On the extension of Bretherton theory for thin liquid films formed around elongated bubbles

Abstract

The formation of thin liquid films around an elongated bubble moving in a capillary is pertinent to many applications. However, development of a theoretical model for the film thickness has been a challenge for several decades. The prominent theory characterizing the film thickness was developed by Bretherton. This theory relates the liquid film thickness in axisymmetric capillaries to the flow capillary number (Ca) for values of up to 0.003. Modified forms of the Bretherton theory have been presented for Ca values as high as 2. However, the validity of these models has not been rigorously examined. In addition, the validity of the Bretherton model itself in non-axisymmetric cross section capillaries remains uncharted. The objective of this paper is to determine whether the Bretherton relation can be extended to a broader range of Ca values and non-axisymmetric cross section channels, wherein the film thickness is not uniform along the channel circumference. A series of experiments are conducted on a set of fluids and different channel sizes and cross-sectional geometries to produce a wide range of viscous, surface tension, and inertial forces. The results show that when inertial forces are significant, and modified Bretherton models fail to predict the film thickness at Ca well below 2. The experimental results also show that depending on fluid properties, another key requirement of Bretherton solution of Landau–Levich equation may not be met. In rectangular cross section channels, the difference in surface tension forces along the longer and shorter channel axes, results in great deviation from the existing lubrication-based theories. This deviation greatly expands with the increase in the channel aspect ratio.