Drop propulsion in tapered tubes

We present the results of a combined experimental and theoretical investigation of the motion of wetting droplets in tapered capillary tubes. We demonstrate that drops may move spontaneously towards the tapered end owing to the Laplace pressure gradient established along their length. The influence of gravity on this spontaneous motion is examined by studying drop motion along a tilted tube with its tapered end pointing upwards. Provided the tube taper varies, an equilibrium height may be achieved in which the capillary force is balanced by the drop’s weight. We deduce the family of tube shapes that support a stable equilibrium. Copyright c © EPLA, 2009 Introduction. – In 1712, Hauksbee observed that a drop of orange oil bridging two non-parallel plates propels itself in the direction of maximum confinement, that is, into the gap [1]. The drop asymmetry is responsible for the motion: this wetting liquid meets the plates with a different curvature on its leading and trailing edges, so that a differential Laplace pressure is established between the drop extremities and drives the liquid to the region of maximum confinement. A similar motion takes place in a tapered tube [2], a geometry that will be the subject of our investigation. Related curvature-driven motions may also arise for wetting drops on conical fibers, for which the sign of the curvature is reversed, so that the drop is propelled away from the cone tip [3]. For multiphase flow in natural porous media, constrictions or enlargements of the pore channels will necessarily generate displacement and reconfiguration of drops or bubbles. Such geometries also exist in man-made microfluidic devices, and can be used to control the topology and dynamics of two-phase systems, such as emulsions or foams [4]. Finally, the motion of drops in confined geometries arises in the biological world; for example, a class of shorebirds transport prey-laden water droplets mouthward by opening and closing their beaks in a tweezering motion [5–7]. We here study the behaviour of liquid slugs in tapered tubes. If inserted into a fluid bath, conical tubes cause the liquid to rise via capillary action [8,9], but Tsori demonstrated that even this simple system displays surprisingly rich behaviour [10,11]. For example, a wetting liquid either rises to a finite height, or to the top of the tube if the cone angle exceeds a critical value. Here we consider a wetting drop initially placed in the widest part of an inclined tapered tube, and examine the equilibrium position of the drop. While one might first expect this equilibrium to be obtainable directly from Laplace’s formula, we here demonstrate that it is potentially unstable, and that its very existence depends on the detailed shape of the tube. Initial observations. – We consider drops (or slugs) in tapered tubes that are open at both ends. The tubes are thin relative to the capillary length, so that the menisci bounding the slugs can be considered hemispherical. The liquid wets the solid and meets it with zero contact angle. This condition implies that nanoscopic wetting films develop and progress ahead of the menisci, eliminating contact angle hysteresis, and so favouring drop motion in response to forces. Here, the forces driving motion are gravity and the capillary force arising from the curvature difference of the two menisci, as follows from the tube’s tapered geometry. The cone opening angle α will be taken small relative to unity. One expects capillary-induced slug motion in a tilted tube to be eventually halted by gravity, as shown in fig. 1. Our experiment consists of determining when a drop equilibrium position exists, then measuring the position zo, as a function of the tube shape and tilt angle. Our capillary cones were created by heating and pulling glass capillary tubes. By fixing a given load at the tube extremity, we generate a constant force that draws and