Moving contact lines and rivulet instabilities. Part 2. Long waves on flat rivulets

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Abstract
A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Long-wave instabilities are investigated for flat rivulets on a vertical wall. Three cases of contact-line conditions are investigated: (i) fixed contact lines, (ii) moving contact lines having fixed contact angles and (iii) moving contact lines whose contact angles vary smoothly with contact-line speeds. For case (i), the straight, unidirectional rivulet is stable below a critical Reynolds number. For case (ii), the rivulet is unconditionally unstable. For case (iii), the variation, G1 ≡ [dα/dUUCL = 0, in contact angle α with contact-line speed UCL stabilizes the rivulet and if G1 is large enough can completely stabilize the flow. The analysis lends support to the idea that contact-angle steepening with contact-line motion is a purely dissipative process.