Non-wetting droplets in capillaries of circular cross-section: Scaling function

Steady motion of long, non-wetting droplets carried by a surrounding liquid in a circular capillary has been the subject of many experimental, theoretical, and numerical simulation studies. Theoretical approaches, even after the application of lubrication approximation in hydrodynamic equations and after neglecting inertia and gravity effects, still lead to a numerical procedure to determine the speed of a droplet or the thickness of the film between a droplet and the wall of the capillary. Here, we develop the lubrication approximation further to introduce an analytical formula for the speed of droplets as a function of the capillary number and of the ratio of the viscosity coefficients of the two immiscible phases. We achieve this by identification of a scaling function within the lubrication approximation. The equations that we propose here corroborate well with the results of numerical simulations of droplet flow in circular capillaries.