J. Fluid Mech. (2009), vol. 640, pp. 27–54. c  Cambridge University Press 2009 doi:10.1017/S0022112009991212 27 Motion of a drop along the centreline of a capillary in a pressure-driven flow ETIENNE LAC† 1 AND J. D. SHERWOOD 2 1 Schlumberger-Doll Research, One Hampshire Street, Cambridge MA 02139-1578, USA 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK (Received 30 May 2008; revised 10 July 2009; accepted 16 July 2009; first published online 2 November 2009) 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. 1. Introduction The main aim of the work presented here and in a second paper (Lac & Sherwood 2009) is to determine how the presence of a liquid drop disturbs the electrical streaming potential generated by pressure-driven flow through a capillary with charged walls. We shall assume that the electric fields generated by the flow are sufficiently small that perturbations to the flow field due to electric stresses are negligible. The first step of the computation is therefore to determine the hydrodynamic behaviour of a drop as it flows through the capillary, in the absence of any electrical effects. This problem is of sufficient importance in its own right that it is discussed separately in this first paper. Electrokinetic effects will be considered by Lac & Sherwood (2009). The motion of a drop in a straight capillary represents an idealized two-phase flow in a porous medium: the geometry of a realistic porous material (e.g. rock) † Email address for correspondence: elac@slb.com