Pinning, de-pinning and re-pinning of a slowly varying rivulet

Abstract

The solutions for the unidirectional flow of a thin rivulet with prescribed volume flux down an inclined planar substrate are used to describe the locally unidirectional flow of a rivulet with constant width (i.e. pinned contact lines) but slowly varying contact angle as well as the possible pinning and subsequent de-pinning of a rivulet with constant contact angle and the possible depinning and subsequent re-pinning of a rivulet with constant width as they flow in the azimuthal direction from the top to the bottom of a large horizontal cylinder. Despite being the same locally, the global behaviour of a rivulet with constant width can be very different from that of a rivulet with constant contact angle. In particular, while a rivulet with constant non-zero contact angle can always run from the top to the bottom of the cylinder, the behaviour of a rivulet with constant width depends on the value of the width. Specifically, while a narrow rivulet can run all the way from the top to the bottom of the cylinder, a wide rivulet can run from the top of the cylinder only to a critical azimuthal angle. The scenario in which the hitherto pinned contact lines of the rivulet de-pin at the critical azimuthal angle and the rivulet runs from the critical azimuthal angle to the bottom of the cylinder with zero contact angle but slowly varying width is discussed. The pinning and de-pinning of a rivulet with constant contact angle, and the corresponding situation involving the de-pinning and re-pinning of a rivulet with constant width at a non-zero contact angle which generalises the de-pinning at zero contact angle discussed earlier, are described. In the latter situation, the mass of fluid on the cylinder is found to be a monotonically increasing function of the constant width

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

“…Note that since a takes the constant valueā for all α whenā ≤ π and for α ≤ α depin whenā > π, and is given by a = π/m for α ≥ α depin whenā > π, the plot of a in Figure 12(a) is identical to the corresponding plot for Newtonian fluid given by Figure 6(b) of Paterson et al [25], but is included here for completeness. Figure 13 shows plots of h given by (9) and (23) for a wide rivulet with prescribed constant semi-widthā = 5 (> π) as a function of y at various stations around the cylinder, including α = α depin ≃ 0.62918π, for N = 1/2.…”

“…[Of course, this behaviour is a special case of the more general scenario in which a rivulet with prescribed constant width de-pins and possibly re-pins at a prescribed non-zero value of the contact angle. This situation was also analysed by Paterson et al [25,Sec. 6] for a Newtonian fluid, but for brevity is not pursued here.]…”

“…In particular, both of these conditions are satisfied if the cylinder is sufficiently large. [25,Sec. 4] showed that a rivulet can have constant width all the way around the cylinder only if the rivulet is sufficiently narrow (specifically only ifā ≤ π), and that for a wider rivulet (specifically forā > π) there is a critical value of α on the lower half of the cylinder, denoted by α depin (π/2 < α depin < π) and given by Figure 1, except for a wide rivulet with prescribed constant semi-width a =ā (> π) and slowly varying contact angle β in 0 ≤ α ≤ α depin , but with zero contact angle β = 0 and slowly varying semi-width a = π/m (π ≤ a ≤ā) in α depin ≤ α ≤ π.…”

“…For their problem, Paterson et al [25] assumed that the contact lines de-pin at α = α depin and that thereafter the rivulet drains from α = α depin to the bottom of the cylinder with zero contact angle β = 0 and slowly varying semi-width a. [Of course, this behaviour is a special case of the more general scenario in which a rivulet with prescribed constant width de-pins and possibly re-pins at a prescribed non-zero value of the contact angle.…”

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.…”