Influence of disjoining pressure on the dynamics of steadily moving long bubbles inside narrow cylindrical capillaries

Abstract

We study the influence of disjoining pressure for moving long bubbles inside cylindrical capillaries. Towards that end, consistent thin-film equations, for the annular region separating the bubble from the channel surface, are presented with specific emphasis on three different attributes: (a) the van der Waals interaction, formalized by the classical Lifshitz form of disjoining pressure; (b) the nonuniformity in film thickness, accommodated by the necessary corrections in the disjoining pressure; and (c) the electrostatic component of disjoining pressure, reminiscent of the electrostatic interactions in the presence of surface charges. The present thin-film analysis appositely uncovers the existence and the breakdown of the two-thirds power law for minimum film thickness behavior. This is attributed to the alteration in the characteristic length scales governing the underlying physics, as quantitatively established by our consistent scaling analysis. In the breakdown regimes, the characteristic length scales are found to be composed of the suitable combinations of the capillary number and the physics driven intrinsic length scales. The characteristics of the breakdown regime reported by us appear to match excellently with reported experimental data in the low capillary number regime. Towards unveiling the possible implications of slope and curvature dependence of disjoining pressure, our analysis reveals that the consequent correction term endorses an order two-thirds power of the capillary number contribution without alerting the governing length scales. The subsequent asymptotic analysis reveals that this correction may be neglected to the leading order approximation. Finally, we consider the electrostatic component of the disjoining pressure which may be realized in the presence of surface charges. Our analysis reveals that the significance of the electrostatic interaction is realized over a very small capillary number regime, leading towards the departure from the two-thirds power law type behavior. Reasonably good agreement is obtained with reported experimental data over this regime.