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arXiv:1010.2920v1 [physics.flu-dyn] 14 Oct 2010
On the depinning of a drop of partially wetting liquid on a
rotating cylinder
Uwe Thiele
Department of Mathematical Sciences,
Loughborough University, Leicestershire LE11 3TU, UK
We discuss the analogy of the behaviour of films and drops of liquid on a rotating
horizontal cylinder on the one hand and substrates with regular one-dimensional
wettability patterns on the other hand. Based on the similarity between the
respective governing long-wave equations we show that a drop of partially wetting
liquid on a rotating cylinder undergoes a depinning transition when the rotation
speed is increased. The transition occurs via a sniper bifurcation as in a recently
described scenario for drops depinning on heterogeneous substrates.
The manuscript was accepted by the J. Fluid Mech. in October
2010 and will be published in the following months.
I. INTRODUCTION
Motivated by the question how much honey can be kept on a breakfast knife by rotating
the knife about its long axis, Moffatt studied the destabilisation of a film on the outside of a
rotating horizontal cylinder and the subsequent development of regular patterns of azimuthal
rings [29]. He also gives a lubrication equation that incorporates gravity and viscosity effects
but neglects surface tension and inertia. It is based on a model by [40] which includes
surface tension. Related questions had been studied before by [56] employing a rotating
horizontal cylinder which is at the bottom immersed in a liquid bath. i.e., in contrast to
[29] the amount of liquid on the cylinder is not an independent control parameter. The
related case of a liquid film covering the inner wall of a horizontal rotating cylinder was
first studied by [37] who discussed experiments performed with water employing an inviscid
model incorporating gravitational and centrifugal forces.
Since the early experimental results engineers and scientists alike have studied flows of
2
free-surface films of liquid on the inner or outer wall of resting or rotating horizontal cylinders
in a number of different settings. The studied systems are of high importance for several
coating and printing processes where rotating cylinders transport the coating material in the
form of liquid films. In the present contribution we will discuss the analogy between film flow
and drop motion on a rotating horizontal cylinder on the one hand and on heterogeneous
substrates with regular wettability patterns on the other hand. First, however, we will give
a short account of the seemingly disconnected fields.
The theoretical analyses given by [56] and [37] for the flow on/in a rotating cylinder
were based on linear stability considerations based on the full hydrodynamic equations for
momentum transport, i.e., the Navier-Stokes equations. Their studies were extended later
on, e.g., by [35] who considered inviscid film flow on the in- and outside of the cylinder
in a unified manner, and [45] whose analysis includes viscous film flow at small and large
Reynolds numbers.
However, most analyses of the non-linear behaviour are based on a long-wave or lubri-
cation approximation [33] valid for the case where the thickness of the liquid film h(θ, t) is
small as compared to the radius of the cylinder R. The resulting evolution equation for the
film thickness profile
∂th=−∂θh3[˜α∂θ(∂θθ h+h)−˜γcos(θ)] + h(1)
was first given by [40] and used by [29] (without capillarity effects, i.e., ˜α= 0). In the
dimensionless Eq. (1), ˜α= 3ε3σ/Rω ˜ηand ˜γ=ε2ρgR/3ω˜ηare the scaled dimensionless
surface-tension and gravity parameter, respectively [18]. It is important to note that both
depend on the angular velocity of the rotation ω, the dynamic viscosity ˜η, the cylinder
radius R, and the ratio εof mean film thickness ¯
hand R. The remaining material constants
are surface tension σand density ρ, and gis the gravitational acceleration. The angle θ
determines the position on the cylinder surface and is measured anti-clockwise starting at
the horizontal position on the right. The force ˜γcos(θ) corresponds to the component of
gravity parallel to the cylinder surface.
The long-wave equation (1) including surface tension effects has been studied in detail by,
e.g., [18, 41] and [23]. Note that for the parameter ranges where Eq. (1) applies, it is valid
without difference for films on the outside and inside of the cylinder (cf., e.g., [2, 18, 31]).
A preliminary comparison of the long-wave approach to results obtained with the Stokes
3
equation is given by [36]. Various extensions of Eq. (1) were developed. See, in particular,
[1] for a discussion of hydrostatic effects. Other higher order terms related to gravity, inertial
and centrifugal effects have been included [2, 8, 14, 15, 19, 25, 31, 32]. In particular, [31] and
[25] present a systematic derivation of several such higher order models and give also detailed
comparisons to a number of previous studies as e.g., [32] , [2] and [8]. The film stability
and non-linear evolution of a film on the outside of a non-isothermal horizontal cylinder was
considered without [43] and with [13] rotation. Most of the aforementioned studies consider
two-dimensional situations. The fully three-dimensionional problem is studied for the cases
without [55] and with [15, 30] rotation.
The second system that we would like to discuss are liquid drops and films on heteroge-
neous solid substrates as studied experimentally, e.g. by [46] and [42]. Although in principle
the heterogeneity may result from substrate topography or chemical heterogeneities we focus
here on the case of a smooth flat substrate with chemical heterogeneities, i.e., we assume
the substrate wettability depends on position. Horizontal heterogeneous substrates are often
considered in connection with micro-patterning of soft matter films via thin film dewetting
[3, 11, 26, 44]. Another application are free-surface liquid channels in microfluidics [16].
Often one employs thin film evolution equations obtained through a long-wave approxima-
tion to study the dynamics of dewetting for an initially flat film or to analyse steady drop
and ridge solutions and their stability on such patterned substrates (see, e.g., [24] and [49]).
Static structures may also be studied using a variational approach [10, 27].
More recently a model system was introduced to study the dynamics of drops on het-
erogeneous inclined substrates. The models apply also to such drops under other driving
forces along the substrate [52]. The corresponding dimensionless evolution equation for the
one-dimensional film thickness profile (i.e., describing a two-dimensional drop) is of the form
∂th=−∂xh3[∂x(∂xxh+ Π(h, x))] + ˜µ,(2)
where ˜µ(h) represents the lateral driving force (µh3in the case of an inclined substrate, where
µis the inclination angle). The position-dependent disjoining pressure Π(h, x) models the
heterogeneous wettability of the substrate. One notices at once that Eqs. (2) and (1) are
rather similar: Identifying x=θ, ˜µ(h) = hand Π(h, x) = ˜αh −˜γsin(x), Eq. (1) may for
˜α= 1 be seen as a special case of Eq. (2). Here, we use the analogy to study drops or films
on a rotating cylinder. In particular, we show that drops on a rotating cylinder may show
4
rather involved depinning dynamics analogeous to the behaviour of drops on heterogeneous
substrates [5, 51, 52].
To illustrate the behaviour we choose the case of drops of a partially wetting liquid on
a substrate with a periodic array of localised hydrophobic wettability defects [52]. On a
horizontal substrate one finds steady drops that are positioned exactly between adjacent
defects, i.e., the drops stay on the most wettable part. At a small lateral driving force µthe
x→ −xsymmetry of the system is broken. However, the drops do not slide continuously
along the substrate as they do in the case of a substrate without defects [53]. Their advancing
contact line is blocked by the next hydrophobic defect, i.e., one finds steady pinned drops
at an upstream position close to the defect. As the driving force is increased, the drops are
pressed further against the defect. As a result the steady drops steepen. At a finite critical
value µcthe driving force allows the drop to overcome the adverse wettability gradient
and the drop depins from the defect and begins to slide. In a large part of the parameter
space spanned by the wettability contrast and the drop size the point of depinning at µc
corresponds to a Saddle-Node Infinite PERiod (sniper) bifurcation. Beyond µcthe drops
do not slide continuously down the incline. They rather perform a stick-slip motion from
defect to defect [51, 52]. However, in another region of the parameter space the depinning
can occur via a Hopf bifurcation, i.e., with a finite frequency. In particular, this happens in
the case of rather thick wetting layers.
The aim of the present contribution is to elucidate under which conditions a similarly
intricate ‘depinning’ behaviour can be found for a pendant drop underneath a horizontal
cylinder when increasing the speed of rotation. In the following section II we will discuss
our scaling and introduce a long-wave model for a film of partially wetting liquid on a
rotating cylinder. It allows (even without gravitation) for the coexistence of drops and a
wetting layer. We discuss several physical situations it can be applied to. Next, section
III presents steady state solutions depending on the non-dimensional rotation velocity for
several thicknesses of the wetting layer and discusses the depinning scenario in the case of
the partially wetting liquid. Finally, we conclude and give an outlook in Section IV.
5
II. MODEL
Before we introduce our long-wave model we discuss our scales. In the non-dimensionali-
sation used in most works studying Eq. (1) the angular velocity of the cylinder rotation ω
is not reflected in a single dimensionless parameter but both dimensionless parameters are
inversely proportional to ω. Furthermore, the time scale depends on ω[18, 40]. This and
a further re-scaling of the equation for steady drop and film states, that absorbs the flow
rate into both dimensionless parameters and also into the thickness scale, implies that the
existence of two steady solutions for identical parameters (thin capillary film and pendant
drop with through-flow), shown analytically by [41] and numerically by [23], actually refers
to solutions of different liquid volume.
Such a scaling is not convenient for our study as we would like to investigate the change in
system behaviour for fixed liquid volume when increasing the rotation speed of the cylinder
from zero, i.e., ωshould only enter a single dimensionless parameter that will be our main
control parameter. A scaling fit for our purpose is the one based on the scales of the gravity-
driven drainage flow employed by [14].
Note also that in particular [23] and to a lesser extent [55] represent some of their solutions
in a way that misrepresents their findings and, in general, the concept of an equation in long-
wave approximation. Whenever one shows the solutions as a thickness profile on a cylinder
one has to pick a particular radius Rand smallness ratio ǫfor the representation (cf. Figs. 3,
5 and 9 below). This is a rather arbitrary choice as the equation in long-wave approximation
is strictly valid only in the limit ǫ→0, ie. it is always only an approximation to a real
physical situation where ǫis finite.
In the present work we focus on solutions without axial variation, i.e., we consider the
two-dimensional physical situation depicted in Fig. 1. Following [14] we use ¯
h,R,ρg¯
h2/η,
ερg¯
h2/η, 3η/ε¯
hρg and ρg¯
hto scale the coordinate zorthogonal to the cylinder surface,
the coordinate x=θR along the surface, the two velocity components vxand vz, time, and
pressure, respectively. The ratio of z−and x−scale is the smallness parameter ε=¯
h/R, i.e.,
the film height is scaled by ¯
h=εR. Here the angle θis defined in clockwise direction starting
at the upper vertical position, i.e., a pendant droplet underneath the resting horizontal
cylinder has its centre of mass at θ=π.
The evolution equation is derived from the Navier-Stokes equations with no-slip boundary
6
θ
ω
g
h
R
FIG. 1: Sketch of a drop of partially wetting liquid coexisting with a thick wetting layer on a
rotating cylinder.
conditions at the (rotating) cylinder, and tangent- and normal force equilibria at the free
surface using the long-wave approximation [14, 33, 40, 43]. Note that here we furthermore
assume that the ratio ¯
h/R and the physical equilibrium contact angle are both of O(ε)
(cf. Section IV). This gives
∂τh=−∂θnh3∂θhBo−1(∂θθh+h)−cos(θ) + e
Π(h)i+e
Ωho,(3)
where τis the non-dimensional time, and
Bo = R3ρg
¯
hσ and e
Ω = ηωR
ρ¯
h2g≥0 (4)
are an effective Bond number and a rotation number for a clock-wise rotation, respectively.
The latter corresponds to the ratio of the rotation velocity at the cylinder surface and the
drainage velocity. The function e
Π(h) is a non-dimensional disjoining pressure that accounts
for the wettability of the liquid film on the cylinder surface [9, 17, 21].
To make the analogy with the depinning droplet on a heterogeneous substrate more
obvious we introduce another timescale t=τ/Bo, disjoining pressure Π = Bo e
Π and rotation
number Ω = Bo e
Ω and obtain
∂th=−∂θh3∂θ[∂θθh+h−Bo cos(θ) + Π(h)] + Ωh,(5)
with
Ω = ηωR
ε3σ.(6)
7
Defining the derivative
∂hf=−Π(h)−h+ Bo cos(θ) (7)
of a ‘local free energy’ f(h, θ) and identifying ˜µ= Ωh, Eq. (5) takes the form of Eq. (2)
and the Bond number takes the role of a ‘heterogeneity strength’. Note that the scaling
used is well adapted for discussing independently the influences of rotation and gravity in
the presence of capillarity. However, it does not allow for a study of the limit of vanishing
surface tension.
As disjoining pressure we choose here the combination of a long- and a short-range power
law Π(h) = Ha/h3(1 −b/h3), where Ha is a dimensionless Hamacker constant. For b > 0
one has a precursor film or wetting layer thickness h0=b1/3[38, 39, 48]. This is valid only
for Ha<0, i.e., when the long-range interaction is destabilising. This corresponds to the
case of a partially wetting liquid. For Ha>0, a rather thick wetting layer can be stabilised
and the b/h6term is of no further relevance and could just as well be dropped.
Note that ∂hfmay contain other terms beside the ones in Eq. (7). For a heated or cooled
cylinder a term (3/2) BoBi Ma log[h/(1 + Bi h)] + 1/(1 + Bi h)] would enter where Bi is the
Biot number and Ma an effective Marangoni number [34, 43, 50]. If the rotating cylinder
forms the inner electrode of a cylindrical capacitor, a term Vo/[h+ (d−h)ǫ]2has to be
included for a dielectric liquid and DC voltage of non-dimensional strength Vo (ǫis the
electric permittivity and dthe distance between the two cylinders (see appendix of [52] and
[22]). The similarity in the behaviour of depinning drops under the influence of different
physical effects in the case that the effective pressure terms ‘look similar’ is discussed by
[52].
Although here we do not consider such other physical effects explicitly, we will explore
the effect of corresponding thick wetting layers by adapting the parameters of our disjoining
pressure accordingly. A thick wetting layer that coexists with droplets might, for instance,
result from the interplay of a destabilising thermal or electrical effect and a stabilising long-
range van der Waals interaction. We obtain adequate values of Ha and bfor our present
Π(h) by relating them to the (non-dimensional) thickness of the wetting layer h0and a static
macroscopic contact angle β0by [see, e.g., 17]
b=h3
0and Ha = −5
3β2
0h2
0.(8)
Note, that β0is the angle in long-wave scaling, i.e., a small physical equilibrium contact
8
angle βeq =εβ0corresponds to a long-wave contact angle β0of O(1). In the limit of a non-
rotating cylinder (i.e. Ω = 0) and assuming the disjoining pressure corresponds to complete
wetting (Ha>0, 1/h6term may be dropped), our equation (5) corresponds to the one given
by [43] for the isothermal case. There, however, first a viscous scaling is used, which they
transform in a second step – a rescaling of time and therefore velocities with the Galileo
number – into the ‘drainage scaling’ used here. Our equation corresponds also to that of
[18] when the disjoining pressure is added to their equation.
To analyse the system behaviour we determine in the following steady-state solutions
that correspond, e.g., to pendant droplets. The behaviour beyond the depinning thresh-
old is analysed using time-stepping algorithms. As we restrict ourselves to the two-
dimensional physical situation an explicit scheme for stiff equations suffices for the latter.
The steady-state solutions are obtained using the continuation techniques of the package
AUTO [12]. The steady and time-periodic solutions are characterized by their L2norm
||δh|| ≡ q(1/2π)R2π
0(h(θ)−1)2dθ and time-averaged L2norm
||δh|| ≡ q(1/2π T )RT
0R2π
0(h(θ)−1)2dθdt, respectively. Tis the time period.
Note finally that Eq. (5) is the result of our choices for the relative order of magnitude of
the involved dimensionless numbers that is based on the choice of physical effects that shall
be discussed (cf. [33]). Once this is accepted, equation (5) is to O(1) asymptotically correct.
Its form as a conservation law implies that R2π
0hdθ = 2π. To O(1) this corresponds to mass
conservation. For a discussion of higher order corrections to this picture see [25].
III. RESULTS
A. Parameters
We base our estimate for realistic Bond and rotation numbers on two liquids typically
studied in the literature: (i) water at 25◦C as in [43] and a silicone oil with σ= 0.021N/m,
η= 1kg/ms and ρ= 1200kg/m3as in [14]. As experimentally feasible configurations we
assume angular rotation velocities ω= 0.1...10s−1, cylinder of radii R= 10−3...10−2m
and smallness ratios ε= 0.01 ...0.1. It is only for the purpose of illustration of steady
droplets on a cylinder that we later use ε= 0.1 in selected figures.
For ω= 1s−1and a smallness ratio ε= 0.1 we find for a cylinder of radius R= 10−2m for
9
material (i) Bo= 136 and Ω = 0.14, whereas material (ii) gives Bo= 561 and Ω = 476. For a
smaller cylinder radius of R= 10−3m one has (i) Bo= 1.36 and Ω = 0.014, and (ii) Bo= 5.61
and Ω = 47.6. Decreasing the smallness ratio to ε= 10−2increases all Bond numbers by
a factor 10 and all rotation numbers by a factor 103. Focusing on cylinder diameters in
the millimetre range, we mainly investigate Bo≤10. Note that experiments may also be
performed with larger cylinders. To keep the Bond number in the interesting range one could
decrease the relevant density by replacing the ambient gas by a second (immiscible) liquid.
A feasible experimental set-up could be a horizontal Taylor-Couette apparatus with rotating
inner cylinder. The theoretical framework would also need to be amended – replacing the
one-layer theory [Eq. (5)] by a closed two-layer model similar to [28].
Note, that silicone oils may have viscosities 2 orders of magnitude smaller or larger than
the chosen value and one is able to vary the velocity of rotation in a wide range. This
implies that there is some flexibility in the choice of Ω. We will find that for Bo= O(1) the
interesting range for the rotation number is Ω = O(1). For instance, with Bo= 1 one finds
that depinning occurs at Ω = 1.68.
As the present work aims at establishing the qualitative correspondence between the
behaviour of a depinning drop on a heterogeneous substrate and the one of a droplet on a
rotating cylinder we restrict our results to one partially wetting case (choosing h0= 0.1 and
β0= 2) and the completely wetting case (β0= 0). For comparison, selected results are also
shown for the contact angle β0= 1.
Note that here the scaling and therefore the discussion of the smallness parameter ǫis
based on ǫ=¯
h/R, i.e., we use the ratio of mean film thickness and cylinder radius. In the
case of pendant drop solutions one also has to keep the maximal film height hmax in mind
when discussing ǫas the scaling becomes questionable when hmax ≫¯
h. This is, however,
normally not the case. In the present work the ratio hmax/¯
his always below 3. Although
an inspection of the figures in [23] shows a ratio hmax /¯
hthat is not much larger, one notices
that there ǫitself (based on ¯
h) seems to be larger than one (see remark above Eq. (II)).
The same applies to Fig. 3 of [55]. There exists no such problem in the region where only
a thin wetting layer of thickness hmin covers the cylinder even if hmin ≪¯
h. Actually, the
long-wave approximation gets better there. In an approach based on matched asymptotics
one could further simplify the governing equation in this region (cf. [7]). However, such an
approach is not taken here as it is of limited use when considering parameter regions that
10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Bo
0
0.2
0.4
0.6
0.8
1
||δh||
1.0
2.0
β0
(i)
(ii)
(iii)
FIG. 2: Solutions on a horizontal cylinder without rotation (Ω = 0) are characterised by their
L2norm as a function of the Bond number for two different equilibrium contact angles β0and
h0= 0.1, ¯
h= 1.0.
show qualitative changes in solution behaviour, e.g., close to the sniper bifurcation discussed
below.
B. Partially wetting case
First, we consider the case of a horizontal cylinder without rotation (Ω = 0) and determine
steady drop and film solutions. Employing the Bond number as control parameter we
obtain families of steady state solutions for two selected values of the equilibrium contact
angle β0. Inspecting Fig. 2 one notices that above a critical value of the Bond number Boc
there exists only a single solution. It corresponds to a symmetric pendant drop located
underneath the cylinder. An example for Bo= 1 and β0= 2.0 is given as solid line in Fig. 3.
However, two additional steady solutions exist below Boc. One of them corresponds to an
unstable symmetric drop sitting on top of the cylinder (dotted line in Fig. 3). The other
one corresponds to an unstable solution that has two minima – a deep one underneath the
cylinder and a shallow one on top of it (dashed line in Fig. 3).
We find that Bocbecomes smaller with decreasing contact angle β0, i.e., the range of Bo
11
0
1
2
3
h(θ)
0 3.14 6.28
θ
-1
0
1
cos θ
(i)
(ii)
(iii)
a
b-10 0 10
x
-10
0
10
y
c
FIG. 3: Steady film thickness profiles on a horizontal cylinder without rotation (Ω = 0) for Bo= 1
and β0= 2.0 (corresponding to marks (i) to (iii) in Fig. 2). The remaining parameters are as in
Fig. 2. Panel (a) gives the profiles as h(θ), (b) gives the space-dependent part of ∂hf(Eq. (7)),
and panel (c) shows the profiles on a cylinder. Note that for illustration purposes we have (rather
arbitrarily) assumed a radius R= 10, i.e. ε= 0.1 (cf. discussion in Section I II A). Line styles in
(c) correspond to those in (a). The cylinder surface is represented by the solid black line.
numbers where steady structures beside the pendant drop exist becomes smaller. The com-
pletely wetting case (β0= 0) is qualitatively different and is discussed below in Section III C.
Increasing β0above β0= 2 the critical value Bocstrongly increases, e.g., for β0= 10 one
finds Boc= 43.3.
Note, that the pendant drop underneath the cylinder [drop on top of cylinder] corresponds
on a horizontal heterogeneous substrate to the drop on the more [less] wettable region as
discussed by [49]. The limit of zero Bond number is analogue to the horizontal homogeneous
substrate.
Next, we increase the dimensionless angular velocity Ω for several selected Bond numbers
and determine the steady state solutions (Fig. 4). Focusing first on Bo= 1 we recognise at
Ω = 0 three solutions as marked by the vertical dotted line in Fig. 2. The curve of largest
norm corresponds to the pendant drop. As Ω increases from zero, its norm and amplitude
decrease slightly, whereas its centre of mass moves towards larger θ, it is ‘dragged along’ by
the rotation without changing its shape much. This can be well appreciated in Fig. 5 which
12
0 1 2 3 4 5
Ω
0
0.2
0.4
0.6
0.8
1
1.2
||δh||
0.5
0.75
1.0
2.5
5.0
10.0
Bo
FIG. 4: Solutions on a horizontal rotating cylinder as a function of the Rotation number Ω for
various Bond number Bo as given in the legend and contact angle β0= 2.0. The drop and film
profiles are characterised by their L2norm. The remaining parameters are h0= 0.1, and ¯
h= 1.0.
0
1
2
3
h(θ)
0.0
0.5
1.0
1.68
0 3.14 6.28
θ
-1
0
1
cos θ
Ωa
b-10 0 10
x
-10
0
10
y
c
FIG. 5: Steady film thickness profiles on a rotating horizontal cylinder for Bond number Bo= 1
and β0= 2.0 for Rotation numbers Ω as given in the legend. All shown solutions are situated on
the stable upper branch in Fig. 4. The remaining parameters are as in Fig. 4. Panel (a) gives the
profiles as h(θ), (b) gives the space-dependent part of ∂hf(Eq. (7)), and panel (c) illustrates the
profiles on a cylinder assuming a radius R= 10. Line styles in (c) correspond to those in (a).
13
11.5 22.5
Ω
0.94
0.96
0.98
1
1.02
||δh||
10-6 10-5 10-4 10-3 10-2 10-1 100
Ω−Ωsn
10-3
10-2
10-1
1/T
FIG. 6: Bifurcation diagram for the depinning transition for a drop on a rotating cylinder with
Bo= 1 and β0= 2.0. Shown are the L2norm for the branch of steady solutions as obtained by
continuation (solid line), selected steady solutions as obtained by direct integration in time (circles)
and the time-averaged L2norm for the unsteady solutions beyond the sniper depinning bifurcation
(triangles). The inset gives for the latter a log-log plot of the dependence of the inverse temporal
period on the distance from the bifurcation Ω −Ωsn . The solid line indicates an exponent of 1/2.
gives selected profiles for the branch of pendant drop solutions. The branch terminates in a
saddle-node bifurcation at Ωsn = 1.68 where it annihilates with one of the unstable branches.
The third branch, i.e., the one of lowest norm, shows for increasing Ω an increase in the
norm, then it undergoes two saddle-node bifurcations at Ω = 1.313 and Ω = 1.329. At
larger Ω the norm decreases again. Note that the two saddle-node bifurcations are a first
sign of rather complicated re-connections which occur when one increases Bo from 1.0 to
2.5. The re-connections involve further (unstable) branches that are not shown in Fig. 4 and
result in the loop structure observed for Bo= 2.5 at about Ω = 1.5. A further indication for
the existence of additional branches is the observation that the value of the norm at Ω = 0
for Bo= 1.0 in Fig. 4 does not agree with the lowest norm at Bo= 1.0 in Fig. 2. As the
re-connections are not relevant for the depinning transition we will here not consider them
further.
At Bond numbers smaller than Bo= 1 the behaviour is qualitatively the same as described
14
(a) (b)
FIG. 7: Space-time plots illustrating the time evolution of co-rotating drops beyond depinning via
a sniper bifurcation (at Ω >Ωsn). Shown is one period in space and time (a) close to depinning
at Rotation number Ω = 1.8 with a temporal period of T= 14.8, and (b) far from depinning at
Ω=3.0 with T= 2.7. The Bond number is Bo= 1 and β0= 2.0.
for Bo= 1. However, the critical Ωsn becomes smaller with decreasing Bo. Above, we have
discussed that the behaviour at Ω = 0 changes qualitatively when decreasing Bo beyond
Boc(Fig. 2). Although this also affects the behaviour at small Ω it has less influence on
the behaviour at larger Ω (Fig. 4). In particular, the saddle-node at Ωsn where the pendant
drop solution ceases to exist, persists for larger Bond number as does the related dynamic
behaviour (see below). Note that for Bo= 10 the saddle-node is at Ωsn ≈6.2 outside the
range of Fig. 4.
An interesting resulting question is what happens to a stable obliquely pendant droplet
at Ω <Ωsn when the rotation velocity is further increased such that it is slightly larger than
Ωsn? Does the solution approach the remaining steady solution? Simulations of Eq. (5)
show that this is not the case. For Ω >Ωsn the droplet moves in a non-stationary way with
the rotating cylinder and corresponds to a space- and time-periodic solution. The resulting
bifurcation diagram for Bo= 1 is given in Fig. 6. It shows that the ‘new’ branch of space- and
time-periodic solutions (characterised by its time-averaged norm) emerges from the saddle
node bifurcation of the steady solutions at Ωsn. The behaviour close to and far from the
saddle-node bifurcation is illustrated in the space-time plots of Fig. 7. Panel (b) shows
that far away from the bifurcation the drop moves continuously with its velocity and shape
varying smoothly. It moves fastest when its maximum passes θ≈π/2 (velocity ≈4.6)
15
and slows down when passing θ≈3π/2 (velocity ≈1.2), i.e., respectively, when gravity
most strongly supports and hinders the motion driven by the cylinder rotation. The ratio
of largest to smallest velocity is about 4:1. When approaching the bifurcation from above,
the droplet still moves with the rotation but the time scales of the slow and the fast phase
become very different. For instance, at Ω = 1.8 (Fig. 7 (a)) the drop moves fastest when
passing θ≈π/2 (velocity ≈3.3). However, when the drop is situated at about θ= 3π/2
it barely moves (velocity ≈0.1) resulting in a velocity ratio of 33:1 between the fastest
and slowest phase. The resulting overall behaviour strongly resembles the stick-slip motion
discussed in the context of contact line motion on heterogeneous substrates [52]. Note that
it also resembles so-called sloshing modes found for a partially liquid-filled rotating cylinder
[19, 54].
The ratio of the velocities diverges when approaching the bifurcation point. In conse-
quence, the frequency 1/T of the periodic drop motion goes to zero. As shown in the inset of
Fig. 6, the dependence corresponds to a power law 1/T ∼(Ω −Ωsn)1/2. This indicates that
the bifurcation at Ωsn is actually a Saddle-Node-Infinite-PERiod bifurcation (or ‘sniper’ for
short; see [47], and discussion by [52]).
The described behaviour is quite generic for the partially wetting case, i.e., it is found
in a wide range of parameters, in particular, Bond number, contact angle and wetting
layer thickness. However, the behaviour changes dramatically when strongly decreasing the
contact angle, i.e., when approaching the completely wetting case.
C. The completely wetting case
After having discussed the intricate depinning behaviour in the partially wetting case,
to emphasise the contrast, we consider briefly the completely wetting case (β0= 0). Note
that the physical setting is then identical to the one employed in most studies of the classic
Moffatt problem [29, 40]. However, the parametrisation used here is different and results
in Eq. (5) with Ha= 0. We analyse the system behaviour as above by studying the steady
state solutions without and with rotation.
In the case without rotation, the first difference one notices is that for fixed Bond number
only one solution exists. It corresponds to the pendant drop solution. With increasing Bo,
i.e., with increasing importance of gravity, its amplitude monotonically increases. Fixing
16
0 1 2 3 4 5
Ω
0
0.2
0.4
0.6
0.8
1
1.2
||δh||
1.0
2.5
5.0
10.0
Bo
FIG. 8: Solutions for a film of completely wetting fluid (β0= 0) on a horizontal rotating cylinder
as a function of the Rotation number Ω for various Bond numbers Bo as given in the legend. The
solutions are characterised by their L2norm. The remaining parameter is ¯
h= 1.0.
0
1
2
3
h(θ)
1e-10
0.5
1.0
2.5
5.0
0 3.14 6.28
θ
-1
0
1
cos θ
Ωa
b-10 0 10
x
-10
0
10
y
c
FIG. 9: Steady film thickness profiles for the case of complete wetting with Bo= 1 for Rotation
numbers Ω as given in the legend. The remaining parameters are as in Fig. 8. Panel (a) gives the
profiles as h(θ), (b) gives the space-dependent part of ∂hf(Eq. (7)), and panel (c) illustrates the
profiles on a cylinder assuming a radius R= 10. Line styles in (c) correspond to those in (a).
17
Bo and increasing the rotation velocity from zero, one finds a monotonic decrease of the
norm (Fig. 8) and height of the drop. Its maximum moves towards larger angular position
θ, i.e., the liquid is dragged along with the rotation. However, in the wetting case there
is no force besides gravity that can favour drops as compared to a flat film. As gravity
acts downwards it only favours pendant drops. Therefore, the drop does not survive as a
‘coherent structure’ when dragged upwards with the rotation: it is flattened and smeared
out. Typical profiles are given in Fig. 9. Note that we do not show the exact case without
rotation, but choose Ω = 10−10 as then a dynamically created wetting layer still exists that
is numerically advantageous (cf. [50]).
When increasing Ω further than shown in Fig. 8 one finds that the thickness profiles
approaches a flat film. For instance, for Bo= 1 the norm decreases below 10−2at Ω ≈70.
IV. CONCLUSION
Based on the observation that the equations in long-wave approximation that govern film
flow and drop motion (i) on or in a rotating cylinder and (ii) on a heterogeneous substrate are
rather similar, we have explored whether the analogy can be exploited, i.e., whether it allows
results obtained for one system to be transferred to the other one. In particular, we have
found that indeed for drops on a rotating cylinder there exists a counterpart of the rather
involved depinning dynamics described recently for drops on heterogeneous substrates.
To study the effect we have introduced an alternative scaling. This has been necessary
because the commonly used scaling contains the angular velocity of the cylinder rotation in
both dimensionless parameters and in the time scale. Together with a further re-scaling of the
steady state equation involving the flow rate [23, 40] this does not allow for an investigation
of either the system behaviour when increasing the rotation speed of the cylinder from
zero or the existence of multiple solutions for fixed rotation speed and liquid volume. The
scaling employed here does allow for such studies as the rotation speed only enters a single
dimensionless parameter. Furthermore, it allows us to discuss the analogy between the two
systems of interest in a rather natural way. In particular, downward gravitation and rotation
speed for the drop on the rotating cylinder correspond to the heterogeneous wettability and
lateral driving force, respectively, for drops on heterogeneous substrates.
Guided by this analogy we have studied the behaviour of drops of partially wetting liquid
18
on the rotating cylinder. As a result it has been shown, how ‘switching on’ gravity (in-
creasing the Bond number) without rotation changes the solution behaviour dramatically as
the ‘gravitational heterogeneity’ along the cylinder surface effectively suppresses multidrop
solutions until only the single pendant droplet underneath the cylinder survives. We have
furthermore found that increasing the rotation from zero to small values, the drop is dragged
along to a stable equilibrium position where downwards gravity and upwards drag compen-
sate. At a critical speed, however, the retaining downwards force is too small and the drop
is dragged above the left horizontal position. It then continuously moves with the rotating
cylinder in a non-constant manner. Close to the transition the drop shows stick-slip motion.
An analysis of the time-dependent behaviour has shown that the frequency related to the
periodic motion goes to zero at the threshold following a power law with power 1/2. This
and the related bifurcation diagram show that the observed transition corresponds to depin-
ning via a sniper bifurcation. The behaviour found is rather generic for the case of partial
wetting. Although, here we have presented results for particular parameter values, it can
be observed in a wide range of parameters, in particular, Bond number, equilibrium contact
angle and wetting layer thickness. However, we have also found that the behaviour changes
dramatically when strongly decreasing the equilibrium contact angle, i.e., when approaching
the completely wetting case that is normally studied in the literature. When increasing the
velocity of rotation in this case we have seen a transition from pendant drops underneath
the cylinder to a nearly uniform film around the cylinder as [14].
The presented results indicate that it may be very fruitful to further explore the analogy
of films/drops on rotating cylinders and heterogeneous substrates. We expect that the
recent exploration of the three-dimensional case for depinning drops [5] and for the various
transitions between ridge, drop and rivulet states [4] based on tools for the continuation of
steady and stationary solutions of the two-dimensional thin film equation (physically three-
dimensional system) [6] is also relevant for the case of a rotating cylinder. A future study
could, e.g., elucidate the relation between the formation of azimuthal rings, pendant ridges
and sets of drops.
The present circular cylinder is an analogy to a sinusoidal wettability profile. Cylinders
with other cross sections, for instance, an elliptical cylinder [20] would correspond to more lo-
calised defects of the corresponding heterogeneous substrate. However, the analogue system
turns out to be more complicated as a rotating non-circular cylinder introduces the aspect
19
of a time-periodic non-harmonic forcing parallel to the substrate into the heterogeneous
substrate system.
Our equation (5) is derived under two conditions: (i) the mean (and also the maximal)
film thickness has to be small as compared to the radius of the cylinder; and (ii) the local
surface slope (e.g., the physical equilibrium contact angle) has to be small. Here we have
assumed that the two related smallness parameters are of the same order of magnitude. In
a next step one may introduce two different smallness parameters and discuss a number of
distinct limits in dependence of their ratio. Such a systematic asymptotic study would also
permit to discuss the influence of other effects on the depinning behaviour like, for instance,
the effects of the hydrostatic pressure [1, 2], centrifugal forces [14] and inertia [19, 25].
V. ACKNOWLEDGEMENTS
I acknowledge support by the EU via grant PITN-GA-2008-214919 (MULTIFLOW).
First steady state solutions where calculated by N. Barranger employing a predecessor of
the present model. I benefited from discussions with E. Knobloch and several group members
at Loughborough University as well as from the input of several referees and the editor.
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