Similarity Solutions for Slender Rivulets with Thermocapillarity

• Published in: Quarterly journal of mechanics and applied mathematics Vol. 56; no. 3; pp. 411 - 439
• Main Authors: Holland, D., Wilson, S. K., Duffy, B. R.
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.08.2003; Oxford Publishing Limited (England)
• Subjects: Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; Physics; Thin films; Inclined plan; Runoff; Digital simulation; Non isothermal condition; Free surface flow; Falling film; Liquid layer; Viscous fluids; Thermocapillarity
• ISSN: 0033-5614; 1464-3855
• DOI: 10.1093/qjmam/56.3.411

We use the lubrication approximation to investigate the steady flow of slender non‐uniform rivulets of a viscous fluid 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. We derive the general equations for a slender rivulet subject to gravity, surface tension, thermocapillarity and a constant surface shear stress. Similarity solutions describing a thermocapillary‐driven rivulet widening or narrowing due to either gravitational or surface‐tension effects on a non‐uniformly heated or cooled substrate are obtained, and we present examples of these solutions when the substrate temperature gradient depends on the longitudinal coordinate according to a general power law. When gravitational effects are strong there is a unique solution representing both a narrowing pendent rivulet and a widening sessile rivulet whose transverse profile always has a single global maximum. When surface‐tension effects are strong there is a one‐parameter family of solutions representing both a narrowing and a widening rivulet whose transverse profile has either a single global maximum or two equal global maxima and a local minimum. Unique similarity solutions whose transverse profiles always have a single global maximum are also obtained for both a gravity‐driven and a constant‐surface‐shear‐stress‐driven rivulet widening or narrowing due to thermocapillarity on a uniformly heated or cooled substrate. The solutions in both cases represent both a narrowing rivulet on a heated substrate and a widening rivulet on a cooled substrate (albeit with infinite width in the gravity‐driven case).