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

Abstract

The flow of a slowly varying rivulet of a power-law fluid with prescribed constant width (i.e. with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a rivulet of a perfectly wetting fluid (which can never have constant width) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively very different from that of a rivulet with prescribed constant contact angle but slowly varying width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N , its mass always decreases monotonically with N . Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.

Rivulet flow of generalized Newtonian fluids

“…where the coefficient k (>0) is found from (69) to be given by The structure of the flow in this case is somewhat similar to that of rivulet flow of a power-law fluid in the strongly shear-thinning limit, as described by Al Mukahal et al [33,34]. Figure 15 shows contour plots of the velocity u for two pairs of values of α and τ av .…”

“…Despite the fact that in practice many of the fluids involved demonstrate significant non-Newtonian behavior, most of the previous work has focused on the simplest case of rivulet flow of a Newtonian fluid. Notable among the limited number of previous studies of non-Newtonian rivulet flow are those by Rosenblat [3], who extended the pioneering work of Towell and Rothfeld [1] to study uniform rivulet flow of a viscoelastic fluid, Wilson et al [13], who extended the pioneering work of Smith [2] and Duffy and Moffatt [9] to study nonuniform rivulet flow of a power-law fluid, Balmforth et al [12] and Wilson et al [14], who studied rivulet flow of a viscoplastic material, Yatim et al [25], who studied unsteady nonuniform rivulet flow of a power-law fluid, and Al Mukahal et al [33,34], who studied locally uniform rivulet flow of a power-law fluid. However, despite a growing body of work on free surface flow of fluids with various non-Newtonian rheologies (see, for example, the recent work by Jossic et al [38] on thin-film flow of an Ellis fluid, Tshehla [39] on thin-film flow of a Carreau fluid, Kheyfets and Kieweg [40] on thin-film flow of an Ellis fluid, Pritchard et al [41] on thin-film flow of a generalized Newtonian fluid, Fomin et al [42] on non-Newtonian rimming flow, and Peralta et al [43] on thin-film flow of a Carreau-Yasuda fluid) there is very little work on rivulet flow of fluids with other than the theoretically convenient but highly idealised power-law rheology.…”

Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid

“…This result is a well known consequence of the flow geometry being "long and thin" [33]. Lubrication flows of non-Newtonian fluids have been studied extensively, for example in the context of rivulets [34,35], but not as much in the microfluidics and FSI contexts. Next, the fluid's only nontrivial momentum balance is in the z-direction:…”

Non-Newtonian Pressure-Governed Rivulet Flows on Inclined Surface

“…Also, studies should be mentioned, where work [38] is perhaps the key paper on the rivulet flow of a viscoplastic fluid. Meanwhile, works [39][40][41][42][43][44][45][46][47][48] present a part of the contributions which stem from the large body of work on Newtonian and non-Newtonian rivulet flows carried out by Duffy and his research team in Scotland over the last couple of decades (not a complete list). In addition, handbook [25] concerns the problem under consideration, for which the theoretical basis has been described in detail, for solving a one-dimensional heat equation such as equations in (6) mentioned above.…”