# Suppression of the Rayleigh-Plateau instability on a vertical fibre coated
with wormlike micelle solutions111Accepted in Soft Matter.
DOI:10.1039/C3SM27940E

F. Boulogne1,2, M.A. Fardin3,4, S. Lerouge3,4,
L. Pauchard1, F. Giorgiutti-Dauphiné1

###### Abstract

We report on the Rayleigh-Plateau instability in films of giant micelles
solutions coating a vertical fibre. We observe that the dynamics of thin films
coating the fibre could be very different from the Newtonian or standard Non-
Newtonian cases. By varying the concentration of the components of the
solutions and depending on the film thickness, we show for the first time that
the Rayleigh-Plateau instability can be stabilized using surfactant solutions.
Using global rheology and optical visualisations, we show that the development
of shear-induced structures is required to stabilize the micellar film along
the fibre. Assuming that the viscoelastic properties of the shear-induced
state can be described by a simple model, we suggest that, in addition to the
presence of shear-induced structures, the latter must have an elastic modulus
greater than a critical value evaluated from a linear stability analysis.
Finally, our analysis provides a way of estimating the bulk elasticity of the
shear-induced state.

_1 UPMC Univ Paris 06, Univ Paris-Sud, CNRS, F-91405. Lab FAST, Bat 502,
Campus Univ, Orsay, F-91405, France.
2<EMAIL_ADDRESS>
3 Laboratoire Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris
Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cédex 13, France.
4 The Academy of Bradylogists _

##  1 Introduction

When a liquid film is coating a fibre, it undergoes spatial thickness
variations. The mechanism of this instability has been understood by Lord
Rayleigh in the 19th century[33]. Due to the energetic cost of free surfaces
(surface tension), the liquid tends to minimize its surface area by breaking a
cylinder into a serie of regularly spaced droplets. This instability can be
observed on free falling jets and it underlies many natural phenomena, like
the dew drops on coweb [7]. The coating of fibres has been extensively studied
by Quéré et al. in the situation where the fibre is drawn out of a bath [31].
They compared their measurements with theoretical predictions from Landau,
Levich and Derjaguin on the film thickness [23, 12].

In the case of a Newtonian fluid flowing down a vertical fibre, flow regimes
have been depicted as a function of physical parameters. For high flow rates
and/or fibre radii, the flow is dominated by inertia and the nature of the
instability is convective [14]. On the contrary, when the fibre size is
smaller than the capillary length and for creeping flow (low Reynolds number),
the instability is absolute and the physics is dominated by surface tension
[14]. In such absolute regimes, we recently studied the flow of polymer
solutions [6]. We investigated experimentally the influence of two non-
Newtonian properties, shear-thinning effect and first normal stress
difference, on the growth rate of the instability and on the morphology of the
drops. The pattern is globally the same as in the case of Newtonian fluids:
the first normal stress difference just tends to slightly decrease the growth
rate and to smooth the drop shape.

Beside polymers, a broad variety of molecules assemble to form non-Newtonian
fluids. For instance, it is well-known that above a Critical Micellar
Concentration (CMC), surfactant molecules can self-assemble to form aggregates
called micelles. The size and the shape of the micelles depend on the
structure of the surfactant molecule, on the surfactant concentration and on
the presence of additives like simple or organic salts [21]. In some range of
parameters, the micelles are giant worms often called ”living polymers”
because thay can entangle like polymers but they can also break and fuse
continuously. Solutions of worm-like micelles present remarkable rheological
properties such as shear-thickening or shear-banding effects, extensively
studied by theoreticians and experimentalists [2, 8]. These non-linear
properties are often associated with the development of out of equilibrium
structures induced by the shear flow. These shear-induced structures present
strong viscoelastic properties leading to specific behaviors like the
oscillation of a falling sphere [22] or the incomplete retraction of a
filament after a pinch-off [36]. These examples highlight the consequence of
the viscoelastic properties of the flow-induced phases.

In this paper, we investigate the flow, at low Reynolds number, along a
vertical fibre of semi-dilute giant micelles solutions. By varying the
concentration of the chemical compounds of the solutions and the thickness of
the film along the fibre, we observe different morphologies for the micellar
film. In some conditions, the film exhibits the expected Rayleigh-Plateau
instability, as it has been observed so far, in Newtonian fluids or
viscoelastic polymer solutions while for other conditions, remarkably, the
micellar film remains stable. Using global rheology and optical
visualisations, we demonstrate that the stabilisation process along the fibre
is connected with the development, above a characteristic shear stress, of
shear-induced structures. We also show that the development of shear-induced
structures is necessary but not sufficient to stabilize the micellar film.
Using a simple viscoelastic model (Kelvin-Voigt) to describe the viscoelastic
properties of the shear-induced state, we suggest that the elastic modulus
characterizing the shear-induced structures has to be greater than a critical
value provided by the linear stability analysis.

The paper is organized as follows. In Section 2, we present the experimental
setup and the chemical system. In Section 3, we report different flow regimes
depending on surfactant and/or salt concentrations and film thicknesses. In
Section 4, we present the rheological behavior of our micellar systems and we
visualize the material under shear to confirm the presence of shear-induced
structures. The next two Sections are devoted to the conditions required to
stabilize the micellar film on the fibre. Section 5 focus on the stress
applied to the film by gravity and its relation with the rheological behaviour
of the solution and Section 6 studies the role of the shear-induced film
elasticity on the Rayleigh-Plateau instability.

##  2 Experimental details

###  2.1 Experimental setups

Figure 1: (a) Notations and schematic view of the vertical fibre and the
injection device. (b-d) Pictures illustrating the different morphologies of
the film flowing along the fibre: (b) Unstable morphology which corresponds to
drops sliding on a quasi static liquid film. (c) Stable film. (d) Film
morphology corresponding to a gel-like block moving down the fibre. The
diameter of the fibre is $0.56$ mm.

The experimental setup used for the study of the Rayleigh-Plateau instability
is depicted in Figure 1(a). An upper tank, of diameter $14$ cm, is connected
to a valve. This valve is composed of two axisymmetric cones. The adjustment
of the gap between these cones controls the flow rate i.e. the film thickness
on the fibre. Guided by a nozzle, the liquid flows along a vertical nylon
fibre. The fibre is about $60$ cm long and has a radius $R=0.28$ mm. Vertical
position and centering of the nozzle are crucial to ensure an axisymmetric
coating. This position is adjusted using an optical support with a precision
of 2.4 arc sec (Model U200-A2K, Newport). Observations are done with a
telecentric lens and a high speed camera (about $500$ images/s) delivering a
resolution of $0.024$ mm/pixel (Fig. 1(b-c)).

The rheological properties of the samples have been characterized using a
stress-controlled rheometer (Anton Paar, MCR 501) at $20.00\pm 0.01^{\circ}$C.
The temperature was adjusted with a Peltier plate. We used a cone and plate
geometry (radius $49.988$mm, angle $0.484^{\circ}$). A solvent trap was used
to limit evaporation during the measurements. Rheological protocol consists in
applying stress sweep, the stress value being imposed during 1000 s at each
step. The flow curve is obtained by averaging the shear rate response over the
last 100 s of each step.

The global rheological measurements have been complemented by rheo-optical
observations of the flow performed in a Taylor-Couette (TC) flow geometry
(gap: $1.13$ mm, length: $4$ cm), adapted to the shaft of a stress-controlled
rheometer (Anton Paar MCR 301). The inner cylinder is rotating while the outer
transparent cylinder is fixed in the laboratory frame. Direct visualizations
were made in the plane velocity gradient/vorticity (i.e. the plane made by the
radial direction and the cylinders axis), the gap of the TC cell being
illuminated with white light. Images of the gap were collected using a CCD
camera. The rheological signal was recorded simultaneously, allowing direct
correlation with optical visualizations [25, 16]. Note that similar rheograms
were obtained with TC and cone and plate geometries.

###  2.2 Materials

We focus on aqueous micellar solutions made of cetyltrimethylammonium bromide
(CTAB) and sodium salicylate (NaSal) purchased from Sigma-Aldrich and used
without further purification. NaSal is an organic salt, sometimes designated
as a co-surfactant since it participates to the micellar structure by taking
place in-between the polar head groups of surfactant molecules [32]. The
addition of this salt also screens repulsive electrostatic interactions
between CTA+ groups and it reduces the spontaneous curvature of the system
[26]. As a result, the micellar uniaxial growth is promoted because of the
large energetic cost of hemispherical endcaps. All the solutions investigated
range in the semi-dilute regime. For the range of concentration chosen here,
the micelles are locally cylindrical and slightly entangled [2].

To prepare chemical solutions, we mixed in pure water (milliQ quality,
resistivity: $18$ M$\Omega$.cm) the weight percentages of NaSal and CTAB until
a complete dissolution. Samples were stored at rest at $23^{\circ}$C in
darkness to avoid any degradation. The solutions were allowed to reach
equilibrium for at least three days in these conditions before experiments
were performed. We carefully checked the homogeneity of solutions before any
measurement. All measurements along the fibre were carried out at $T=20.0\pm
0.5^{\circ}$C.

The equilibrium surface tension of the solutions studied in this paper is
$36.0\pm 0.5$ mN/m (Wilhelmy plate method with a Krüss tensiometer), a value
consistent with measurements reported in the literature for solutions above
the critical micellar concentration [11].

##  3 Morphologies of the liquid film along a vertical fibre

We focused on low Reynolds number flows ($Re\lesssim 1$) dominated by surface
tension ($R<l_{c}=\sqrt{\frac{\gamma}{\rho g}}=1.9$ mm), for which the flow is
absolutely unstable [14]. We studied the flow along the vertical fibre of a
large set of wormlike micelles solutions. Depending at once on the surfactant
concentration [CTAB], the salt concentration [NaSal] and the film thickness
$h$, we observe three possible morphologies of the film along the fibre as
displayed in Fig. 1(b-d):

  * •

The film can be unstable with respect to the Rayleigh-Plateau instability. The
associated morphology corresponds to drops sliding on a quasi-static liquid
film (Fig. 1(b)).

  * •

The film can be stable and in this case, the associated morphology corresponds
to a flat film (Fig. 1(c)).

  * •

The film can be subjected to some kind of fracture leading to separate blocks
moving down the fiber (Fig. 1(d)). The morphology is essentially flat but the
film is discontinuous.

In order to illustrate the effect of the various parameters on the flow along
the fibre, we construct a flow-phase diagram (see Fig. 2a) in the plane
([CTAB], [NaSal]), allowing to identify different domains.

At low NaSal concentrations, the apparent viscosity is low ($\lesssim 50$
mPa.s) which corresponds to inertial regimes [14]. At high NaSal
concentrations, due to a high zero shear viscosity ($>100$ Pa.s), the fluid
hardly or could not flow though our experimental device (”No flow” domain).
These two domains are excluded from the present study which, focuses on the
intermediate range of NaSal concentrations, typically between $0.2$ and $0.5$
wt.%. In this intermediate range, three domains, denoted by $\cal{A}$,
$\cal{B}$ and $\cal{C}$, can be distinguished according to the sequence of
morphologies observed as a function of the film thickness.

Figure 2: (a): Flow-phase diagram showing domains depending on surfactant
(CTAB) and cosurfactant (NaSal) concentrations. Five domains are identified:
Inertial (green $+$), ”No flow” (magenta $\times$), $\cal{A}$ (blue
$\triangle$), $\cal{B}$ (red $\Box$) and $\cal{C}$ (orange $\circ$). (b):
Schematic representations of the different sequences of film morphologies as a
function of the film thickness associated with the different domains
$\cal{A}$, $\cal{B}$ and $\cal{C}$ in the flow-phase diagram.

  * •

Domain $\cal{A}$

In this domain, the micellar solutions behave as usual Newtonian fluids [14]
or polymer solutions [6] for all the film thicknesses accessible with our
experiment: the film is first flat and then, the Rayleigh-Plateau instability
develops (Fig. 1(b)).

  * •

Domain $\cal{B}$

In this domain, regardless of the film thickness, the micellar solutions do
not exhibit the Rayleigh-Plateau instability. For sufficiently small
thicknesses (i.e. low flow rates), the film remains flat and continuous over
the whole fibre extent (about $60$ cm). From this stable morphology, an
increase of the film thickness leads to a breaking of the liquid thread into
blocks of a few centimeters long, sliding along the fibre (Fig. 1 (d)). Note
that, visually the blocks seem to have a gel-like character with a large
elasticity, in contrast to the solutions at rest. Similar viscoelastic threads
were observed in the case of pinch-off and filament retraction in extensional
flow of wormlike micelles [36, 3].

  * •

Domain $\cal{C}$

In between the two domains $\cal{A}$ and $\cal{B}$, the domain $\cal{C}$ is
characterized by the existence of the three morphologies (see Fig. 1(b-d)).
For small thicknesses, the films are destabilized by the Rayleigh-Plateau
instability with patterns similar to those described in domain $\cal{A}$. As
its thickness is increased, the film adopts a stable morphology corresponding
to a flat continuous liquid thread. Finally, for sufficiently large
thicknesses, the flat and continuous liquid thread breaks into gel-like blocks
as described in domain $\cal{B}$.

Figure 3: From a vertical fibre where a stable film is flowing down, we tilt
the fibre horizontaly leading to the development of the Rayleigh-Plateau
instability.

The different sequences of morphologies described above and observed
respectively in the domains $\cal{A}$, $\cal{B}$ and $\cal{C}$ are schematized
in Fig. 2(b).

When a stable morphology is observed, we have noticed that if we tilt the
fibre horizontally, after a few seconds, drops can appear (figure 3). This
observation suggests that the flow under the action of gravity is necessary to
stabilize the film. Particularly, this shows that the flow properties of the
micellar solutions are involved.

Note that this effect is different from the stabilizing process depicted by
Quéré [30] with Newtonian fluids. In his experiments, stable films flowing
down a vertical fibre were obtained by saturating the Rayleigh-Plateau
instability thanks to the advection of the flows. This saturation (i.e. flat
film) occurs if the film thickness is smaller than
$h_{c}=\frac{R^{3}}{l_{c}^{2}}$. In our experiment, $h_{c}=6\mu$m while
$h>200\mu$m, and the flow is absolutely unstable .

##  4 Rheo-optical properties under simple shear flow

In contrast to the free jets dominated by extensionnal flow [9, 15, 4], the
flow along the fibre is dominated by shear. We thus performed rheological
experiments under simple shear flow in order to characterize the rheological
properties of the micellar solutions used in this study. The measurement of
the flow properties is combined with direct optical visualisations of the
samples illuminated with white light in order to detect possible changes in
the structure of the micellar fluid. In the different domains $\cal{A}$,
$\cal{B}$ and $\cal{C}$ identified in the flow-phase diagram, we observe the
same type of flow behaviour. Figure 4(a) displays a representative flow curve
gathered from a sample belonging to domain $\cal{C}$ and obtained in stress-
controlled mode. The choice of this mode has been motivated by the fact that
the maximum stress in the liquid flowing down the fibre can be easily
estimated (See Section 5). The response of the samples to simple shear flow is
highly nonlinear, with successive transitions, from shear-thickening to
different degrees of shear-thinning. Such complex evolution of the shear
stress as a function of the shear rate has already been observed for dilute
micellar solutions, well-known to exhibit a shear-thickening transition [20,
19]. Following Refs. [20, 19], we divide the flow curve into four distinct
regimes, defined by “critical” shear stresses, denoted $\sigma_{c}$,
$\sigma_{s}$ and $\sigma_{f}$.

  * •

Regime I ($\sigma<\sigma_{c}$)

This regime corresponds to the primary response of the micellar solution to
exceedingly low shear stresses. For dilute systems, we expect a linear
increase of the shear stress as a function of the shear rate, associated with
a Newtonian behaviour while for semi-dilute systems, a slight decrease of the
viscosity as a function of the shear rate is usually observed, due to partial
alignment of the micelles by the flow [18, 20]. Unfortunately, reaching this
regime requires shear stresses that cannot be imposed by the rheometer, what
leads to the lack of experimental data points in Fig. 4. Note that in these
conditions, $\sigma_{c}$ is only roughly defined.

  * •

Regime II ($\sigma_{c}<\sigma<\sigma_{s}$)

This regime is characterized by a shear-thickening transition, the apparent
viscosity of the material increasing with the shear rate (see Fig. 4). This
transition is associated with a re-entrant behavior of the flow curve since
the shear rate first decreases and then increases with the shear stress in
this regime. Due to the re-entrant character of the transition, this regime
can only be observed at imposed stress. The upper boundary of this regime is
noted $\sigma_{s}$ and corresponds to the shear stress for which the apparent
viscosity reaches a maximum. Images of the sample in the shear-thickening
regime do not exihibit any changes compared to the situation at rest. The gap
of the TC cell appears homogeneous suggesting that the changes in the
structure of the fluid associated with the shear-thickening transition, occur
at sub-micronic scale.

  * •

Regime III ($\sigma_{s}<\sigma<\sigma_{f}$)

In this range of applied stresses, the flow becomes shear-thinning and direct
visualisations show that the gap of the TC cell remains homogeneous (see Fig.
6 (a-c)).

  * •

Regime IV ($\sigma>\sigma_{f}$)

This regime is characterized by a strong degree of shear-thinning associated
with the existence of a stress plateau, the onset of which is denoted by a
“critical” shear stress $\sigma_{f}$. The stress plateau generates a jump in
the shear rate resulting in a sharp decrease of the apparent viscosity. The
stress $\sigma_{f}$ is determined by increasing the shear stress step by step
with an appropriate sampling as illustrated in Fig. 5. The plateau is
characterized by a large jump in the shear rate at imposed stress, or
equivalently a strong increase of the shear rate versus time at fixed stress
(see inset in Fig. 5) and a large increase of the first normal stress
difference $N_{1}$, indicative of nonlinear viscolelastic properties of the
material under shear. Note that below or above $\sigma_{f}$, variations of the
shear rate are lower than $10$% over the duration of the measure ($1000$ s).
For the sample in Fig. 5, we estimate $\sigma_{f}=4.1\pm 0.3$ Pa. In this
regime, direct visualisations in the TC geometry reveal a drastic change in
the structure of the fluid (see Fig.6 (d-e)). A fraction of the sample becomes
slightly turbid leading to variations in the refraction index observable with
visible light. These observations suggest that, for $\sigma>\sigma_{f}$, a new
phase characterized by a length scale in the order of a few microns nucleates
in the gap of the TC device [35]. We observed that the nucleation occurs from
the edges of the cylinders and progressively extends along the inner cylinder,
the growth being favoured at the bottom of the cell which has a conic shape.
At fixed stress, the turbidity fluctuations observed on the pictures evolve in
time with a complex dynamics. The proportion of the induced phase increases
with the imposed shear stress, and for sufficiently high shear stresses, the
induced phase invades the whole gap (see Fig.6 (f-g)). This scenario is
reminiscent of the shear-banding transition even if there is no clear evidence
for an organisation into two bands separated by a well-defined interface [24].
By monitoring the shear-rate as a function of time, we noticed that the
appearance of the induced phase is correlated with the presence of a kink in
the $\dot{\gamma}(t)$ curve which appears sooner as the stress increases. This
kink is also observed in the cone-plate geometry for $\sigma>\sigma_{f}$ (See
inset in Figure 5) and is then the signature for the appearance of the induced
phase.

Figure 4: (a): Shear stress $\sigma$ and (b): apparent viscosity $\eta$ as a
function of the shear rate $\dot{\gamma}$ for a solution of CTAB ($2.55$ wt%)
and NaSal ($0.32$wt.%). The temperature is T=$20.00\pm 0.01^{\circ}$C. The
rheological protocol consists in stress sweep experiment: at each step, the
shear stress is kept fixed during $1000$ s. The resulting shear rate
$\dot{\gamma}(t)$ reaches a stationnary value after a time period that depends
on the applied shear stress. The flow curve is obtained by averaging the
measured shear rate over the last $100$ s of each step. $\sigma_{c}$,
$\sigma_{s}$ and $\sigma_{f}$ denotes characteristic shear stresses that
delimit different regimes in the flow curve. Figure 5: Determination of the
stress plateau value ($\sigma_{f}$). Stress $\sigma$ (red $+$) and first
normal stress difference $N_{1}$ (green $\times$) versus shear rate for
$2.55$wt.% CTAB and $0.32$wt.% NaSal (T=$20.00\pm 0.01^{\circ}$C). At each
step, the shear stress is imposed during $1000$ s and the measured shear rate
results from the average over the last $100$ s of each step. Inset: Time
evolution of the shear rate $\dot{\gamma}$ for a fresh sample at $\sigma=4.5$
Pa. The black arrow indicates the signature linked to the apparition of SIS.

Beyond this purely phenomenological description, our observations deserve
further comments. As mentioned above, similar complex succession of
rheological transitions in the flow curve have already been observed in dilute
micellar solutions of TTAA/NaSal [20, 19]. However, the direct observations
reported by the authors largely differ from the present study: Indeed, in the
shear-thickening regime (regime II), the fluid phase of regime I was found to
coexist with a viscous phase of gel-like shear-induced structures while in
regime III, the induced gel-like phase invaded the whole gap of the flow
geometry. Furthermore, the rheological response in regime III was also
different, the apparent viscosity being constant, while in our case the system
is shear-thinning. This behaviour was ascribed to the existence of a plug flow
with slip at the walls. Finally, above $\sigma_{f}$, fracture of the gel-like
phase followed by flow instabilities were observed.

Here we do not observe any phase coexistence in regime II. However, this does
not rule out the fact that the shear-thickening transition is associated with
a change of the structure of the system but rather suggests that this change
of the structure occurs at sub-micronic scale which prevents direct
visualisation. Our situation could correspond to the minimal scenario for the
shear-thickening transition in micellar systems, namely the uniaxial growth of
the micelles [1, 29, 27, 24]. On the other hand, we clearly observe a phase
coexistence in regime IV, above $\sigma_{f}$, the induced phase being
characterized by a large first-normal stress difference. Note that, at
equilibrium, the samples under investigation are only weakly entangled: their
concentration is associated with the very beginning of the semi-dilute regime,
in a “transition” range where the rheological behaviour is between pure shear-
thickening of dilute rod-like micellar systems, and pure shear-banding of
strongly entangled wormlike micelles [17, 24]. In this context, the shear-
banding-like behaviour observed in the present study is likely to correspond
to a transition from the state induced during the shear-thickening regime
(regime II) towards another induced “phase” structured at larger scale (regime
IV).

Note that in the literature on wormlike micelles, the shear-induced state
associated with the shear-thickening transition was called SIS for shear-
induced structure [28], or SIP for shear-induced phase [5] interchangeably.
Recent studies [37] have also used FISP to denote irreversible “flow-induced
structured phase” obtained in extensional flow. However, in the present paper,
we make no distinction between the expressions “shear-induced structures” and
“shear-induced phase” and in the following we will refer to SIS as the
reversible structures (or “phase”) induced at larger scale in regime IV.

Figure 6: Views of the gap of a Couette cell in the plane velocity
gradient/vorticity (noted $(\vec{\nabla}\vec{v},\vec{\nabla}\times\vec{v})$)
for different applied shear stresses. Each image is taken after a time $t\sim
1000$ s. The solution is composed of $2.55$wt.% CTAB and $0.32$wt.% NaSal. The
top and bottom sides correspond to the outer and inner wall respectively. For
$\sigma>\sigma_{f}=4$ Pa, structures appear near the inner wall. Arrows
indicate some turbidity fluctuations indicative of SIS with micronic
characteristic length scale.

##  5 Stress induced by gravity and development of shear-induced structures

As described in section 3, the film morphology depends on the film thickness
$h$. Our working hypothesis is that the transition between unstable to stable
morphologies along the fibre might be linked to the presence of the SIS. To
clarify and to go further on that point, we will focus on solutions in domain
$\cal{C}$. This choice is motivated by the fact that solutions which belong to
this domain present all three morphologies. We first study the influence of
the CTAB concentration. We use a set of chemical solutions for which CTAB
concentration is in the range $[1.8,3.2]$wt.% keeping
$[\textrm{NaSal}]=0.32$wt.% constant. For each solution, we keep increasing
the flow rate. The results are reported in figure 7 where we observe
successively unstable, stable and block morphologies as the film thickness
increases.

Figure 7: Film morphologies for different film thicknesses and various
surfactant concentrations ($[\textrm{NaSal}]=0.32$wt.%). The black line is a
fit of film thicknesses $h_{f}=\sigma_{f}/(\rho g)$ ($\square$) related to
stress plateau values. It separates unstable and stable flow regimes.

To explain this transition between stable and unstable morphologies, and since
the rheology is controlled by the applied shear stress, we compare the stress
plateau $\sigma_{f}$ extracted from the rheological curves to the
gravitational stress $\sigma_{g}$ applied on the film. This stress
$\sigma_{g}=\rho gh$ ranges between 2 and 8 Pa in our experiments and is then
comparable to the value of the stress plateau $\sigma_{f}$. We define the film
thickness $h_{f}$ related to the stress plateau $\sigma_{f}$ as
$h_{f}=\sigma_{f}/(\rho g)$.

Figure 7 illustrates the succession of film morphologies as a function of the
film thickness $h$ for different surfactant concentrations and a fixed salt
concentration. We also report, for each sample, the value
$h_{f}=\sigma_{f}/(\rho g)$ gathered from the rheological experiments.
Remarkably, the unstable-stable transition is found to occur as $h\simeq
h_{f}$. Thus, we can conclude that the condition $\sigma_{g}>\sigma_{f}$ is
necessary to stabilize the film. This suggests that the stabilization of the
film is connected to the presence of the SIS as $\sigma_{f}$ is the
characteristic stress value for the development of these structures.

Consequently, the SIS are able to be formed, before the Rayleigh-Plateau
instability acts, if $\sigma>\sigma_{f}$. This fixes the first condition to
stabilize the film. As we will see in the next section, this condition is
necessary but not sufficient.

##  6 Effect of the SIS elasticity on the instability growth rate

In this section, we study a second set of chemical solutions in order to
complete our understanding. The question we attempt to answer is: why do films
become suddenly always unstable if the concentrations are slightly changed
near the border between domains $\cal{A}$ and $\cal{C}$? To cross these two
domains, we work with a constant CTAB concentration
($[\textrm{CTAB}]=2.55$wt.%) while NaSal concentrations vary in the range
$[0.25,0.4]$wt.%. Figure 8 displays the film morphology as a function of the
thickness for different salt concentrations covering domains $\cal{A}$ and
$\cal{C}$ at fixed surfactant concentration. Based on rheological data, the
thickness of the film $\sigma_{f}/(\rho g)$ beyond which the SIS are likely to
develop is also reported. A good agreement is again observed in domain
$\cal{C}$ between the expected thickness computed from rheological results and
the effective thickness above which the films are stable. However, there is a
discrepancy for solutions in domain $\cal{A}$ since films thicker than $h_{f}$
are unstable. According to our analysis, the SIS still develop but they do not
stabilize the film.

Figure 8: Film morphologies for different film thicknesses and various salt
concentrations ($[\textrm{CTAB}]=2.55$wt.%). The black line is a fit of film
thicknesses $h_{f}=\sigma_{f}/(\rho g)$ related to stress plateau values.

To understand these observations, we focus on two effects: the surface tension
which is responsible for the instability and the bulk elasticity which tends
to slow down the film deformations.

It is worth noting that the constitutive relation describing the SIS is still
unknown. Since SIS present strong elastic properties, we attempt a description
of the SIS by a simple Kelvin-Voigt model allowing analytic calculations [34,
10]. The Kelvin-Voigt model takes into account the viscous dissipation at
short time scales and elasticity at long time scales. The stress tensor
$\sigma_{ij}$ is function of two parameters: viscosity $\eta_{0}$ and elastic
modulus $G_{0}$:

$\sigma_{ij}(t)=G_{0}\epsilon_{ij}+\eta_{0}\dot{\epsilon}_{ij}$ (1)

where $\epsilon_{ij}$ is the strain and the dot represents the time
derivative.

To obtain the dispersion relation for this constitutive equation, we perform a
linear stability calculation. The Fourier transform of the expression (1)
reduces the equation to

$\sigma_{ij}(\omega)=\eta(\omega)\dot{\epsilon}_{ij}(\omega)$ (2)

where $\omega$ is a complex frequency and with
$\eta(\omega)=\frac{iG_{0}}{\omega}+\eta_{0}$.

Experimentally, we observe that the film thicknesses $h(z)$ and the fibre
radius $R$ have the same order of magnitude. Under this consideration, we can
calculate the dispersion relation in our geometry[13].

In the cylindrical reference frame $(r,\theta,z)$ (See figure 1(a)), the
$z$-component of the momentum balance, taking into account the lubrication
approximation, becomes:

$\rho g-\partial_{z}p+\frac{\eta}{r}\partial_{r}(r\partial_{r}u)=0$ (3)

where $u$ is the fluid velocity along the fibre ($z$ direction). The boundary
conditions are: no-slip on the fibre ($u(r=R)=0$) and zero tangential stress
at the liquid-air interface ($\partial_{r}u(r=R+h)=0$). The pressure gradient
is caused by surface curvatures and it is expressed from Laplace’s law:
$\partial_{z}p=-\gamma\left(\frac{\partial_{z}h}{(R+h)^{2}}+\partial_{zzz}h\right)$
(assuming $\partial_{z}h\ll 1$). Solving this equation, the fluid velocity
profile is $u(r)=\frac{\partial_{z}p-\rho
g}{4\eta}\left[(r^{2}-R^{2})-2(R+h)^{2}\ln\left(\frac{r}{R}\right)\right]$.

Incompressibility leads to:

$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial z}=0$ (4)

where the flow rate per unit length is
$q=\frac{1}{2\pi(R+h)}\int_{R}^{R+h}u(r)2\pi rdr$.

To conduct a linear stability analysis, we develop the film thickness as
$h=h_{0}+h_{1}e^{i(kz-\omega t)}$ where $k$ is the real wave number. Thus, we
derive the following dispersion relation

$\omega=k\frac{\rho gR^{2}\psi(\alpha)}{16\eta(\omega)}-i\frac{\gamma
Rh_{0}^{3}\phi(\alpha)}{3\eta(\omega)(R+h_{0})}\left(k^{4}-\frac{k^{2}}{(R+h_{0})^{2}}\right)$
(5)

with $\alpha=h_{0}/R$. Two dimensionless functions reflect the geometry:
$\psi(x)=\frac{-x(2+x)(6+5x(2+x))+12(1+x)^{4}\ln(1+x)}{(1+x)^{2}}$ and
$\phi(x)=\frac{3(4(x+1)^{4}\ln(x+1)-x(x+2)(3x(x+2)+2))}{16x^{3}}$

The imaginary part of $\omega$ corresponds to the instability growth rate
noted $\Omega(k)$:

$\Omega(k)=\frac{-\gamma
Rh_{0}^{3}\phi(\alpha)}{3\eta_{0}(R+h_{0})}\left(k^{4}-\frac{k^{2}}{(R+h_{0})^{2}}\right)-\frac{G_{0}}{\eta_{0}}$
(6)

For $\Omega(k)<0$, the system is stable, whereas for $\Omega(k)>0$ the system
is unstable. If $G_{0}=0$, the solution for Newtonian fluids is recovered. The
evolution of the growth rate with $k$ is plotted in figure 9.

We observe that the bulk elasticity ($G_{0}$) plays a stabilizing effect on
the film as illustrated in figure 9: the marginal stability curve is shifted
to the negative $\Omega$. If $\textrm{max}(\Omega)=\Omega(k_{max})<0$ (blue
curve in figure 9), all modes are damped by the elasticity resulting in a
stable film.

Figure 9:  Growth rate of the instability $\Omega$ vs the wave number $k$ from
the Kelvin-Voigt model illustrating the stabilizing effect of the bulk
elasticity $G_{0}$. The case $G_{0}=0$ (in green) is the marginal stability
curve for Newtonian fluids (viscosity $\eta_{0}$). As $G_{0}$ increases, the
marginal curves (in blue) is shifted to the negative growth rate domain.

Note that, in this model, the wave number for the maximum of the growth rate
is $k_{max}=\frac{1}{\sqrt{2}(R+h_{0})}$ and is independent of the elastic
modulus.

Solving the condition $\Omega(k_{max})=0$ defines a critical elastic modulus
$G_{0}^{c}$:

$G_{0}^{c}(h)=\frac{\gamma h^{3}R\phi(h/R)}{12(R+h)^{5}}$ (7)

Experimentally, we can estimate $G_{0}^{c}$ from the border between domains
$\cal{A}$ and $\cal{C}$. From the data presented in figure 8, we can evaluate
for $[\textrm{CTAB}]\simeq 2.55\textrm{wt.}\%\simeq 70$mM,
$[\textrm{NaSal}]\simeq 0.33\textrm{wt.}\%\simeq 20$mM: $G_{0}\simeq
G_{0}^{c}(h_{f})=1.0$ Pa (with $h=h_{f}=0.4$ mm).

Thus the bulk elasticity of the SIS has to be sufficient to shift the curve of
the growth rate to the negative values, in order to stabilize the film. To
summarize, we have shown that the elastic shear induced structures are
developed in our flow if $\rho gh>\sigma_{f}$. If this condition is fulfilled,
we expect a stable film only if the bulk elasticity is greater than
$G_{0}^{c}$. As the result, we can explain the apparent discrepancy of the
unstable region in figure 8. From the transient zone to the unstable one (i.e.
increasing salt concentration), the bulk elasticity decreases such as
$G_{0}<G_{0}^{c}$.

The value $G_{0}^{c}=1$ Pa can be compared to oscillatory shear flow tests. As
suggested by the flow curve in figure 4, the linear regime could not be
reached with our rheometer. The first test consists in the measurement of the
elastic modulus versus time for a shear stress amplitude of $5\times 10^{-4}$
Pa at a frequency $f=1$Hz. The elastic modulus increases in time from $\sim
10^{-2}$ Pa to a few Pascal ($[\textrm{CTAB}]=2.55$ wt.% and
$[\textrm{NaSal}]=0.32$ wt.%). In a second test, a constant shear stress
$\sigma>\sigma_{f}$ is applied to the sample followed by an oscillating stress
of small amplitude. Even if the SIS has time to partially relax, such
experiments can provide a reasonnable order of magnitude of the moduli
($G^{\prime}$ and $G"$) of the SIS at high angular velocities. We found a
value for the elastic modulus around $1.5$ Pa, consistent with the value
$G_{0}^{c}=1$ Pa gathered from the flow along the fibre and considering that
the viscoelastic properties of the SIS can be modelled by a Kelvin-Voigt
model. Our results suggest that the elastic modulus of the solutions under
shear is larger than the one for the solution at rest and that the value
required to stabilize the flow could be deduced from the Kelvin-Voigt model.

##  7 Conclusion

In this paper, we studied the flow of CTAB/NaSal solutions on a vertical
fibre. Structure modifications occuring from the chemical composition and the
shear flow provides a large variety of flowing regimes. The flow on the fiber
can actually be unstable and similar to the flow of newtonian fluids: the
capillary driven Rayleigh-Plateau instability produces drops sliding on the
fibre. For another set of solutions, the film can stay uniform along the fiber
provided the film thickness satisfies $h>h_{f}=\sigma_{f}/(\rho g)$. From
rheo-optical measurements, we found that the condition for stabilizing the
flow is that the characteristic stress on the film should be larger than a
critical stress $\sigma_{f}$. This critical stress has been identified as the
onset of emergence of micronic structures induced by the shear-flow, i.e.,
shear-induced structures (SIS). A last flowing regime is observed for high
film thicknesses and in a given range of concentrations: in this last regime,
the film breaks in a series of gel-like blocks sliding along the fibre.

Our analysis suggests that the bulk elasticity of the SIS is responsible for
the decrease of the instability growth rate until inhibited the Rayleigh-
Plateau instability for a negative growth rate. Two conditions are required to
prevent the destabilization of the film: the presence of the SIS and a
sufficient elasticity of theses structures to inhibit the instability driven
by the surface tension. A linear analysis of stability assuming a Kelvin-Voigt
model for SIS, evidences a stable flow provided the elastic modulus of the SIS
is higher than a critical value. By comparing with experimental results on
systems which can go through an unstable to stable regime, we can estimate the
value of the elastic modulus value of these SIS. In the future, the inspection
of the micronic SIS can provide a better understanding of the micellar
structure inducing the bulk elasticity.

##  8 Acknowledgment

The authors thank Triangle de la Physique for the rheometer (Anton Paar, MCR
501) and Fédération Paris VI for the high-speed camera. Also, thanks to Jérôme
Delacotte, Christophe Clanet and Marina Moreno Luna for discussions.

## References

  * [1] J.-F Berret, R. Gamez-Corrales, Y. Séréro, F. Molino, and P. Lindner.  Shear-induced micellar growth in dilute surfactant solutions.  Europhys. Lett., 54:605–611, 2001.
  * [2] J.F Berret.  Rheology of wormlike micelles: equilibrium properties and shear banding transitions, pages 667–720.  Springer, 2006.
  * [3] A. Bhardwaj, E. Miller, and J. Rothstein.  Filament stretching and capillary breakup extensional rheometry measurements of viscoelastic wormlike micelle solutions.  Journal of Rheology, 51:693–719, 2007.
  * [4] P.P Bhat, S. Appathurai, M.T Harris, M. Pasquali, G.H McKinley, and O.A Basaran.  Formation of beads-on-a-string structures during break-up of viscoelastic filaments.  Nature Physics, 6:625–631, 2010.
  * [5] P. Boltenhagen, Y. Hu, E.F. Matthys, and D.J. Pine.  Inhomogeneous structure formation and shear-thickening in worm-like micellar solutions.  Europhys. Lett., 38:389 – 394, 1997.
  * [6] F. Boulogne, L. Pauchard, and F. Giorgiutti-Dauphiné.  Instability and morphology of polymer solutions coating a fiber.  Journal of Fluid Mechanics, 704:232, 2012.
  * [7] C.V Boys.  Soap Bubbles: Their Colors and Forces Which Mold Them.  Thomas Y. Crowell Company, 1959.
  * [8] M. E. Cates and S. M. Fielding.  Rheology of giant micelles.  Advances in Physics, 55:799–879, 2006.
  * [9] C. Clasen, J. Eggers, M.A Fontelos, J. LI, and G.H McKinley.  The beads-on-string structure of viscoelastic threads.  Journal of Fluid Mechanics, 556:283–308, 2006.
  * [10] F. Closa, F. Ziebert, and E. Raphaël.  Effects of in-plane elastic stress and normal external stress on viscoelastic thin film stability.  Mathematical Modelling of Natural Phenomena, 7:6–19, 2012.
  * [11] J. J. Cooper-White, R. C. Crooks, and D. V. Boger.  A drop impact study of worm-like viscoelastic surfactant solutions.  Colloids and Surfaces A: Physicochemical and Engineering Aspects, 210:105–123, 2002.
  * [12] B. Derjaguin.  On the thickness of the liquid film adhering to the walls of a vessel after emptying.  Acta Physicochim. URSS, 20:349–352, 1943.
  * [13] C. Duprat, C. Ruyer-Quil, and F. Giorgiutti-Dauphiné.  Spatial evolution of a film flowing down a fiber.  Physics of Fluids, 21:042109, 2009.
  * [14] C. Duprat, C. Ruyer-Quil, S. Kalliadasis, and F. Giorgiutti-Dauphiné.  Absolute and convective instabilities of a viscous film flowing down a vertical fiber.  Phys. Rev. Lett., 98:244502, 2007.
  * [15] J. Eggers and E. Villermaux.  Physics of liquid jets.  Reports on Progress in Physics, 71:036601, 2008.
  * [16] M. A. Fardin, B. Lasne, O. Cardoso, G. Grégoire, M. Argentina, J. P. Decruppe, and S. Lerouge.  Taylor-like vortices in shear-banding flow of giant micelles.  Phys. Rev. Lett., 103:028302, 2009.
  * [17] V. Hartmann and R. Cressely.  Influence of sodium salicylate on the rheological behaviour of an aqueous ctab solution.  Colloids and Surfaces A: Physicochemical and Engineering Aspects, 121:151–162, 1997.
  * [18] V. Hartmann and R. Cressely.  Simple salts effects on the characteristics of the shear thickening exhibited by an aqueous micellar solution of ctab/nasal.  Europhys. Lett., 40:691–696, 1997.
  * [19] Y. T. Hu, P. Boltenhagen, E. Matthys, and D. J. Pine.  Shear thickening in low-concentration solutions of wormlike micelles. ii. slip, fracture, and stability of the shear-induced phase.  Journal of Rheology, 42:1209–1226, 1998.
  * [20] Y. T. Hu, P. Boltenhagen, and D. J. Pine.  Shear thickening in low-concentration solutions of wormlike micelles. i. direct visualization of transient behavior and phase transitions.  Journal of Rheology, 42:1185–1208, 1998.
  * [21] J. N. Israelachvili.  Intermolecular and Surface Forces.  Academic Press, 2011.
  * [22] A. Jayaraman and A. Belmonte.  Oscillations of a solid sphere falling through a wormlike micellar fluid.  Phys. Rev. E, 67:065301, 2003.
  * [23] L. Landau and B. Levich.  Dragging of a liquid by a moving plate.  Acta Physicochim. URSS, 17:42–54, 1942.
  * [24] S. Lerouge and J.-F Berret.  Shear-induced transitions and instabilities in surfactant wormlike micelles.  Advances in Polymer Science, 230:1–71, 2010.
  * [25] S. Lerouge, M.A Fardin, M. Argentina, G. Grégoire, and O. Cardoso.  Interface dynamics in shear-banding flow of giant micelles.  Soft Matter, 4:1808–1819, 2008.
  * [26] Z. Lin, J. J. Cai, L. E. Scriven, and H. T. Davis.  Spherical-to-wormlike micelle transition in ctab solutions.  The Journal of Physical Chemistry, 98:5984–5993, 1994.
  * [27] R. Oda, V. Weber, P. Lindner, D.J Pine, E. Mendes, and F. Schloesser.  Time-resolved small-angle neutron scattering study of shear-thickening surfactant solutions after the cessation of flow.  Langmuir, 14:4859–4863, 2000.
  * [28] D. Ohlendorf, W. Interthal, and H. Hoffmann.  Surfactant systems for drag reduction: Physico-chemical properties and rheological behaviour.  Rheol. Acta, 25(5):468–486, 1986.
  * [29] B. Prötzl and J. Springer.  Light scattering experiments on shear induced structures of micellar solutions.  J. Colloid Interface Sci., 190:327–333, 1997.
  * [30] D. Quéré.  Thin films flowing on vertical fibers.  Europhysics Letters, 13:721, 1990.
  * [31] D. Quéré.  Fluid coating on a fiber.  Annual Review of Fluid Mechanics, 31:347–384, 1999.
  * [32] URK Rao, C. Manohar, B. S. Valaulikar, and R. M. Iyer.  Micellar chain model for the origin of the visoelasticity in dilute surfactant solutions.  The Journal of Physical Chemistry, 91:3286–3291, 1987.
  * [33] L. Rayleigh.  On the instability of jets.  Proceedings of the London Mathematical Society, s1-10:4–13, 1878\.
  * [34] S. A. Safran and J. Klein.  Surface of viscoelastic thin films.  J. Phys. II France, 3:749–757, 1993.
  * [35] Beth A. Schubert, Norman J. Wagner, Eric W. Kaler, and Srinivasa R. Raghavan.  Shear-induced phase separation in solutions of wormlike micelles.  Langmuir, 20(9):3564–3573, 2004.
  * [36] L. B. Smolka and A. Belmonte.  Drop pinch-off and filament dynamics of wormlike micellar fluids.  Journal of Non-Newtonian Fluid Mechanics, 115:1–25, 2003.
  * [37] Mukund Vasudevan, Eric Buse, Donglai Lu, Hare Krishna, Ramki Kalyanaraman, Amy Q Shen, Bamin Khomami, and Radhakrishna Sureshkumar.  Irreversible nanogel formation in surfactant solutions by microporous flow.  Nature Materials, 9(5):436–441, 2010.