A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow

Abstract

We use the lubrication approximation to analyse three closely related problems involving a thin rivulet or ridge (i.e. a two-dimensional droplet) of fluid subject to a prescribed uniform transverse shear stress at its free surface due to an external airflow, namely a rivulet draining under gravity down a vertical substrate, a rivulet driven by a longitudinal shear stress at its free surface, and a ridge on a horizontal substrate, and find qualitatively similar behaviour for all three problems. We show that, in agreement with previous numerical studies, the free surface profile of an equilibrium rivulet/ridge with pinned contact lines is skewed as the shear stress is increased from zero, and that there is a maximum value of the shear stress beyond which no solution with prescribed semi-width is possible. In practice, one or both of the contact lines will de-pin before this maximum value of the shear stress is reached, and so we consider situations in which the rivulet/ridge de-pins at one or both contact lines. In the case of de-pinning only at the advancing contact line, the rivulet/ridge is flattened and widened as the shear stress is increased from its critical value, and there is a second maximum value of the shear stress beyond which no solution with a prescribed advancing contact angle is possible. In contrast, in the case of de-pinning only at the receding contact line, the rivulet/ridge is thickened and narrowed as the shear stress is increased from its critical value, and there is a solution with a prescribed receding contact angle for all values of the shear stress. In general, in the case of de-pinning at both contact lines there is a critical “yield” value of the shear stress beyond which no equilibrium solution is possible and the rivulet/ridge will evolve unsteadily. In an Appendix we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations

A rivulet of a power-law fluid with constant contact angle draining down a slowly varying substrate

“…where Γ and B denote the Gamma and Beta functions, respectively. To complete the solution, we must solve (17) with Q =Q to obtain the semi-width a; this depends on the form of f N (ma) in (18), and therefore also on the form of λ N in (19). Figure 2 shows a plot of λ N as a function of N, illustrating that λ N increases monotonically with N, satisfying…”

“…3 as dotted and dashed curves, respectively. In summary, the solutions for the pressure p and velocity u in the steady unidirectional flow of a thin uniform rivulet are given by (9) and (13), respectively, in which the free surface profile h is given by (11), with the semi-width a to be determined from flux condition (17) with Q =Q when N, α, andQ are prescribed. In the special case of a Newtonian fluid (N = 1), Eqs.…”

“…8 Subsequently, this work has been extended to include, for example, non-planar substrates, [9][10][11][12] thermocapillary effects, 13 thermoviscosity effects, 14 and the presence of an external airflow. [15][16][17][18] Recently, the pinning, de-pinning, and re-pinning of a rivulet have also been studied. 19 In addition, similarity solutions have been obtained and analysed for steady 20,21 and unsteady 22 flows of non-uniform rivulets.…”

Stability of a rivulet flowing in a microchannel

“…The lubrication and thinfilm approximations have been used to calculate the basic flows driven not only by the gravitational force (Perazzo and Gratton, 2004;Paterson et al, 2013) but also by other factors, such as a prescribed uniform transverse shear stress at its free surface (Sullivan et al, 2012). Under those approximations, rivulets flowing over a vertical plane (Mechkov et al, 2008) and under a sloping plate (Benilov, 2009) have proved to be stable as long as their triple contact lines are fixed.…”

Droplet Impact and Solidification on Hydrophilic and Superhydrophobic Substrates in Icing Conditions

“…This flow is also known as rivulets [8]. Runback flow over the wing of an aircraft is one of the vital issues in the icing problematics in the aerospace industry [9][10].…”