Skip to main content Accessibility help

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

  • ETIENNE LAC (a1) and J. D. SHERWOOD (a2)


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.


Corresponding author

Email address for correspondence:


Hide All
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, 97132.
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), 537543.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (2), 166188.
Cox, B. G. 1962 On driving a viscous fluid out of a tube. J. Fluid Mech. 14 (1), 8196.
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.
Edvinsson, R. K. & Irandoust, S. 1996 Finite-element analysis of Taylor flow. AIChE J. 42 (7), 18151823.
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), 24202428.
Goldsmith, H. L. & Mason, S. G. 1963 The flow of suspensions through tubes: II. Single large bubbles. J. Colloid Interface Sci. 18, 237261.
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), 705710.
Hetsroni, G., Haber, S. & Wacholder, E. 1970 The flow in and around a droplet moving axially within a tube. J. Fluid Mech. 41 (4), 699705.
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), 361384.
Hodges, S. R., Jensen, O. E. & Rallison, J. M. 2004 The motion of a viscous drop through a cylindrical tube. J. Fluid Mech. 501, 279301.
Hyman, W. A. & Skalak, R. 1972 Viscous flow of a suspension of liquid drops in a cylindrical tube. Appl. Sci. Res. 26, 2752.
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids, 1st ed. Springer.
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.
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86 (4), 727744.
Martinez, M. J. & Udell, K. S. 1989 Boundary integral analysis of the creeping flow of long bubbles in capillaries. J. Appl. Mech. 56 (1), 211217.
Martinez, M. J. & Udell, K. S. 1990 Axisymmetric creeping motion of drops through circular tubes. J. Fluid Mech. 210, 565591.
Olbricht, W. L. 1996 Pore-scale prototypes of multiphase flow in porous media. Ann. Rev. Fluid Mech. 28, 187213.
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), 13471354.
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2 (3), 340352.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Preziosi, L., Chen, K. & Joseph, D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323356.
Ratulowski, J. & Chang, H.-C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1 (10), 16421655.
Soares, E. J., Carvalho, M. S. & Souza Mendes, P. R. 2005 Immiscible liquid–liquid displacement in capillary tubes. J. Fluids Engng 127, 2431.
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.
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, 223263.
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10 (2), 161165.
Tsai, T. M. & Miksis, M. J. 1994 Dynamics of a drop in a constricted capillary tube. J. Fluid Mech. 274, 197217.
Westborg, H. & Hassager, O. 1989 Creeping motion of long bubbles and drops in capillary tubes. J. Colloid Interface Sci. 133 (1), 135147.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed