capillary number and viscosity ratio, and computed values of drop speed and extra pressure loss are in excellent agreement with small deformation theories (Hestroni et al. 1970; Hyman & Skalak 1972a). For this drop size range, significant deformation will result only for Ca > 0.25. The onset of a re-entrant cavity is predicted at the trailing end of the drop for Ca ≈ 0.75. Drop speed and meniscus shape become independent of drop size for radius ratios as small as 1.10. The extra pressure loss can be positive or negative depending mainly on the viscosity ratio, however a relatively inviscid drop can cause a positive extra pressure loss when capillary forces are significant. Computed values for extra pressure loss and drop speed are in good agreement with the experimental data of Ho & Leal (1975) for drops of sizes comparable with the tube radius.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 565 - 591
Copyright
© 1990 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

Brenner, H. 1971 Pressure drop due to the motion of neutrally buoyant particles in duct flows. II. Spherical droplets and bubbles. Indust. Engng Chem. Fund. 10, 537–543.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Chi, B. K. 1986 The motion of immiscible drops and the stability of annular flow. Ph.D. thesis, Cal. Inst. of Tech., Pasadena, California.
Cox, B. G. 1962 On driving a viscous fluid out of a tube. J. Fluid Mech. 14, 81–98.Google Scholar
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Paris: Victor Dalmont.
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on the creeping motion of bubbles. Chem. Engng Sci. 21, 681–685.Google Scholar
Dullien, F. A. L. 1979 Porous Media: Fluid Transport and Pore Structure. Academic.
Fairbrother, F. & Stubbs, A. 1935 Studies in electroendosmosis. Part VI. The bubble-tube methods of measurement. J. Chem. Sci. 1, 527–529.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1963 The flow of suspensions through tubes: II. Single large bubbles. J. Colloid Sci. 18, 237–261.Google Scholar
Haberman, W. L. & Sayre, R. M. 1958 Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David Taylor Model Basin Rep. 1143. Washington D. C.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hestroni, G., Haber, S. & Wacholder, E. 1970 The flow field in and around a droplet moving axially within a tube. J. Fluid Mech. 41, 689–705.Google Scholar
Ho, B. P. & Leal, L. G. 1975 The creeping motion of liquid drops through a circular tube of comparable diameter. J. Fluid Mech. 71, 361–384.Google Scholar
Hyman, W. A. & Skalak, R. 1972a Viscous flow of a suspension of liquid drops in a cylindrical tube. Appl. Sci. Res. 26, 27–52.Google Scholar
Hyman, W. A. & Skalak, R. 1972b Non-Newtonian behavior of a suspension of liquid drops in tube flow. AIChE J. 18, 149–154.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach.
Martinez, M. J. 1987 Viscous flow of drops and bubbles in straight and constricted tubes. Ph.D. thesis, University of California, Berkeley.
Martinez, M. J. & Udell, K. S. 1989 Boundary integral analysis of the creeping flow of long bubbles in capillaries. Trans. ASME E: J. Appl. Mech. 56, 211–217.Google Scholar
Olbricht, W. L. & Leal, L. G. 1983 The creeping motion of immiscible drops through a converging/diverging tube. J. Fluid Mech. 134, 329–355.Google Scholar
Payatakes, A. C. 1982 Dynamics of oil ganglia during immiscible displacement in water-wet porous media. Ann. Rev. Fluid Mech. 14, 365–393.Google Scholar
Payatakes, A. C. & Neira, M. A. 1977 Model of the constricted unit cell type for isotropic granular porous media. AIChE J. 23, 922–930.Google Scholar
Payatakes, A. C., Tien, C. & Turian, R. M. 1973 A new model for granular porous media: Part II. Numerical solution of steady state Newtonian flow through periodically constricted tubes. AIChE J. 19, 67–76.Google Scholar
Prothero, J. & Burton, A. C. 1961 The physics of blood flow in capillaries. J. Biophys. 2, 525–572.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
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Statist. Comput. 6, 542–561.Google Scholar
Rizzo, F. J. & Shippy, D. J. 1968 A formulation and solution procedure for the general nonhomogeneous elastic inclusion problem. Intl J. Solids Struct. 4, 1161–1179.Google Scholar
Scheidegger, A. E. 1974 The Physics of Flow through Porous Media, 3rd edn. University of Toronto Press.
Schwartz, L. W., Princen, H. M. & Kiss, A. D. 1986 On the motion of bubbles in capillary tubes. J. Fluid Mech. 172, 259–275.Google Scholar
Shen, E. I. & Udell, K. S. 1985 A finite element study of low Reynolds number two-phase flow in cylindrical tubes. Trans. ASME E: J. Appl. Mech. 52, 253–256.Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.Google Scholar
Teletzke, G. F. 1983 Thin liquid films: molecular and hydrodynamic implications. Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota.
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.Google Scholar