Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

Abstract

A novel family of three-dimensional travelling-wave similarity solutions describing a steadily translating slender dry patch in an infinitely wide thin fluid film on an inclined planar substrate is obtained, the flow being driven by gravity and/or a prescribed constant shear stress on the free surface of the film. For both driving mechanisms, the dry patch has a parabolic shape (which may be concave up or concave down the substrate), and the film thickness increases monotonically away from the contact lines to its uniform far-field value. The two most practically important cases of purely gravity-driven flow and of purely surface-shear-stress-driven flow are analysed separately.

Viscous free-surface flows past cylinders

“…Dry regions may also arise owing to instabilities associated with surface tension effects, particularly when the film is very thin [19][20][21][22][23][24][25]. Yatim et al [26] showed that in the absence of any capillary effects or obstructions to the flow, a dry region, which may develop for example owing to isolated heating, has parabolic shape and migrates steadily downslope.…”

Surface-tension- and injection-driven spreading of a thin viscous film

“…Further analysis of gravity-driven rivulet flow of Newtonian and power-law fluids down an inclined plane is given in Yatim, Wilson & Duffy (2010), whereas shear-stress-driven flow is considered in Yatim, Duffy & Wilson (2012). Corresponding travelling-wave similarity solutions are found in Yatim, Duffy & Wilson (2013). Momoniat, Ravindran & Roy (2010) consider a similar problem but for radially symmetric spreading with an annular injection slot and investigate the flow behaviour numerically.…”

Modelling the wetting of a solid occlusion by a liquid film

“…Indeed the dry region developing downstream of the hydrophobic patch has an important detrimental effect to the overall heat transfer. It has been studied experimentally by Podgorski et al (1999), Rio and Limat (2006), and Sebilleau et al (2009) and theoretically, a fine example being the papers of Wilson and co-workers describing the shape of a slender dry patch in a liquid film draining under gravity down an inclined plane, (Wilson et al, 2001;Yatim et al, 2013). Much work related to the motion of a contact line on a substrate with variable wettability was performed numerically in the context of the lubrication approximation which reduces the three-dimensional problem fluid dynamics problem to a two-dimensional one by averaging quantities across the thickness of the fluid layer.…”