Liquid trains in a tube

– Trains of juxtaposed drops in a tube are described and found to move spontaneously, in wetting conditions, because of their asymmetry. We focus on the coating properties of these devices, and show in particular that highly viscous species can be (self-) transported, thanks to the lubricating film left by the first drop. Different properties of these systems are finally displayed, which stress their versatility towards microfluidics applications. Coating a solid with a liquid is a question of practical interest together with a classical field of interfacial hydrodynamics [1–8]. It is generally achieved by forcing a relative motion between a solid and a liquid: if a wetting liquid is displaced by a non-wetting fluid in a tube [3, 4], it leaves a film behind. The thickness h of this film results from a balance between viscosity η (which favours the film, because of the no-slip condition at the solid/liquid interface) and surface tension γ (since the film deposition implies a deformation of the free interface). The capillary number Ca compares viscous and capillary effects: Ca = ηV/γ (V denoting the velocity of deposition), and thus determines the film thickness h, as shown by Landau, Levich, Derjaguin and Bretherton [1–3]: h = 1.34rCa , (1) where r is the tube radius. Bretherton law (eq. (1)) is obeyed if the film is thin enough to neglect geometric effects (h < 0.1r) [4] and thick enough to avoid the influence of long-range forces (h > 100 nm) [7]. If the displacing fluid is of much higher viscosity than the coating one, eq. (1) remains valid, except for the coefficient which is multiplied by a factor 2 (about 1.6), as shown by Schwartz et al. [5]. Film deposition can occur spontaneously, instead of being forced. Very generally, spontaneous motion in a tube is caused by an asymmetry between the advancing and the receding interfaces. In classical capillary rise, for example, the front is convex and the rear flat. Similarly, Weislogel designed a tube made with two materials of different wettability and showed that a drop spanning both sides of the boundary between these materials moves towards the most wettable side [9]. In the same spirit, Bain [10] and Ondarçuhu [11] designed self-running drops, made of a solvent containing species likely to react with the solid. These reactants