Developed liquid film passing a trailing edge: small-scale analysis and the `teapot effect' at large Reynolds numbers

TL;DR

This paper analytically investigates the small-scale flow behavior of a viscous liquid film passing a trailing edge, revealing capillary ripples and clarifying the flow detachment process related to the teapot effect at high Reynolds numbers.

Contribution

It provides a detailed analytical analysis of the flow near the trailing edge, predicting capillary ripples and elucidating the flow detachment mechanism, including the influence of wetting and edge geometry.

Findings

Prediction of Rayleigh-type capillary ripples upstream of the edge

Increase in ripple wavelength and amplitude as Weber number decreases

Clarification of the flow detachment process and its relation to the teapot effect

Abstract

Recently, the authors considered a thin steady developed viscous liquid wall jet passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity acting vertically in the asymptotic slender-layer limit (J. Fluid Mech. 850, pp. 924--953, 2018). We revisit the capillarity-driven short-scale viscous--inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of O(1). The aspect in the main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller then the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, unprecedented in the rational theories of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the so-called teapot effect.