Thickness of residual wetting film in liquid-liquid displacement

Abstract

Core-annular flow is common in nature, representing, for example, how streams of oil, surrounded by water, move in petroleum reservoirs. Oil, typically a nonwetting fluid, tends to occupy the middle (core) part of a channel, while water forms a surrounding wall-wetting film. What is the thickness of the wetting film? A classic theory has been in existence for nearly 50 years offering a solution, although in a controversial manner, for moving gas bubbles. On the other hand, an acceptable, experimentally verified theory for a body of one liquid flowing in another has not been available. Here we develop a hydrodynamic, testable theory providing an explicit relationship between the thickness of the wetting film and fluid properties for a blob of one fluid moving in another, with neither phase being gas. In its relationship to the capillary number Ca, the thickness of the film is predicted to be proportional to Ca2 at lower Ca and to level off at a constant value of ∼20% the channel radius at higher Ca. The thickness of the film is deduced to be approximately unaffected by the viscosity ratio of the fluids. We have conducted our own laboratory experiments and compiled experimental data from other studies, all of which are mutually consistent and confirm the salient features of the theory. At the same time, the classic law, originally deduced for films surrounding moving gas bubbles but often believed to hold for liquids as well, fails to explain the observations.