Predicting droplet velocity in a Hele-Shaw cell
Résumé
We study the motion of low viscous non-wetting droplet in a Hele-Shaw cell, while it is pushed by an external phase of imposed flow rate, at low capillary numbers. In this regime, the droplet's mobility, defined as the ratio between the droplet velocity and the external phase mean velocity, evolves non linearly with the capillary number, a signature of the different dissipation mechanisms at play. Experiments are performed with surfactant free air bubbles in fluorinated oil, and with surfactant laden fluorinated oil droplets in water. We propose a model based on a power balance which takes into account the dissipation in the thin wetting film trapped between the bubble (or the drop) and the channel wall. The full topography of this thin film is obtained theoretically for the bubble case. By contrast, the presence of surfactants in the drop case induces uncontrolled boundary conditions at the interface, thus imposing to use the experimental topography measured in the previous paper [Reichert et al., J. Fluid. Mech., 850 p.708 (2018)]. Remarkably, the model reproduces the experimental velocities and shows that the velocity can be strongly affected by a stagnant cap effect at the rear of the drop, even if localized in less than a few percents of the total film area.
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