Elastocapillary interaction between a long rectangular membrane and a liquid drop

We report elastocapillary interaction between a long rectangular membrane fixed along its central axis and a liquid drop dispensed at one of its ends. The introduction of the drop results in the elastocapillary-driven wrapping of the membrane along its width and a concomitant flow in the resulting conduit along its length. Depending upon the drop size (d) and capillary length scale (L_c), we identified general criteria for achieving complete wrapping of the membrane in the dry state from energy considerations. For small droplets satisfying d ≲ L_c, we find that the critical membrane length (W_c) required for complete wrapping is proportional to the elastocapillary length scale (L_ec). In the case of large droplets with d > L_c, the wrapping behavior depends on the ratio of membrane width to elastocapillary length scale (W/L_ec) and the ratio of capillary length scale to the elastocapillary length scale (L_c/L_ec). Our study suggests that the critical membrane width for complete wrapping is smaller in the wet state compared to that in the dry state, which can be attributed to the existence of a transmembrane pressure in the wet state. The effect of membrane thickness and width and drop volume on the length and cross-section of the wrapped conduit and attached width of the wrapped membrane is studied. For small droplets, the resulting elastocapillary flow exhibits an inertial regime at small times, followed by a Washburn regime at intermediate times, and finally an inertial regime, and for large droplets, only an inertial regime is observed throughout.