Viscous drag friction acting on a fluid drop confined in between two plates

Dealing with a small amount of liquid has become increasingly important in recent applications in many fields such as biology, chemistry and medicine. In such a context, viscous drag friction acting on fluid drops in confined geometries is an indispensable fundamental issue, as the Stokes' friction law for a sphere in the bulk is useful in many physical processes. We study here, in a quasi two-dimensional confined space (i.e., in a Hele-Shaw cell), such viscous drag friction opposing gravitational drive. As a result, we establish in a clear way scaling laws for the viscous drag friction in different regimes. These scaling laws replace, in the confined geometry, the well-known Stokes' friction law. The proposed laws are unexpectedly simple in spite of the potential subtle effects of liquid thin films existing between the drop and the cell plates, thanks to the principle of minimal dissipation in viscous hydrodynamics. PACS numbers: 47.55.D- Drops and bubbles; 89.75.Da Systems obeying scaling laws; 68.15.+e Liquid thin films; 83.50.Lh Slip boundary effects; 47.57.Bc Foams and emulsions