On the transition to dripping of an inverted liquid film

TL;DR

This paper investigates the transition to dripping in gravity-driven liquid films on inclined plates at zero Reynolds number, comparing lubrication and full Stokes models to predict the critical conditions for dripping.

Contribution

It provides a detailed comparison of lubrication and full Stokes models, revealing their accuracy and limitations in predicting dripping onset in inclined liquid films.

Findings

Lubrication models predict the critical inclination angle accurately.

Full models show bifurcation curves with turning points or singularities.

Asymptotic analysis explains the approach to equilibrium profiles.

Abstract

The transition to dripping in the gravity-driven flow of a liquid film under an inclined plate is investigated at zero Reynolds number. Computations are carried out on a periodic domain assuming either a fixed fluid volume or a fixed flow rate for a hierarchy of models: two lubrication models with either linearised curvature or full curvature (the LCM and FCM, respectively), and the full equations of Stokes flow. Of particular interest is the breakdown of travelling-wave solutions as the plate inclination angle is increased. For any fixed volume the LCM reaches the horizontal state where it attains a cosine-shaped profile. For sufficiently small volume, the FCM and Stokes solutions attain a weak Young-Laplace equilibrium profile, the approach to which is described by an asymptotic analysis generalising that of Kalliadasis & Chang (1994) for the LCM. For large volumes, the bifurcation curves for the FCM and Stokes model have a turning point so that the fully inverted state is never reached. For fixed flow rate the LCM blows up at a critical angle that is well predicted by asymptotic analysis. The bifurcation curve for the FCM either has a turning point or else reaches a point at which the surface profile has an infinite slope singularity, indicating the onset of multi-valuedness. The latter is confirmed by the Stokes model which can be continued to obtain overturning surface profiles. Overall the thin-film models either provide an accurate prediction for dripping onset or else supply an upper bound on the critical inclination angle.