On a slender dry patch in a liquid film draining under gravity down an inclined plane

Abstract

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“…Particular forms of these equations have been studied previously for isothermal flow by Wilson et al [6] who considered a dry patch in a gravity-driven film for the cases of both strong gravitational and strong surface-tension effects. In the case of strong gravitational effects the two gravity terms in (3) (that is, the terms in S) are dominant with the gravity term dominating the flux condition (9).…”

“…Following the approach of Wilson et al [6] we seek similarity solutions of (3) and (9) for h of the form…”

“…where [6] showed how the isothermal rivulet solutions can be modified locally near the contact line to accommodate a fixed-contact-angle condition by incorporating sufficiently strong slip at the solid/fluid interface into the model. Unfortunately this procedure does not work for the isothermal dry-patch solutions and so is not attempted here.…”

“…Podgorski et al [3] independently developed a model similar to that of Wilson [5] based on a balance between surface-tension forces and the weight of the capillary ridge, and found good agreement between their theoretical predictions and experimental measurements. Recently, Wilson, Duffy and Davis [6] obtained similarity solutions describing a slender dry patch in a gravity-driven film for the cases of both strong gravitational and strong surface-tension effects. When gravitational effects are strong the dry patch has a parabolic shape and the transverse profile of the free surface has a monotonically increasing shape, and when surface-tension effects are strong the dry patch has a quartic shape and the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it.…”

“…In this paper we follow the approach of Wilson et al [6] and use the lubrication approximation to investigate slender dry patches in an infinitely wide thin film of viscous fluid flowing steadily on an inclined plane that is either heated or cooled relative to the surrounding atmosphere. Four non-isothermal situations in which thermocapillary effects play a significant role are considered.…”