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Paterson, Colin and Wilson, Stephen and Duffy, Brian (2013) Pinning, de-pinning and re-pinning ofa slowly varing rivulet. European Journal of Mechanics - B/Fluids, 41 (septem). pp. 94-108. ISSN0997-7546

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European Journal of Mechanics B/Fluids 41 (2013) 94–108

Contents lists available at SciVerse ScienceDirect

European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Pinning, de-pinning and re-pinning of a slowly varying rivuletC. Paterson, S.K. Wilson ∗,1, B.R. DuffyDepartment of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow, G1 1XH, United Kingdom

a r t i c l e i n f o

Article history:Received 9 May 2012Received in revised form8 February 2013Accepted 18 February 2013Available online 5 March 2013

Keywords:Rivulet flowThin-film flowContact linesPinningDe-pinningRe-pinning

a b s t r a c t

The solutions for the unidirectional flow of a thin rivulet with prescribed volume flux down an inclinedplanar 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 subsequentde-pinning of a rivuletwith constant contact angle and thepossible de-pinning and subsequent re-pinningof a rivulet with constant width as they flow in the azimuthal direction from the top to the bottom of alarge horizontal cylinder. Despite being the same locally, the global behaviour of a rivulet with constantwidth can be very different from that of a rivulet with constant contact angle. In particular, while a rivuletwith constant non-zero contact angle can always run from the top to the bottom of the cylinder, thebehaviour of a rivulet with constant width depends on the value of the width. Specifically, while a narrowrivulet can run all the way from the top to the bottom of the cylinder, a wide rivulet can run from thetop of the cylinder only to a critical azimuthal angle. The scenario in which the hitherto pinned contactlines of the rivulet de-pin at the critical azimuthal angle and the rivulet runs from the critical azimuthalangle 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 situationinvolving the de-pinning and re-pinning of a rivuletwith constantwidth at a non-zero contact anglewhichgeneralises 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 constantwidth.

© 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

The gravity-drivendraining of a rivulet of fluid downan inclinedsubstrate is a fundamental fluid mechanics problem of enduringinterest, not least because of the wide range of industrial devicesand processes to which it is relevant, including heat exchangers(see, for example, Vlasogiannis et al. [1]), trickle-bed reactors (see,for example, Maiti, Khanna and Nigam [2]), various coating pro-cesses (see, for example, Kistler and Schweizer [3]), and even thecleaning of the long and narrow tubes found in endoscopes (see,for example, Labib et al. [4]). In particular, the pioneering studiesby Towell and Rothfeld [5], Allen and Biggin [6], Bentwich et al. [7],and Davis and co-workers [8–10] have led to a substantial bodyof subsequent work on unidirectional (i.e. rectilinear) rivulet flow.For example, Schmuki and Laso [11] considered the stability ofrivulet flow, Kuibin [12], Alekseenko, Geshev and Kuibin [13] and

∗ Corresponding author. Tel.: +44 141 548 3820; fax: +44 141 548 3345.E-mail addresses: [email protected] (C. Paterson),

[email protected] (S.K. Wilson), [email protected] (B.R. Duffy).1 Presently also a Visiting Fellow in the Oxford Centre for Collaborative Applied

Mathematics (OCCAM), University of Oxford, Mathematical Institute, 24–29 St.Giles’, Oxford OX1 3LB, United Kingdom.

Alekseenko, Bobylev and Markovich [14] considered rivulet flowon the underside of an inclined cylinder, Perazzo and Gratton [15]and Tanasijczuk, Perazzo and Gratton [16] studied sessile and pen-dent rivulet flow, Myers, Liang and Wetton [17] and Wilson andDuffy [18] considered rivulet flow subject to a constant longitudi-nal shear stress, and Benilov [19] considered rivulet flow down aninclined substrate and found that sessile and sufficiently narrowpendent rivulets are always stable but that sufficiently wide pen-dent rivulets are stable only when the incline is sufficiently steep.Duffy andMoffatt [20] used the solution for the unidirectional flowof a thin rivulet with non-zero contact angle and prescribed vol-ume flux to describe the locally unidirectional flowof a rivuletwithconstant non-zero contact angle but slowly varying width down aslowly varying substrate. In particular, they studied rivulet flow inthe azimuthal direction from the top to the bottom of a large hori-zontal cylinder, and showed that the rivulet becomes wide and flatnear the top of the cylinder, but narrow and deep near the bottomof the cylinder. Subsequently Duffy andWilson [21] performed thecorresponding analysis for a rivulet with zero contact angle and, inparticular, showed that such rivulets can occur only on the lowerhalf of the cylinder. Various other physical effects, including lo-cally non-planar substrates, thermocapillary effects, viscoplasticityeffects, thermoviscosity effects, and a constant longitudinal shearstress, have also been considered (see [21–27]).

0997-7546/$ – see front matter© 2013 Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.euromechflu.2013.02.006


C. Paterson et al. / European Journal of Mechanics B/Fluids 41 (2013) 94–108 95

a b

Fig. 1. Sketches of the scaled semi-width a/π as a function of the scaled azimuthal angle α/π for a rivulet with (a) constant non-zero contact angle β = β > 0 and (b)constant zero contact angle β = β = 0.

a b

Fig. 2. Plots of (a) αmin/π and (b) amin/π as functions of the constant contact angle β when Q = 1, together with their asymptotic behaviour in the limits β → 0+ givenby (6) and β → ∞ given by (7), shown with dotted lines.

In the present work we take a rather different approachfrom the earlier studies and show how the solutions for theunidirectional flow of a thin rivulet with prescribed volume fluxdown an inclined planar substrate can be used to describe thelocally unidirectional flow of a rivulet with constant width butslowly varying contact angle (i.e. pinned contact lines) as well asthe possible pinning and subsequent de-pinning of a rivulet withconstant contact angle and thepossible de-pinning and subsequentre-pinning of a rivulet with constant width as they flow in theazimuthal direction from the top to the bottomof a large horizontalcylinder. In particular, we find that, despite being the same locally,the global behaviour of a rivulet with constant width can bevery different from that of a rivulet with constant contact angledescribed by Duffy and Moffatt [20] and Duffy and Wilson [21].

One specific situation in which flow of the type considered inthe present work can occur is in the falling-film horizontal-tubeevaporators used in a variety of industrial processes, includingrefrigeration, desalination and petroleum refining. The reviewarticle on falling-film evaporation by Ribatski and Jacobi [28]describes how partial film dry-out may occur as a result of a non-uniform distribution of the fluid on the tubeswithin an evaporator.This non-uniformity can be caused by the gas flow within the

evaporator or by uneven draining of the fluid from one tube ontothe tube below it in a bundle of horizontal tubes. Typically thisdraining occurs in one of three main flow regimes, namely acontinuous sheet of fluid, an array of separate columns of fluid, orindividual drops that drip intermittently. Mitrovic [29] describesvarious flow regimes and compares the various experimentallydetermined correlations for the boundaries of the regions in whichthe different flow regimes occur in Reynolds number–Kapitzanumber parameter space. In particular, as Mitrovic [29] shows inhis Fig. 2(h), in the columnar flow regime the fluid in each columncan drain around the tubes in an array of separate rivulets or ringsof fluid. The flow of both a two-dimensional sheet of fluid and asingle three-dimensional column of fluid, falling onto the top of,and draining round to the bottom of, a horizontal cylinder wasstudied numerically by Hunt [30,31].

As well as evidently being of direct relevance to falling-filmevaporators, the results obtained in the present work may alsobe relevant to a variety of other practical contexts, such asthe rings of fluid on the outer surface of a uniformly rotatinghorizontal cylinder observed byMoffatt [32] and recently analysedby Leslie,Wilson andDuffy [33], and the banded films of condensedammonia–water mixtures on the outer surface of a stationaryhorizontal cylinder observed by Deans and Kucuka [34].


96 C. Paterson et al. / European Journal of Mechanics B/Fluids 41 (2013) 94–108

a b

Fig. 3. Sketches of the contact angle β as a function of the scaled azimuthal angle α/π for (a) a ‘‘narrow’’ rivulet with constant semi-width a = a < π and (b) a ‘‘wide’’rivulet with constant semi-width a = a > π . For brevity, the marginal case a = a = π is not shown.

a b

Fig. 4. Plots of (a) αmin/π and (b) βmin as functions of the scaled constant semi-width a/π when Q = 1, together with their asymptotic behaviour in the limits a → 0+

given by (12) and a → ∞ given by (13), shown with dotted lines.

2. Unidirectional flow of a thin rivulet

Consider the steady unidirectional flow of a thin symmetricrivulet with semi-width a and volume flux Q (> 0) down a planarsubstrate inclined at an angle α (0 ≤ α ≤ π ) to the horizontal. Weassume that the fluid is Newtonian with constant viscosity µ, den-sity ρ and coefficient of surface tension γ , and choose Cartesiancoordinates Oxyz with the x axis down the line of greatest slope,the y axis horizontal, and the z axis normal to the substrate z = 0.The velocity u = u(y, z)i and the pressure (relative to its ambi-ent value) p = p(y, z) satisfy the familiar mass-conservation andNavier–Stokes equations subject to the usual normal and tangen-tial stress balances and the kinematic condition at the free surfacez = h(y), the no-slip condition at the substrate z = 0, and the con-dition of zero thickness at the contact lines (i.e. h(±a) = 0). Thecontact angle is denoted by β = ∓h′(±a) (≥ 0), where the dashdenotes differentiation with respect to argument, and the maxi-mum thickness of the rivulet, which always occurs at y = 0, isdenoted by hm = h(0). We non-dimensionalise y and awith ℓ, z, hand hm with δℓ, uwith U = δ2ρgℓ2/µ, Q with δℓ2U = δ3ρgℓ4/µ,and p with δρgℓ, where g is the magnitude of gravitationalacceleration, ℓ = (γ /ρg)1/2 is the capillary length, and δ is the

transverse aspect ratio. There is some freedom regarding the defi-nition of δ. When β > 0 we could define δ using the value of thecontact angle by choosing δ = β , corresponding to taking β = 1without loss of generality. Alternatively, we could define δ usingthe prescribed value of the flux, denoted by Q (> 0), by choosingδ = (µQ/ρgℓ4)1/3, corresponding to taking Q = 1 without loss ofgenerality. However, for the moment we leave δ unspecified andretain both β and Q in order to keep the subsequent presentationas general as possible.

In the general case of non-zero contact angle β > 0 Duffyand Moffatt [20] showed that at leading order in the limit of smalltransverse aspect ratio δ → 0 (i.e. for a thin rivulet) the governingequations are readily solved to yield the velocity u = sinα(2h −

z)z/2, the pressure p = cosα(h − z) − h′′, the free surface shape

h(y) = β ×

coshma − coshmym sinhma

for 0 ≤ α <π

2,

a2 − y2

2afor α =

π

2,

cosmy − cosmam sinma

forπ

2< α ≤ π,

(1)



Fig. 5. Plot of the scaled critical azimuthal angle αc/π as a function of the scaledconstant semi-width a/π .

the maximum thickness of the rivulet

hm = β ×

1m

tanhma

2

for 0 ≤ α <

π

2,

a2

for α =π

2,

1m

tanma

2

for

π

2< α ≤ π,

(2)

and the volume flux

Q =β3 sinα

9m4f (ma), (3)

where m = | cosα|1/2. The function f = f (ma) appearing in (3) is

given by

f (ma) =

15ma coth3 ma − 15 coth2 ma − 9ma cothma + 4for 0 ≤ α <

π

2,

1235

(ma)4 for α =π

2,

−15ma cot3 ma + 15 cot2 ma − 9ma cotma + 4for

π

2< α ≤ π,

(4)

and satisfies f ∼ 12(ma)4/35 → 0 as ma → 0, f ∼ 6ma − 11 →

∞ asma → ∞ for 0 ≤ α < π/2, and f ∼ 15π(π − ma)−3→ ∞

asma → π for π/2 < α ≤ π .In the special case of zero contact angle β = 0 we recover

the solution for a perfectly wetting fluid described by Duffy andWilson [21], namely that there is no solution for 0 ≤ α ≤ π/2, but

a =π

m, h =

hm

2(1 + cosmy), Q =

5π sinαh3m

24m

forπ

2< α ≤ π. (5)

3. A rivulet with constant contact angle

3.1. The general case of non-zero contact angle β = β > 0

Duffy and Moffatt [20] used the solution (1)–(4) to describethe locally unidirectional flow with prescribed flux Q = Q down

a slowly varying substrate, specifically the flow in the azimuthaldirection from the top α = 0 to the bottom α = π of a largehorizontal cylinder, of a rivulet with constant non-zero contactangle β = β > 0 but slowly varying semi-width a. Note thathere and henceforth ‘‘slowly varying’’ means that the longitudinalaspect ratio ϵ = ℓ/R, where R is the radius of the cylinder, satisfiesϵ ≪ δ so that ϵ/δ → 0 in the limit ϵ → 0. Imposing the conditionsof prescribed flux, Q = Q with Q given by (3), and of constantnon-zero contact angle, β = β > 0, yields a non-linear algebraicequation for the semi-width a which can, in general, be solvedonly numerically or asymptotically. Fig. 1(a) shows a sketch of thescaled semi-width a/π as a function of the scaled azimuthal angleα/π when β = β > 0. For all values of Q there is a slowly varyingrivulet that runs all the way from α = 0 [where a = O(α−1) → ∞

and hm → 1+ as α → 0+] to α = π [where a → π− andhm = O(π − α)−1/3

→ ∞ as α → π−]. The rivulet doesnot have top-to-bottom symmetry; its semi-width a has a singleminimum, denoted by a = amin (< π) and occurring at α = αmin,on the lower half of the cylinder (i.e. for π/2 < α ≤ π ), andits maximum thickness hm either increases monotonically or hasa single maximum and a single minimum on the upper half of thecylinder (i.e. for 0 ≤ α < π/2). Furthermore, in the limit of smallflux, Q → 0+, the rivulet satisfies a = O(Q 1/4) and hm = O(Q 1/4)while in the limit of large flux, Q → ∞, it satisfies a = O(Q ) andhm = O(1) on the upper half of the cylinder, a = O(Q 1/4) andhm = O(Q 1/4) at α = π/2, and a = O(1) and hm = O(Q 1/3)on the lower half of the cylinder. Since the location and value ofthe minimum semi-width are important in what follows, Fig. 2shows plots of αmin/π and amin/π as functions of the constantcontact angle β , and shows that both aremonotonically decreasingfunctions of β satisfying

αmin ∼ π −

40β3

81π2Q

15

→ π− and

amin ∼ π −5π4

40β3

81π2Q

25

→ π− (6)

as β → 0+, and

αmin ∼π

2+

29

105Q4β3

12

→π

2

+

and

amin ∼

105Q4β3

14

→ 0+ (7)

as β → ∞.

3.2. The special case of zero contact angle β = β = 0

Duffy and Wilson [21] used the solution (5) to describe thecorresponding flow of a rivulet with zero contact angle β = β = 0(i.e. a perfectly wetting fluid). Specifically, imposing the conditionof prescribed flux, Q = Q with Q given by (5), yields an explicitsolution for the maximum thickness hm = hm0, where

hm0 =

24Qm5π sinα

13

=

24Q | cosα|

1/2

5π sinα

13

. (8)

Fig. 1(b) shows a sketch of the scaled semi-width a/π as a functionof the scaled azimuthal angle α/π when β = β = 0. For allvalues of Q there is a slowly varying rivulet on the lower half ofthe cylinder with monotonically decreasing semi-width a = π/mand monotonically increasing maximum thickness hm = hm0 thatruns from α = π/2+ [where a = O(α − π/2)−1/2

→ ∞ andhm = O(α − π/2)1/6 → 0+ as α → π/2+] to α = π [where



a b

c

Fig. 6. Plots of (a) the contact angle β , (b) the scaled semi-width a/π , and (c) the maximum thickness hm as functions of the scaled angle α/π for a/π =

0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 5, 10 when Q = 1 for a rivulet whose contact lines de-pin at zero contact angle β = β = 0. The corresponding solutions for a rivuletwith zero contact angle β = 0 given by (5) and (8) are shown with dashed lines (visible only in part (c)). De-pinning occurs at α = αc for a/π > 1, and the points at whichthis happens are denoted by dots.

a → π+ and hm = O(π − α)−1/3→ ∞ as α → π−]. Note

that, unlike in the general case of non-zero contact angle β > 0,in which there is an infinite mass of fluid on the cylinder, in thespecial case of zero contact angle β = β = 0 the mass of fluid onthe cylinder, denoted by M and non-dimensionalised with δρℓ2R,is given by

M =

π

π2

+a

−ah dy dα =

π

π2

πhm

mdα =

6π2Q

5

13

C, (9)

where the constant C is given by

C =

π

0

dα

(sinα)13

=

√π Γ

13

Γ

56

≃ 4.2065. (10)

4. A rivulet with constant width

The solutions (1)–(5) can also be used to describe the locallyunidirectional flowwith prescribed fluxQ = Q from the topα = 0

to the bottom α = π of a large horizontal cylinder of a rivuletwith constant semi-width a = a (> 0) (i.e. pinned contact lines) butslowly varying contact angle β (≥ 0). Imposing the conditions ofprescribed flux, Q = Q with Q given by (3), and of constant semi-width, a = a, yields an explicit solution for the contact angle β ,namely

β =

9Qm4

f (ma) sinα

13

=

9Q cos2 α

f (| cosα|1/2a) sinα

13

. (11)

The solution (11) reveals that, unlike in the case of constant contactangle described in Section 3 (in which the dependence of a on Qis not straightforward), in this case β is simply proportional toQ 1/3 for all values of α and a. Moreover, as in the case of constantcontact angle, the rivulet does not have top-to-bottom symmetry.Inspection of the solution (11) also reveals that, unlike in the case ofconstant contact angle (in which the behaviour is qualitatively thesame for all values of the contact angle), the behaviour of the rivuletis qualitatively different for a ‘‘narrow’’ rivulet with a = a < π ,



a b

Fig. 7. Cross-sectional profiles h(y) when Q = 1 in the cases (a) a = 2 (< π) for α = π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8 and (b) a = 5 (> π) for α = π/8, π/4,3π/8, π/2, αc ≃ 1.9766, 3π/4, 7π/8. For clarity, the two parts of this figure use the same vertical but different horizontal ranges.

ba

Fig. 8. Sketches of a slowly varying rivulet with prescribed flux Q with (when not de-pinned with zero contact angle β = 0 and slowly varying semi-width a) constantsemi-width a = a and slowly varying contact angle β that runs from the top α = 0 to the bottom α = π of a large horizontal cylinder, in the cases (a) a < π , in which therivulet is never de-pinned, and (b) a > π , in which the rivulet is de-pinned and has zero contact angle in the interval αc < α ≤ π .

in the marginal case a = a = π , and for a ‘‘wide’’ rivulet witha = a > π . We shall therefore describe the behaviour of therivulet in each of these three cases separately in the next threesubsections.

4.1. A narrow rivulet with a = a < π

Fig. 3(a) shows a sketch of the contact angle β as a function ofthe scaled azimuthal angle α/π for a narrow rivulet with constantsemi-width a = a < π . When a < π for all values of Q thereis a slowly varying rivulet that runs all the way from α = 0 toα = π , and its contact angle β has a single minimum, denotedby β = βmin and occurring at α = αmin, on the lower half of thecylinder (i.e. for π/2 < α < π ),2 and its maximum thickness hm

2 Note that αmin is independent of Q .

has a single minimum on the upper half of the cylinder (i.e. for0 < α < π/2). Since the location and value of the minimumcontact angle are important in what follows, Fig. 4 shows plots ofαmin/π and βmin as functions of the scaled constant semi-widtha/π , and shows that αmin is a monotonically increasing function ofa and βmin is a monotonically decreasing function of a satisfying

αmin ∼π

2+

2a2

9→

π

2

+

and βmin ∼

105Q4a4

13

→ ∞ (12)

as a → 0+, and

αmin ∼ π −

4(π − a)

5π

12

→ π− and

βmin ∼65

9Q 2

20π

16

(π − a)56 → 0+ (13)



a b

c

Fig. 9. Plots of (a) the semi-width a, (b) the contact angle β , and (c) the maximum thickness hm as functions of the scaled angle α/π for β = 0, 0.25, 0.5, 0.75,βc ≃ 1.0249, 1.25, 1.5, 1.75, 2 when Q = 1 for a rivulet whose contact lines pin at a = a = 2 (< π). The corresponding solutions for a rivulet with constant semi-width a = 2 are shown with dashed lines. Pinning and de-pinning occur for β > βc ≃ 1.0249, and the points at which this happens are denoted by dots.

as a → π−. The rivulet becomes deep near the top and the bottomof the cylinder according to

β ∼

9Q

f (a)α

13

→ ∞ and

hm ∼

9Q

f (a)α

13

tanha2

→ ∞ (14)

as α → 0+, and

β ∼

9Q

f (a)(π − α)

13

→ ∞ and

hm ∼

9Q

f (a)(π − α)

13

tana2

→ ∞ (15)

as α → π− (so that the thin-film approximation ultimately fails inthese limits); also β and hm take the O(1) values

β =

105Q4a4

13

and hm =

105Q32a

13

(16)

at α = π/2. In the limit of a narrow rivulet, a → 0+, the rivuletbecomes narrow and deep everywhere according to

β ∼

105Q

4a4 sinα

13

→ ∞ and

hm ∼

105Q

32a sinα

13

→ ∞. (17)

4.2. The marginal case a = a = π

In themarginal case a = a = π (not shown in Fig. 3 for brevity)the rivulet behaves qualitatively the same as in the case of a narrowrivulet with a = a < π described in Section 4.1, except that, sincein this case β = 0 at α = π , instead of satisfying (15) the rivuletbecomes deep with zero contact angle and finite semi-width πnear the bottom of the cylinder according to

β ∼

3π2Q (π − α)5

320

13

→ 0+ and



a b

c

Fig. 10. Plots of (a) the semi-width a, (b) the contact angle β , and (c) the maximum thickness hm as functions of the scaled angle α/π for β = 0, 0.25, . . . , 1.5 when Q = 1for a rivulet whose contact lines pin at a = a = 5 (> π). The corresponding solutions for a rivulet with constant semi-width a = 5 are shownwith dashed lines. Pinning andde-pinning occur for all β > 0, and the points at which this happens are denoted by dots. The vertical dashed lines show the scaled critical azimuthal angle αc/π ≃ 0.6292at which de-pinning occurs for all β > 0.

hm ∼

24Q

5π(π − α)

13

→ ∞ (18)

as α → π−.

4.3. A wide rivulet with a = a > π

Fig. 3(b) shows a sketch of the contact angle β as a function ofthe scaled azimuthal angle α/π for a wide rivulet with constantsemi-width a = a > π . Unlike when a ≤ π , when a > π for allvalues of Q there is a slowly varying rivulet that runs from α = 0only as far as a critical azimuthal angle α = αc on the lower halfof the cylinder (i.e. for π/2 < α < π ), 3 and its contact angle β isa monotonically decreasing function of α, attaining its minimum

3 Note that αc is independent of Q .

physically realisable value of zero at α = αc, where the criticalazimuthal angle αc is given by solvingma = π to obtain

αc = cos−1

−π2

a2

for a > π (19)

and is a monotonically decreasing function of a satisfying αc =

π + O(a − π)1/2 → π− as a → π+ and αc = π/2 + O(a−2) →

π/2+ as a → ∞. Fig. 5 shows the scaled critical azimuthal angleαc/π plotted as a function of the scaled constant semi-width a/π .The rivulet again becomes deep near the top of the cylinderaccording to (14) and again β and hm take the O(1) values givenby (16) at α = π/2. At α = αc the rivulet has zero contact angleβ = 0, semi-width a = a, and maximum thickness hm = hmc,where

hmc =

24Q

5a sinαc

13

=

24aQ

5√a4 − π4

13

. (20)



a b

c

Fig. 11. Cross-sectional profiles h(y) when Q = 1 in the cases (a) a = 2 (< π) and β = 0.5 (< βc ≃ 1.0249) for α = π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8,(b) a = 2 (< π) and β = 1.5 (> βc) for α = π/8, αpin ≃ 0.9028, 3π/8, π/2, 5π/8, 3π/4, 7π/8, and (c) a = 5 (> π) and β = 1 for α = π/16, αpin ≃ 0.4345, π/4, 3π/8,π/2, αc ≃ 1.9766, 3π/4, 7π/8. For clarity, in part (b) no profiles are shown in the interval αdepin ≃ 2.9923 ≤ α ≤ π , and the three parts of this figure use the same verticalbut different horizontal ranges.

In particular, as α → α−c we find that β → 0+ according to

β =

3(a4 − π4)Q

40a2

13

(αc − α) + O(αc − α)2, (21)

a ≡ a, and hm → h−mc according to

hm = hmc +(a4 + π4)hmc

6π2√a4 − π4

(α − αc) + O(α − αc)2. (22)

However, beyond α = αc the solution for β given by (11) is nolonger physically realisable because it always predicts that h < 0somewhere in the interval y = −a to y = +a, and so an alternativedescription of the behaviour beyond α = αc is required. Physicallyit is possible that the rivulet simply falls off the cylinder at α = αcor that the flow becomes unsteady beyond α = αc. However,an alternative (and possibly more likely) scenario in which steadyrivulet flow still occurs is that the hitherto pinned contact lines

of the rivulet de-pin at α = αc, and that the rivulet runs fromα = αc to the bottom of the cylinder α = π with zero contactangle according to the solution in the case β = 0 given by (5)and (8), with monotonically decreasing semi-width a = π/m(π ≤ a ≤ a) and monotonically increasing maximum thicknesshm = hm0 (≥ hmc). In particular, as α → α+

c we find that β ≡ 0,a → a− according to

a = a −a√a4 − π4

2π2(α − αc) + O(α − αc)

2, (23)

and hm → h+mc according to (22), so that the solutions in α < αc

and α > αc join continuously (but not smoothly) at α = αc. Thelatter scenario is a special case of the behaviour which will be dis-cussed in Section 6, in which we consider the de-pinning and re-pinning of a rivulet with constant width at a prescribed (and, ingeneral, non-zero) value of the contact angle β = β (≥ 0). In par-ticular, when the rivulet de-pins at zero contact angle β = β = 0



a b

c

Fig. 12. Sketches of a slowly varying rivulet with prescribed flux Q with (when not pinned with constant semi-width a = a and slowly varying contact angle β) constantnon-zero contact angle β = β > 0 and slowly varying semi-width a that runs from the top α = 0 to the bottom α = π of a large horizontal cylinder, in the cases (a) a < π

and 0 < β < βc , in which the rivulet is never pinned, (b) a < π and β > βc , in which the rivulet is pinned in the interval αpin < α < αdepin , and (c) a > π , in which therivulet is pinned in the interval αpin < α < αc and has zero contact angle in the interval αc ≤ α ≤ π .

it becomes deep with zero contact angle and finite semi-width πnear the bottom of the cylinder according to

a = π +π

4(π − α)2 + O(π − α)4 → π+ and

hm ∼

24Q

5π(π − α)

13

→ ∞ (24)

as α → π−, and in the limit of a wide rivulet on the upper half ofthe cylinder, a → ∞, (in which αc → π/2+) the rivulet becomeswide and flat on the upper half of the cylinder according to

β ∼

3Qm3

2a sinα

13

→ 0+ and hm ∼

3Q

2a sinα

13

→ 0+ (25)

and is given by the solution in the case β = 0 given by (5) and (8)on the lower half of the cylinder.

4.4. Rivulet profiles

The behaviour for both a ≤ π and a > π is illustrated in Fig. 6,which shows plots of the contact angle β , the scaled semi-widtha/π , and the maximum thickness hm as functions of the scaledangle α/π for a range of values of a/π when Q = 1. In particular,Fig. 6 shows that de-pinning occurs at α = αc for a/π > 1. Fig. 7shows typical cross-sectional profiles of the rivulet in the cases (a)a = 2 (< π) and (b) a = 5 (> π), and Fig. 8 shows sketches ofthe rivulet in the same two cases, namely (a) a < π , in which the

rivulet is never de-pinned, and (b) a > π , in which the rivulet isde-pinned and has zero contact angle in the interval αc < α ≤ π .

5. Pinning and de-pinning of a rivulet with constant contactangle β = β at a = a

As we described in Section 3, the semi-width a of a slowlyvarying rivulet with constant non-zero contact angle β = β > 0is unbounded at α = 0 (i.e. the rivulet is infinitely wide at the topof the cylinder), has a single minimum value of a = amin (< π)at α = αmin on the lower half of the cylinder and takes the valuea = π atα = π , while in the special case of zero contact angle β =

β = 0 the semi-width is unbounded at α = π/2 and decreases tothe value a = π at α = π . In practice, however, there could be aminimum physically realisable value of the semi-width, denotedby a = a, at which the contact lines become pinned. Evidentlythe behaviour of the rivulet in this situation will be qualitativelydifferent for a ≤ π and a > π .

5.1. a ≤ π

When a ≤ amin (< π) the semi-width is always greater thanor equal to a and hence pinning and de-pinning do not occur, andso the rivulet behaves exactly as described in Section 3. However,when amin < a < π the rivulet runs from α = 0 with constantnon-zero contact angle β = β > 0 but decreasing semi-widtha as described in Section 3 until it reaches the value a = a at



a b

c

Fig. 13. Plots of (a) the contact angle β , (b) the scaled semi-width a/π , and (c) the maximum thickness hm as functions of the scaled angle α/π for a/π =

0.2, 0.4, 0.6, ac/π ≃ 0.6446, 0.8, 1, 1.2, 1.4 when Q = 1 for a rivulet whose contact lines de-pin at non-zero contact angle β = β = 1. The corresponding solutionsfor a rivulet with constant contact angle β = β = 1 are shownwith dashed lines. De-pinning and re-pinning occur for a/π > ac/π ≃ 0.6446, de-pinning but no re-pinningoccurs for a/π > 1, and the points at which this happens are denoted by dots.

α = αpin(0 < αpin < αmin) at which the contact lines pin. Therivulet then runs from α = αpin with constant semi-width a = abut varying contact angleβ as described in Section4until it reachesα = αdepin(αmin < αdepin < π) at which the contact lines de-pin.The rivulet then runs from α = αdepin to α = π with constantcontact angle β = β > 0 but increasing semi-width a as describedin Section 3. Here α = αpin and α = αdepin are the appropriatesolutions of the equation Q = Q with Q given by (3), a = a andβ = β . In the marginal case a = π we have αdepin = π and so de-pinning does not occur. Expressed in another way, in the generalcase of constant non-zero contact angle β > 0 pinning and de-pinning occur when β > βc, where the value of βc correspondsto amin = a. In the special case of pinning at zero contact angleβ = β = 0 we have a = π/m ≥ π ≥ a, and so pinning and de-pinning do not occur. In the limit β → ∞ we have αpin → 0+ andαdepin → π− and so recover the solution for a rivuletwith constantsemi-width a = a ≤ π described in Section 4.

The behaviour when a ≤ π is illustrated in Fig. 9, which showsplots of the semi-width a, the contact angle β , and the maximum

thickness hm as functions of the scaled angle α/π for a range ofvalues of β when Q = 1 and a = 2 (< π). In particular, Fig. 9shows that in this case pinning and de-pinning occur for β > βc ≃

1.0249.

5.2. a > π

When a > π , as in the case amin < a ≤ π , the rivulet runsfrom α = 0 with constant non-zero contact angle β = β > 0 butdecreasing semi-width a as described in Section 3 until it reachesthe value a = a at α = αpin (0 < αpin < αc) at which thecontact lines pin. The rivulet then runs fromα = αpin with constantsemi-width a = a but decreasing contact angle β as described inSection 4 until, unlike in the case amin < a ≤ π , it reaches thecritical azimuthal angle α = αc (π/2 < αc < π ) at which thecontact angle β reaches the value zero and the contact lines de-pin. The rivulet then runs from α = αc to α = π with zero contactangle β = 0, decreasing semi-width a = π/m and increasingmaximum thickness hm = hm0 as described in Section 3. In the



0

a

b c

Fig. 14. Cross-sectional profiles h(y) when Q = 1 in the cases (a) β = 1 and a = 2 (< ac ≃ 2.0252) for α = π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8, (b) β = 1 anda = 2.5 (ac < a < π) for α = π/8, π/4, αdepin ≃ 1.2834, 3π/8, π/2, 5π/8, 3π/4, 7π/8, and (c) β = 1 and a = 5 (> π) for α = π/16, αdepin ≃ 0.4345, π/4, 3π/8, π/2,5π/8, 3π/4, 7π/8. For clarity, in part (b) no profiles are shown in the interval αrepin ≃ 3.0814 ≤ α ≤ π , and the three parts of this figure use the same vertical but differenthorizontal ranges.

special case of pinning at zero contact angle β = β = 0 pinningand de-pinning do not occur. In the limit β → ∞ we have αpin →

0+ and so recover the solution for a rivulet with constant semi-width a = a > π described in Section 4.

The behaviourwhen a > π is illustrated in Fig. 10, which showsplots of the semi-width a, the contact angle β , and the maximumthickness hm as functions of the scaled angle α/π for a range ofvalues of β when Q = 1 and a = 5 (> π). In particular, Fig. 10shows that pinning and de-pinning occur for all β > 0, and that inthis case de-pinning occurs at the scaled critical azimuthal angleαc/π ≃ 0.6292 for all β > 0.


Fig. 11 shows typical cross-sectional profiles of the rivulet inthe cases (a) a = 2 (< π) and β = 0.5 (< βc ≃ 1.0249), (b)a = 2 (< π) and β = 1.5 (> βc), and (c) a = 5 (> π) andβ = 1, and, in order to clarify what might appear to be a rathercomplicated situation, Fig. 12 shows sketches of the rivulet in the

same three cases, namely (a) a < π and 0 < β < βc, in which therivulet is never pinned, (b) a < π and β > βc, in which the rivuletis pinned in the interval αpin < α < αdepin, and (c) a > π , in whichthe rivulet is pinned in the interval αpin < α < αc and has zerocontact angle in the interval αc ≤ α ≤ π .

6. De-pinning and re-pinning of a rivulet with constant widtha = a at β = β

In Section 5 we described the pinning and de-pinning of arivulet with constant contact angle β = β at a = a. In thissection we describe the corresponding situation involving the de-pinning and re-pinning of a rivulet with constant width a = a atβ = β > 0. As we described in Section 4, for a narrow rivuletwith a < π the contact angle β of a slowly varying rivulet withconstant semi-width a = a is unbounded at α = 0 and α = π ,and has a single minimum value of β = βmin at α = αmin on thelower half of the cylinder, while for a wide rivulet with a > π thecontact angle is unbounded at α = 0 and decreases to the value



a b

c

Fig. 15. Sketches of a slowly varying rivulet with prescribed flux Q with (when not de-pinned with non-zero constant contact angle β = β > 0 and slowly varying semi-width a) constant semi-width a = a and slowly varying contact angle β that runs from the top α = 0 to the bottom α = π of a large horizontal cylinder, in the cases (a)a < ac < π , in which the rivulet is never de-pinned, (b) ac < a < π , in which the rivulet is de-pinned in the interval αdepin < α < αrepin , and (c) a > π , in which the rivuletis de-pinned in the interval αdepin < α ≤ π .

zero at α = αc. In Section 4 we showed how there can be steadyflow of a wide rivulet all the way from α = 0 to α = π when thecontact lines de-pin when the contact angle reaches its minimumphysically realisable value of zero, i.e. at α = αc. More generally,however, the contact lines could de-pin at a non-zero value of thecontact angle, denoted by β = β > 0. Evidently, as in Section 5,the behaviour of the rivulet in this situation will be qualitativelydifferent for a ≤ π and a > π .

6.1. a ≤ π

When a < π and β < βmin the contact angle is always greaterthan or equal to β and hence de-pinning and re-pinning do notoccur, and so the rivulet behaves exactly as described in Section 4.However, when a < π and β ≥ βmin the rivulet runs from α = 0with constant semi-width a = a but decreasing contact angle β asdescribed in Section 4 until it reaches the valueβ = β atα = αdepin(0 < αdepin < αmin) at which the contact lines de-pin. The rivuletthen runs from α = αdepin with constant contact angle β = βbut varying semi-width a as described in Section 3 until it reaches

α = αrepin (αmin < αrepin < π ) at which the contact lines re-pin.The rivulet then runs from α = αrepin to α = π with constantsemi-width a = a but increasing contact angle β as described inSection 4. Here α = αdepin and α = αrepin are the appropriatesolutions of the equation Q = Q with Q given by (3), a = a andβ = β . In the marginal case a = π we have αrepin = π and so re-pinning does not occur. Expressed in another way, de-pinning andre-pinning occur when a > ac, where the value of ac correspondsto βmin = β . In the limit a → ∞ we have αdepin → 0+ andαrepin → π− and so recover the solution for a rivulet with constantnon-zero contact angle β = β > 0 described in Section 3.

6.2. a > π

When a > π , as in the case a ≤ π , the rivulet runs from α = 0with constant semi-width a = a but decreasing contact angle βas described in Section 4 until it reaches the value β = β > 0 atα = αdepin (0 < αdepin < αc) at which the contact lines de-pin. Therivulet then runs from α = αdepin with contact angle β = β > 0but varying semi-width a as described in Section 3 until, unlike in



Fig. 16. The mass of fluid on the cylinder M for a rivulet whose contact lines de-pin at contact angle β = β plotted as a function of the logarithm of the scaledsemi-width log(a/π) for β = 0, 1/2, 1, 2 when Q = 1, together with its leadingorder asymptotic behaviour in the limits a → 0+ given by (27) and a → ∞ whenβ = 0 given by (28), shown with dotted lines. The triangles indicate the slopes3Q/β2

= 12, 3, 3/4, confirming the leading order asymptotic behaviour in thelimit a → ∞ when β > 0 given by (29). De-pinning and re-pinning occur fora/π > ac/π , de-pinning but no re-pinning occurs for a/π > 1, and the points atwhich de-pinning first occurs are denoted by dots.

the case a ≤ π , it reaches α = π . In the limit a → ∞ we haveαdepin → 0+ and so again recover the solution for a rivulet withconstant non-zero contact angle β = β > 0 described in Section 3.


The behaviour for β > 0 for both a ≤ π and a > π is illustratedin Fig. 13, which shows plots of the contact angle β , the scaledsemi-width a/π , and the maximum thickness hm as functions ofthe scaled angle α/π for a range of values of a/π when Q = 1and β = 1. In particular, Fig. 13 shows that in this case de-pinningand re-pinning occur for a/π > ac/π ≃ 0.6446 and de-pinningbut no re-pinning occurs for a/π > 1. Fig. 14 shows typical cross-sectional profiles of the rivulet in the cases (a) β = 1 and a = 2 (<ac ≃ 2.0252) (b) β = 1 and a = 2.5 (ac < a < π), and (c) β = 1and a = 5 (> π), and, in order to clarify what might again appearto be a rather complicated situation, Fig. 15 shows sketches of therivulet in the same three cases, namely (a) a < ac < π , in whichthe rivulet is never de-pinned, (b) ac < a < π , in which the rivuletis de-pinned in the interval αdepin < α < αrepin, and (c) a > π , inwhich the rivulet is de-pinned in the interval αdepin < α ≤ π .

6.4. Mass of fluid on the cylinder

The mass of fluid on the cylinderM is given by

M =

π

0

+a

−ah dy dα =

π2

0

2β(ma cothma − 1)m2

dα

+

π

π2

2β(1 − ma cotma)m2

dα. (26)

Fig. 16 showsM plotted as a function of the logarithm of the scaledsemi-width log(a/π) for a range of values of β , and shows that M

is a monotonically increasing function of a. Fig. 16 also shows thatin the limit of a narrow rivulet, a → 0+, M → 0+ according to

M ∼

70a2Q

9

13

C → 0+, (27)

while in the limit of awide rivulet on the upper half of the cylinder,a → ∞, M → ∞ according to

M ∼

3a2Q2

13

C → ∞ (28)

when β = 0 and

M ∼3Qβ2

log a → ∞ (29)

when β > 0, where the constant C is again given by (10).

7. Conclusions

In the present work we showed how the solutions for theunidirectional flow of a thin rivulet with prescribed volumeflux down an inclined planar substrate can be used to describethe locally unidirectional flow of a rivulet with constant width(i.e. pinned contact lines) but slowly varying contact angle as wellas the possible pinning and subsequent de-pinning of a rivuletwithconstant contact angle and thepossible de-pinning and subsequentre-pinning of a rivulet with constant width as they flow in theazimuthal direction from the top α = 0 to the bottom α = π ofa large horizontal cylinder. We found that, despite being the samelocally, the global behaviour of a rivulet with constant width canbe very different from that of a rivulet with constant contact angledescribed by Duffy and Moffatt [20] and Duffy and Wilson [21].Specifically, while a rivulet with constant non-zero contact angleβ = β > 0 can always run from the top to the bottom of thecylinder, the behaviour of a rivulet with constant width a dependson the value of a. In particular, while a narrow rivuletwith constantsemi-width a = a ≤ π can run all the way from the top to thebottom of the cylinder, a wide rivulet with constant semi-widtha = a > π can run from the top of the cylinder only to a criticalazimuthal angleα = αc given by (19). In Section 4wediscussed thescenario in which the hitherto pinned contact lines of the rivuletde-pin at α = αc and the rivulet runs from α = αc to the bottomof the cylinder with zero contact angle but slowly varying semi-width a = π/m, as sketched in Fig. 8.

In Section 5 we described the pinning and de-pinning of arivulet with constant contact angle β = β at a = a. In particular,we showed that when a ≤ π the rivulet is pinned in the intervalαpin < α < αdepin for β > βc, but that when a > π the rivulet ispinned in the interval αpin < α < αc and has zero contact anglein the interval αc ≤ α ≤ π for all β > 0, as sketched in Fig. 12. InSection 6 we described the corresponding situation involving thede-pinning and re-pinning of a rivulet with constant semi-widtha = a at a non-zero contact angle β = β > 0 which generalisesthe de-pinning at zero contact angle discussed in Section 4. Inparticular, we showed that when a ≤ π the rivulet is de-pinnedin the interval αdepin < α < αrepin for a > ac, but that whena > π the rivulet is de-pinned in the interval αdepin < α ≤ π ,as sketched in Fig. 15. In the latter situation, the mass of fluid onthe cylinder was found to be a monotonically increasing functionof the constant semi-width a.

Acknowledgements

The first author (CP) gratefully acknowledges the financialsupport of the University of Strathclyde via a Postgraduate



Research Scholarship. Part of this work was undertaken whilethe corresponding author (SKW) was a Visiting Fellow in theDepartment of Mechanical and Aerospace Engineering, School ofEngineering and Applied Science, Princeton University, USA, andpart of it was undertaken while he was a Visiting Fellow in theOxford Centre for Collaborative Applied Mathematics (OCCAM),Mathematical Institute, University of Oxford, United Kingdom. Thispublication was based on work supported in part by Award NoKUK-C1-013-04, made by King Abdullah University of Science andTechnology (KAUST). A preliminary version of thiswork, discussingonly the de-pinning of a rivuletwith constantwidth at zero contactangle, appears as OCCAM preprint number 12/43 [35].

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