Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T21:49:22.027Z Has data issue: false hasContentIssue false
  • English
  • Français

Motion of a drop along the centreline of a capillary in a pressure-driven flow

Published online by Cambridge University Press:  02 November 2009

ETIENNE LAC  and
J. D. SHERWOOD
ETIENNE LAC*
Affiliation:
Schlumberger-Doll Research, One Hampshire Street, Cambridge MA 02139-1578, USA
J. D. SHERWOOD
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ < 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ < 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (λ > 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 27 - 54
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Bai, R., Chen, K. & Joseph, D. D. 1992 Lubricated pipelining: stability of core-annular flow. Part 5. Experiments and comparison with theory. J. Fluid Mech. 240, 97–132.CrossRefGoogle Scholar
Brenner, H. 1971 Pressure drop due to the motion of neutrally buoyant particles in duct flows. II. Spherical droplets and bubbles. Ind. Engng Chem. Fundam. 10 (4), 537–543.CrossRefGoogle Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (2), 166–188.CrossRefGoogle Scholar
Cox, B. G. 1962 On driving a viscous fluid out of a tube. J. Fluid Mech. 14 (1), 81–96.CrossRefGoogle Scholar
Dupont, J.-B., Legendre, D. & Fabre, J. 2007 Motion and shape of long bubbles in small tube at low Re-number. In Intl Conf. Multiphase Flow, Leipzig, Germany, July 9–13, 2007.Google Scholar
Edvinsson, R. K. & Irandoust, S. 1996 Finite-element analysis of Taylor flow. AIChE J. 42 (7), 1815–1823.CrossRefGoogle Scholar
Giavedoni, M. D. & Saita, F. A. 1997 The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid: a simultaneous solution of the governing equations. Phys. Fluids 9 (8), 2420–2428.CrossRefGoogle Scholar
Goldsmith, H. L. & Mason, S. G. 1963 The flow of suspensions through tubes: II. Single large bubbles. J. Colloid Interface Sci. 18, 237–261.Google Scholar
Greenstein, T. & Happel, J. 1968 Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube. J. Fluid Mech. 34 (4), 705–710.CrossRefGoogle Scholar
Hetsroni, G., Haber, S. & Wacholder, E. 1970 The flow in and around a droplet moving axially within a tube. J. Fluid Mech. 41 (4), 699–705.CrossRefGoogle 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 (2), 361–384.Google Scholar
Hodges, S. R., Jensen, O. E. & Rallison, J. M. 2004 The motion of a viscous drop through a cylindrical tube. J. Fluid Mech. 501, 279–301.CrossRefGoogle Scholar
Hyman, W. A. & Skalak, R. 1972 Viscous flow of a suspension of liquid drops in a cylindrical tube. Appl. Sci. Res. 26, 27–52.Google Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids, 1st ed. Springer.Google Scholar
Lac, E. & Sherwood, J. D. 2009 Streaming potential generated by a drop moving along the centreline of a capillary. J. Fluid Mech. (in press) doi:10.1017/S002211200999156X.CrossRefGoogle Scholar
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86 (4), 727–744.CrossRefGoogle Scholar
Martinez, M. J. & Udell, K. S. 1989 Boundary integral analysis of the creeping flow of long bubbles in capillaries. J. Appl. Mech. 56 (1), 211–217.CrossRefGoogle Scholar
Martinez, M. J. & Udell, K. S. 1990 Axisymmetric creeping motion of drops through circular tubes. J. Fluid Mech. 210, 565–591.Google Scholar
Olbricht, W. L. 1996 Pore-scale prototypes of multiphase flow in porous media. Ann. Rev. Fluid Mech. 28, 187–213.Google Scholar
Olbricht, W. L. & Kung, D. M. 1992 The deformation and breakup of liquid drops in low Reynolds number flow through a capillary. Phys. Fluids A 4 (7), 1347–1354.CrossRefGoogle Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2 (3), 340–352.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Preziosi, L., Chen, K. & Joseph, D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323–356.CrossRefGoogle Scholar
Ratulowski, J. & Chang, H.-C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1 (10), 1642–1655.Google Scholar
Soares, E. J., Carvalho, M. S. & Souza Mendes, P. R. 2005 Immiscible liquid–liquid displacement in capillary tubes. J. Fluids Engng 127, 24–31.Google Scholar
Soares, E. J. & Thompson, R. L. 2009 Flow regimes for the immiscible liquid-liquid displacement in capillary tubes with complete wetting of the displaced liquid. J. Fluid Mech. (accepted for publication) doi:10.1017/S0033223.9991546.CrossRefGoogle Scholar
Stone, H. & Leal, L. G. 1989 The influence of initial deformation on drop breakup in subcritical time-dependent flows at low Reynolds numbers. J. Fluid Mech. 206, 223–263.CrossRefGoogle Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10 (2), 161–165.CrossRefGoogle Scholar
Tsai, T. M. & Miksis, M. J. 1994 Dynamics of a drop in a constricted capillary tube. J. Fluid Mech. 274, 197–217.CrossRefGoogle Scholar
Westborg, H. & Hassager, O. 1989 Creeping motion of long bubbles and drops in capillary tubes. J. Colloid Interface Sci. 133 (1), 135–147.Google Scholar