Rivulet flow round a horizontal cylinder subject to a uniform surface shear stress

Paterson, Colin; Wilson, S. K.; Duffy, Brian

doi:10.1093/qjmam/hbu018

Cited by 12 publications

(7 citation statements)

References 35 publications

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“…Note that the function f (ma) was first obtained by Duffy and Moffatt [27] in their pioneering study of the case λ = 0, while the function g(ma) was first obtained by Sullivan et al [28] in their study of rivulet flow in the presence of a uniform longitudinal surface shear stress τ . There is, in fact, a close relationship between the present problem and the corresponding problem of a rivulet in the presence of a uniform longitudinal surface shear stress studied by Wilson and Duffy [29], Sullivan et al [28], Wilson, Sullivan and Duffy [30], and Paterson et al [31]. Specifically, while the velocity for the shear-stress problem given by…”

Section: Problem Formulationsupporting

confidence: 64%

Rivulet flow down a slippery substrate

2020

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A detailed analysis of small-scale locally unidirectional gravity-driven rivulet flow with prescribed volume flux down an inclined slippery substrate for a rivulet with either constant width (i.e., pinned contact lines) or constant contact angle is undertaken. In particular, we determine the effect that varying the Navier slip length λ (i.e., the strength of the slip at the solid-fluid interface) has on the rivulet. The present analysis shows that the shape and size of the rivulet and the velocity within it depend strongly on the value of λ. Increasing the value of λ reduces the viscous resistance at the substrate and hence leads to a larger velocity within the rivulet, and so the prescribed flux is achieved with a smaller rivulet. In particular, in the limit of strong slip, λ → ∞, for a rivulet of a perfectly wetting fluid and a rivulet with constant width the velocity becomes large and plug-like like O(λ 1/2) 1 and the rivulet becomes shallow like O(λ −1/2) 1, while for a rivulet with positive constant contact angle the velocity becomes large and plug-like like O(λ 2/3) 1 and the rivulet becomes narrow like O(λ −1/3) 1 and shallow like O(λ −1/3) 1.

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Section: Problem Formulationsupporting

confidence: 64%

Rivulet flow down a slippery substrate

2020

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“…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…”

Section: B Solutionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2015

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Locally unidirectional steady gravity-driven flow of a thin rivulet of a power-law fluid with prescribed volume flux down a locally planar substrate is considered. First the solution for unidirectional flow of a uniform rivulet down a planar substrate is obtained, and then it is used to obtain the solution for a slowly varying rivulet with prescribed constant (nonzero) contact angle down a slowly varying substrate, specifically flow in the azimuthal direction around the outside of a large horizontal circular cylinder. The solution is shown to depend strongly on the value of the power-law index of the fluid. For example, a rivulet of strongly shear-thinning fluid "self-channels" its flow down a narrow central channel between two "levees" of slowly moving fluid that form at its sides, and in the central channel there is a "plug-like" flow except in a boundary layer near the substrate. On the other hand, in a rivulet of a strongly shear-thickening fluid the velocity profile is linear except in a boundary layer near the free surface. Another notable qualitative departure from Newtonian behaviour is that, whereas the mass of a rivulet of a Newtonian or a shear-thinning fluid is theoretically infinite, the mass of a rivulet of a shear-thickening fluid is finite

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“…In the present geometry, the variation of the hydrostatic pressure is stabilizing for θ ∈ [0, π/2[ and destabilizing for θ ∈ ]π/2, π]. On the lower external part of the cylinder, not considered in this study, the liquid will accumulate in rivulets due to the Rayleigh-Taylor instability, eventually forming dripping droplets (Duffy & Moffatt 1995;Indeikina et al 1997;Takagi & Huppert 2010;Lin et al 2012;Leslie et al 2013;Paterson et al 2014). The problem being symmetric with respect to θ = 0, only half of the cylinder is considered.…”

Section: Problem Formulation and Governing Equationsmentioning

confidence: 85%

Fingering instability on curved substrates: optimal initial film and substrate perturbations

et al. 2019

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We investigate the stability of a thin Newtonian fluid spreading on a horizontal cylinder under the action of gravity. The capillary ridge forming at the advancing front is known to be unstable with respect to spanwise perturbations, resulting in the formation of fingers. In contrast to the classic case of a flow over an inclined plane, the gravity components along a cylindrical substrate vary in space and the draining flow is time-dependent, making a modal stability analysis inappropriate. A linear optimal transient growth analysis is instead performed to find the optimal spanwise wavenumber. We not only consider the optimal perturbations of the initial film thickness, as commonly done in the literature, but also the optimal topographical perturbations of the substrate, which are of significant practical relevance. We found that, in both cases, the optimal gains are obtained when the perturbation structures are the least affected by the time horizon. The optimal spanwise wavenumber is found to be dependent on the front location, due to the dependence of the characteristic length of the capillary ridge on its polar location.

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Rivulet flow round a horizontal cylinder subject to a uniform surface shear stress

Cited by 12 publications

References 35 publications

Rivulet flow down a slippery substrate

Rivulet flow down a slippery substrate

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

Fingering instability on curved substrates: optimal initial film and substrate perturbations

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