2.1 Problem settings

We seek to describe the thin liquid film that is created when a perfectly wetting horizontal fibre is vertically pulled out of a liquid bath at a constant velocity $U$ (in the frame of the laboratory). The liquid under consideration is incompressible and Newtonian, of dynamic viscosity $\unicode[STIX]{x1D702}$ , density $\unicode[STIX]{x1D70C}$ and surface tension $\unicode[STIX]{x1D6FE}$ . As pictured in figure 1, the fibre is parallel to the $z$ -axis and invariant by translation along this axis. It is supposed to have a rectangular cross-section, of width $2H$ , such that the liquid film is pinned at $y=\pm H$ . The film is assumed to be symmetric with respect to the vertical $x$ -axis, so that only the half-film thickness $h(x,t)$ will be considered in the following.

Figure 1.

Sketches of the liquid film under consideration, showing the notation and in particular the locations $x_{0}$ and $L(t)$ where the boundary conditions are imposed. The initial configuration is presented in (a), where a static meniscus connects the fibre, at the initial position $L_{0}$ , to the liquid bath located at $x=0$ . The configuration in (b) represents the liquid film at a later time, during pulling at a constant velocity $U$ .

It is also assumed that the film is uniform in the lateral direction, i.e. perpendicular to the $xy$ -plane, such that edge effects are ruled out. During film pulling, the vertical and horizontal velocity fields, respectively denoted $u(x,y,t)$ and $v(x,y,t)$ and the pressure field $P(x,y,t)$ obey the following governing equations and boundary conditions:

1. (i) Navier–Stokes (momentum) equations

   (2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}(\unicode[STIX]{x2202}_{t}u+u\unicode[STIX]{x2202}_{x}u+v\unicode[STIX]{x2202}_{y}u)=-\unicode[STIX]{x2202}_{x}P+\unicode[STIX]{x1D702}(\unicode[STIX]{x2202}_{xx}u+\unicode[STIX]{x2202}_{yy}u)-\unicode[STIX]{x1D70C}g, & \displaystyle\end{eqnarray}$$

   (2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}(\unicode[STIX]{x2202}_{t}v+u\unicode[STIX]{x2202}_{x}v+v\unicode[STIX]{x2202}_{y}v)=-\unicode[STIX]{x2202}_{y}P+\unicode[STIX]{x1D702}(\unicode[STIX]{x2202}_{xx}v+\unicode[STIX]{x2202}_{yy}v), & \displaystyle\end{eqnarray}$$

2. (ii) continuity equation

   (2.2) $$\begin{eqnarray}\unicode[STIX]{x2202}_{x}u+\unicode[STIX]{x2202}_{y}v=0,\end{eqnarray}$$

3. (iii) symmetry conditions at $y=0$

   (2.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle v=0, & \displaystyle\end{eqnarray}$$

   (2.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{y}u=0, & \displaystyle\end{eqnarray}$$

4. (iv) kinematic condition at $y=h(x,t)$

   (2.4) $$\begin{eqnarray}v=\unicode[STIX]{x2202}_{t}h+u\unicode[STIX]{x2202}_{x}h,\end{eqnarray}$$

5. (v) tangential and normal stress balances at $y=h(x,t)$

   (2.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D702}}{1+(\unicode[STIX]{x2202}_{x}h)^{2}}[(1-(\unicode[STIX]{x2202}_{x}h)^{2})(\unicode[STIX]{x2202}_{y}u+\unicode[STIX]{x2202}_{x}v)-4\unicode[STIX]{x2202}_{x}h\unicode[STIX]{x2202}_{x}u]=0, & \displaystyle\end{eqnarray}$$

   (2.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{0}-P-\frac{2\unicode[STIX]{x1D702}}{1+(\unicode[STIX]{x2202}_{x}h)^{2}}[(1-(\unicode[STIX]{x2202}_{x}h)^{2})\unicode[STIX]{x2202}_{x}u+\unicode[STIX]{x2202}_{x}h(\unicode[STIX]{x2202}_{y}u+\unicode[STIX]{x2202}_{x}v)]=2\unicode[STIX]{x1D6FE}K, & \displaystyle\end{eqnarray}$$
   in which the continuity equation (2.2) has been used, and where $P_{0}$ is the atmospheric pressure and $K$ is the mean curvature of the interface expressed as
   (2.6) $$\begin{eqnarray}K(x,t)=\frac{1}{2}\frac{\unicode[STIX]{x2202}_{xx}h}{[1+(\unicode[STIX]{x2202}_{x}h)^{2}]^{3/2}}.\end{eqnarray}$$

The continuity equation (2.2) can then be integrated along the $y$ -coordinate, using the Leibniz integral rule and the boundary conditions (2.3a ) and (2.4), such that it yields the mass conservation equation,

where

is the average velocity in the film at a given height $x$ and time $t$ .

2.2 Lubrication approximation and velocity decomposition

We first pose that the characteristic lengths in the $x$ - and $y$ -directions are, respectively, the capillary length $\ell _{c}=\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}$ and the fibre half-width $H$ , such that the aspect ratio $\unicode[STIX]{x1D700}$ is defined as

The smallness of $\unicode[STIX]{x1D700}$ corresponds to small space modulations of the dependent variables inherent to the lubrication approximation.

Next, it is well established (see e.g. Howell Reference Howell1996) that when no shear is applied at the interfaces, i.e. the right-hand side of (2.5a ) is zero, the flow at leading order is purely extensional (i.e. a plug flow) and that the vertical velocity is uniform and equal the average velocity $\bar{u}$ . Therefore, the vertical velocity field is split into a plug flow contribution $\bar{u}(x,t)$ and a shear contribution $u_{1}(x,y,t)$ as

where the dominant contribution $\bar{u}$ scales as $U$ , while the shear contribution $u_{1}(x,y,t)$ is assumed to be of order $\unicode[STIX]{x1D700}^{2}U$ . Conservation imposes that

A similar velocity decomposition has been proposed by Schwartz & Roy (Reference Schwartz and Roy1999) aiming at modelling surfactant-stabilised liquid films. For this purpose, they rather imposed a no-slip condition $u_{1}(x,h(x,t),t)=0$ for the shear correction, such that the leading-order velocity contribution coincides with the interfacial velocity $u_{s}(x,t)$ , instead of $\bar{u}(x,t)$ in the present case. The justification of Schwartz and Roy’s approach relies in the fact that $u_{s}$ explicitly appears in the transport equation of surfactant at the interface, while such equation is absent in the case of pure liquids.

The continuity equation imposes the horizontal velocity $v$ to be of order $\unicode[STIX]{x1D700}U$ , the pressure $P$ is scaled by $\unicode[STIX]{x1D702}\,U/\ell _{c}$ and the time $t$ by $\ell _{c}/U$ . The system of equations (2.1)–(2.5), in which (2.9) and (2.10) are used, is then non-dimensionlised and truncated at $O(\unicode[STIX]{x1D700}^{2})$ . Making the resulting equations dimensional again yields the following approximated system:

1. (i) momentum equations

   (2.12a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}\,\unicode[STIX]{x2202}_{yy}u_{1}=\unicode[STIX]{x2202}_{x}P+\unicode[STIX]{x1D70C}g+\unicode[STIX]{x1D70C}\,(\unicode[STIX]{x2202}_{t}\bar{u}+\bar{u}\unicode[STIX]{x2202}_{x}\bar{u})-\unicode[STIX]{x1D702}\,\unicode[STIX]{x2202}_{xx}\bar{u}, & \displaystyle\end{eqnarray}$$

   (2.12b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{y}P=0, & \displaystyle\end{eqnarray}$$

2. (ii) continuity equation

   (2.13) $$\begin{eqnarray}\unicode[STIX]{x2202}_{x}\bar{u}+\unicode[STIX]{x2202}_{y}v=0,\end{eqnarray}$$

3. (iii) symmetry conditions at $y=0$

   (2.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle v=0, & \displaystyle\end{eqnarray}$$

   (2.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{y}u_{1}=0, & \displaystyle\end{eqnarray}$$

4. (iv) tangential and normal stress balances at $y=h(x,t)$

   (2.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{y}u_{1}=4\unicode[STIX]{x2202}_{x}h\,\unicode[STIX]{x2202}_{x}\bar{u}-\unicode[STIX]{x2202}_{x}v, & \displaystyle\end{eqnarray}$$

   (2.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle P=P_{0}-2\unicode[STIX]{x1D702}\,\unicode[STIX]{x2202}_{x}\bar{u}-2\unicode[STIX]{x1D6FE}K. & \displaystyle\end{eqnarray}$$

The above system is closed by the unchanged conservation equation (2.7). Note that we did not truncate at $O(\unicode[STIX]{x1D700}^{2})$ the expression (2.6) for the mean curvature, despite the higher-order correction at the denominator, as done for liquid bridges by Vincent, Duchemin & Le Dizès (Reference Vincent, Duchemin and Le Dizès2014) for example. This keeps the model compatible with the solution of the static meniscus near the bath, where the slope is not small anymore, and allows describing the entire film shape, from the bath up to the fibre, without having to use a matching procedure between the film region and the static meniscus.

Integrating (2.12b ) leads to a constant pressure in the horizontal direction, whose expression is then given by (2.15b ). In order now to account for van der Waals interactions leading to film rupture, we use the concept of disjoining pressure introduced by Derjaguin & Kusakov (Reference Derjaguin and Kusakov1936). These authors stated that the pressure in a thin film is modified by the molecular interactions as compared with the surrounding bulk phase such that (2.15b ) is rewritten as follows

where $\unicode[STIX]{x1D6F1}(x,t)$ is the disjoining pressure. Classical theory predicts that the attractive London–van der Waals interaction for a film of thickness $2h$ can be expressed as (Israelachvili Reference Israelachvili2011)

where $A_{H}>0$ is the non-retarded Hamaker constant. Note that even though intermolecular forces are bulk contributions, they are equivalently introduced as surface forces in the frame of the lubrication approximation, as thoroughly discussed by Ivanov (Reference Ivanov1988).

Finally, using (2.13) and (2.14a ), the horizontal velocity can be expressed as

and the shear component of the vertical velocity $u_{1}$ is obtained by integration of (2.12a ) using (2.11) and (2.14b ),

Substituting (2.18) and (2.19) into (2.15a ), while using (2.16) and (2.17), finally yields

The dimensional system of equations to be solved consists of equations (2.6), (2.7) and (2.20), for the three unknowns, $K$ , $h$ and $\bar{u}$ .

2.3 Non-dimensionalised problem

Applying the following transformations,

equations (2.7) and (2.20) become in dimensionless form

where the dimensionless mean curvature (2.6) is rewritten as

In addition to $\unicode[STIX]{x1D700}$ , the dimensionless numbers governing the above system of equations are the capillary number, the Weber number and the Hamaker number, respectively defined as

The first two terms in (2.22b ) account for inertia while the others account, respectively, for capillary pressure, gravity, intermolecular forces and extensional viscous stress.

The dimensional physico-chemical parameters we implement in the simulation are those of the V1000 silicone oil that will be used in the experiments presented in § 5, namely $\unicode[STIX]{x1D702}=1.00~\text{Pa}~\text{s}$ , $\unicode[STIX]{x1D70C}=970~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D6FE}=21.1~\text{mN}~\text{m}^{-1}$ and $A_{H}=4.4\times 10^{-20}~\text{J}$ , yielding a capillary length $\ell _{c}=1.49~\text{mm}$ . For a given liquid, the physical parameters that can be changed experimentally are the pulling velocity $U$ and the fibre half-width $H$ . In the simulation, these parameters will be explored in the ranges $0.001~\text{mm}~\text{s}^{-1}\leqslant U\leqslant 30~\text{mm}~\text{s}^{-1}$ and $0.1~\unicode[STIX]{x03BC}\text{m}\leqslant H\leqslant 1000~\unicode[STIX]{x03BC}\text{m}$ , respectively. Consequently, the dimensionless numbers defined in (2.24) will vary in the following ranges:

It is worth noting that the Hamaker number ${\mathcal{A}}$ can be expressed as a function of $\unicode[STIX]{x1D700}$ as

is called the dimensionless Hamaker constant and solely depends on the properties of the liquid. Similarly, the Weber number $We$ can be expressed as a function of the capillary number $Ca$ and of the liquid properties. Hence, for a given liquid, only $Ca$ and $\unicode[STIX]{x1D700}$ can be varied independently.

3.1 Initial solutions and boundary conditions

We consider the initial configuration pictured in figure 1(a), where the fibre is located at a (dimensionless) position $L_{0}$ above the surface of the liquid bath such that it forms a static liquid bridge. Heller (Reference Heller2008) has shown that for $L_{0}\leqslant 2$ , a solution exists, whose shape results from a balance between the capillary pressure gradient $\unicode[STIX]{x1D6FE}\unicode[STIX]{x2202}_{x}K$ and the hydrostatic pressure gradient $\unicode[STIX]{x1D70C}g$ . Integrating the corresponding dimensionless balance

with the boundary conditions $K_{stat}(0)=0$ , $h_{stat}^{\prime }(0)=-\infty$ and $h_{stat}(L_{0})=1$ , yields an analytical expression for the shape of the static meniscus,

which is valid in the range $0<L_{0}\leqslant 2$ .

The dimensionless initial solutions corresponding to the static meniscus are therefore

where we approximate the initial condition (3.3c ) on $\bar{u}$ using the value resulting from (2.22a ) in stationary regime.

At first sight, the system of equations (2.22)–(2.23) should require five boundary conditions: two on the thickness $h$ , two on the vertical velocity $\bar{u}$ and one on the mean curvature $K$ . However, the order of differentiation on $\bar{u}$ can by reduced by one by defining $\unicode[STIX]{x1D706}=\unicode[STIX]{x2202}_{x}h$ as an intermediate variable. Using the mass conservation $h\unicode[STIX]{x2202}_{x}\bar{u}=-\unicode[STIX]{x2202}_{t}h-\bar{u}\unicode[STIX]{x1D706}$ in equation (2.22b ), only first-order spatial derivative for the variables $h$ , $\unicode[STIX]{x1D706}$ , $K$ and $\bar{u}$ are left in the system. The four necessary boundary conditions are then imposed as follows.

A position $x_{0}$ is fixed in the static meniscus, close to the surface of the liquid bath $x=0$ such that $0<x_{0}<L_{0}$ , as sketched in figure 1. The following dimensionless boundary conditions are set at that location:

where the prime denotes the derivative with respect to the $x$ -coordinate. The remaining boundary conditions are imposed on the fibre, moving upwards at a constant velocity $U$ , and thus located at the dimensionless position $L(t)=L_{0}+t$ : We show in appendix A that the results are independent of the arbitrary position $x_{0}$ taken in the static meniscus and do not vary significantly with the initial position $L_{0}$ of the fibre, as long as $L_{0}\leqslant \sqrt{2}$ . These parameters will henceforth be set to $x_{0}=0.1$ and $L_{0}=\sqrt{2}$ , as justified in appendix A.

The system of partial differential equations (2.22)–(2.23), supplemented by the initial solutions (3.3) and boundary conditions (3.4)–(3.5), is solved using the direct solver MUMPS in COMSOL 5.2. Since the problem involves a moving boundary, the fibre which is lifted at a constant velocity to create the film, the domain geometry is changing with time. The mesh is thus deformed as computed using the arbitrary Lagrangian Eulerian (ALE) algorithm.

Figure 2.

The dimensionless break-up height $L_{0}^{\ast }$ of a static film, deduced from (3.6), is plotted as a function of the aspect ratio $\unicode[STIX]{x1D700}=H/\ell _{c}$ (solid line). The points correspond to the break-up height computed from our model when pulling at a capillary number $Ca=10^{-8}$ , i.e. in a quasi-static regime. The corresponding Weber number is $We\approx 3\times 10^{-18}$ and the Hamaker constant is set to $A_{H}=4.4\times 10^{-20}~\text{J}$ .

3.2 Static limit

We first check that the static limit is recovered for very small values of $Ca$ in (2.22). Thanks to the analytical expression of the thickness profile (3.2) in the static case, it can be shown (Heller Reference Heller2008) that the point of minimal thickness in the static film is located at $x=\sqrt{2}$ for $\sqrt{2}<L_{0}\leqslant 2$ . The (dimensionless) maximum height $L_{0}^{\ast }$ of a static film is then given by $h_{stat}(\sqrt{2})=0$ , hence the implicit analytical expression for $L_{0}^{\ast }$ :

In figure 2, we compare the maximum height $L_{0}^{\ast }$ of a static film given by (3.6) to the break-up height of a quasi-static film pulled at $Ca=10^{-8}$ using our model, for different fibre half-widths, i.e. different values of $\unicode[STIX]{x1D700}$ . The static limit is successfully recovered in our simulation.

3.3 Comparison with the two-dimensional case

We now compare the results of our lubrication model in the dynamic regime with those reported in Heller (Reference Heller2008), who computed the Stokes equations in a two-dimensional and deformable domain. Note that both inertia and van der Waals forces were neglected in Heller’s simulation, so we temporarily set $We=0$ and ${\mathcal{A}}=0$ in (2.22) for the sake of consistency.

Figure 3 compares the drainage dynamics of films pulled at various velocities for a fixed aspect ratio $\unicode[STIX]{x1D700}=0.01$ , using our model (solid lines with symbols) and the two-dimensional simulation results of Heller (Reference Heller2008) (dotted lines). The minimum of the film thickness $\text{min}(h)$ is plotted as a function of time, normalised by the time $t_{1\,\%}$ when the minimal thickness reaches 0.01, namely 1 % of its initial value. This normalisation allows us to compare the drainage dynamics obtained for different values of the capillary number.

Figure 3.

The minimum film thickness $\text{min}(h)$ is represented as a function of time $t$ , normalised by the time $t_{1\,\%}$ when $\text{min}(h)$ reaches 1 % of its initial value. For a given capillary number, the solid line with symbols is the drainage curve obtained from our lubrication approach, while the dotted line is the prediction of the two-dimensional simulations reported by Heller (Reference Heller2008). In both cases, the aspect ratio is fixed to $\unicode[STIX]{x1D700}=0.01$ and inertia and van der Waals forces are omitted ( $We=0$ and $A_{H}=0$ ).

The comparison presented in figure 3 shows that the lubrication approach is in good agreement with the corresponding two-dimensional simulations, especially at small capillary numbers $Ca\leqslant 10^{-4}$ . A small discrepancy (less than 10 %) is observed at capillary numbers $Ca\geqslant 10^{-2}$ , where the lubrication approximation predicts a slightly slower drainage than the two-dimensional simulations, but the overall shape of the drainage curve is preserved.

4.1 Thickness and velocity profiles

Typical film thickness profiles $h(x)$ are represented in figure 4 in (a) linear and (b) log scales, at different times during pulling. At $t=0$ , a static meniscus spans between the liquid bath at $x=0$ and the fibre, which is initially located at $L_{0}=\sqrt{2}$ and set into motion at a constant velocity $U$ ( $U=1~\text{mm}~\text{s}^{-1}$ in figure 4). As the fibre goes up, the liquid film grows in height and thins, as shown in figure 4(a,b). Once the film has become sufficiently thin, the van der Waals term ${\mathcal{A}}\,\unicode[STIX]{x2202}_{x}h/h^{3}$ comes into play in (2.22b ), as we will see in a more quantitative way in § 4.3. This term ultimately triggers the film break-up, which is characterised by a very rapid and localised drop of the film thickness, as demonstrated by the magenta curve at $t=2.0955$ in figure 4(b).

Figure 4(c) displays the spatial variation of the average velocity $\bar{u}$ in the film at different times during film pulling, for the same parameters as in figure 4(a,b). The initial velocity profile (black curve) differs slightly from the initial condition (3.3c ) and is numerically converged towards at time $t=0$ to satisfy (2.22)–(2.23). For $t>0.4$ , the velocity in the static meniscus ( $x\lesssim \sqrt{2}$ ) goes back to zero and the initial velocity profile is forgotten. At all times $t>0$ , the average velocity $\bar{u}$ is negative in the lower part of the film and positive in its upper part. Consequently, there is a location in the film where $\bar{u}=0$ , which is close to the minimum in the thickness profile. The liquid is expelled from this minimum with a flow rate that seems to diverge at the point of rupture (magenta curve, at $t=2.0955$ ). The flow in the vicinity of the point of minimal thickness is further studied in appendix B, where the streamlines are computed for various times during film pulling.

Figure 5.

The minimum thickness $\text{min}(h)$ is plotted as a function of time in dimensionless units, for various values of the pulling velocity $U$ (solid lines) and a fixed fibre half-width $H=50~\unicode[STIX]{x03BC}\text{m}$ ( $\unicode[STIX]{x1D700}=0.034$ ) and Hamaker constant $A_{H}=4.4\times 10^{-20}~\text{J}$ . For $U\gtrsim 0.2~\text{mm}~\text{s}^{-1}$ ( $Ca\gtrsim 0.01$ ), a ‘slow’ drainage regime is first observed, followed by a sharp decrease in thickness, corresponding to film rupture. The transition from the drainage to the rupture regime happens when the film reaches a critical thickness, which can be defined as the intersection between a power-law fit of the end of the drainage regime (dashed line) and the rupture regime (dotted lines).

4.2 Film drainage and rupture

We now follow the film thickness at the point where it is minimal as a function of time. The thinning dynamics at this point is displayed in figure 5 (solid lines) for different values of the pulling velocity $U$ . For velocities above $0.1~\text{mm}~\text{s}^{-1}$ , the thinning dynamics exhibits two distinct parts: a drainage regime, where the thickness gently decreases with time, and a rupture regime, where the thickness drops abruptly.

In practice, the simulation is stopped when the minimal thickness $\text{min}(h)$ reaches $10^{-6}$ , which corresponds to a subangstrom film thickness for the example displayed in figure 4 (where $H=50~\unicode[STIX]{x03BC}\text{m}$ ). The dimensionless film lifetime $\unicode[STIX]{x1D70F}$ is then defined as the time when $\text{min}(h)=10^{-6}$ and the corresponding dimensionless break-up height is given by $L^{\ast }=L_{0}+\unicode[STIX]{x1D70F}=\sqrt{2}+\unicode[STIX]{x1D70F}$ . Note that this definition does not depend on the cutoff value $10^{-6}$ as long as the rupture regime, where the slope is almost vertical, has been reached. Thus, the break-up height of the film is an observable, which will be studied in details in § 4.4. Note that the model also gives a prediction for the location of the puncture in the film, which we compare with experimental observations in appendix C.

Figure 6.

The dimensionless (respectively dimensional) critical thickness for rupture $h^{\ast }$ (respectively $H\times h^{\ast }$ ) is plotted as a function of the control parameters, one being varied at a time with the others fixed to the following values: $\unicode[STIX]{x1D700}=0.034$ ( $H=50~\unicode[STIX]{x03BC}\text{m}$ ),  $Ca=0.47$ ( $U=10~\text{mm}~\text{s}^{-1}$ ) and ${\mathcal{A}}^{\prime }=A_{H}/16\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}\ell _{c}^{2}=1.84\times 10^{-14}$ ( $A_{H}=4.4\times 10^{-20}~\text{J}$ ).

4.3 Critical thickness for rupture

The transition from the drainage regime to the rupture regime in figure 5 occurs around a critical film thickness, when the van der Waals forces come into play. We define this critical thickness for rupture, denoted $h^{\ast }$ , as the intersection between a power-law fit of the end of the drainage regime (dashed line in figure 5) and the rupture regime (dotted lines in figure 5). The variation of the critical thickness $h^{\ast }$ with the capillary number $Ca$ , the aspect ratio $\unicode[STIX]{x1D700}$ and the dimensionless Hamaker constant ${\mathcal{A}}^{\prime }=A_{H}/16\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}\ell _{c}^{2}$ are respectively displayed in figure 6(a,c,e). The dimensional counterparts of these data sets are shown in figure 6(b,d,f), where the dimensional critical thickness $H\times h^{\ast }$ is plotted as a function of the pulling velocity $U$ , the fibre half-width $H$ and the Hamaker constant $A_{H}$ .

Figure 6(a,b) show that the critical thickness $h^{\ast }$ decreases with the pulling velocity for capillary numbers in the range $0.01\lesssim Ca\lesssim 1$ where the drainage and rupture regimes are clearly distinguished (see figure 5). For $Ca\lesssim 0.01$ , the critical thickness is no longer defined in our approach since the drainage and rupture regimes merge. The decrease of $h^{\ast }$ with $Ca$ shows that entrainment is a stabilising factor for the film, in the sense that the destabilising effect of van der Waals forces is delayed, i.e. comes into play for thinner films.

Figure 6(c) indicates that the critical thickness also decreases with the aspect ratio  $\unicode[STIX]{x1D700}$ . This behaviour can be qualitatively understood from the fact that the scale of the van der Waals contribution in (2.22b ) is set by the Hamaker number ${\mathcal{A}}={\mathcal{A}}^{\prime }/\unicode[STIX]{x1D700}^{3}$ . For a given dimensionless Hamaker constant ${\mathcal{A}}^{\prime }$ , the larger the aspect ratio, the smaller the Hamaker number and so the smaller the thickness required for ${\mathcal{A}}\,\unicode[STIX]{x2202}_{x}h/h^{4}$ to be of leading order, hence the decrease of $h^{\ast }$ with $\unicode[STIX]{x1D700}$ . More quantitatively, the variation of the critical thickness compares favourably with the power law $h^{\ast }\propto \unicode[STIX]{x1D700}^{-0.96}$ for $\unicode[STIX]{x1D700}=10^{-2}{-}10^{-1}$ , as will be justified later. The critical thickness varies more slowly than $\unicode[STIX]{x1D700}^{-1}$ below $\unicode[STIX]{x1D700}\sim 10^{-1}$ , but faster than $\unicode[STIX]{x1D700}^{-1}$ for $\unicode[STIX]{x1D700}\gtrsim 10^{-1}$ , hence the non-monotonic behaviour of the dimensional critical thickness $H\times h^{\ast }$ as a function of the fibre half-width $H$ , as shown in figure 6(d). This change in behaviour for $\unicode[STIX]{x1D700}\gtrsim 10^{-1}$ is likely due to the $\unicode[STIX]{x1D700}^{2}$ term in the denominator of the mean curvature (2.23), which becomes non-negligible.

Finally, figure 6(e,f) display the variation of the critical thickness with the dimensionless Hamaker constant ${\mathcal{A}}^{\prime }=A_{H}/16\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}\ell _{c}^{2}$ , which is a property of the liquid. The parameters $Ca$ and $\unicode[STIX]{x1D700}$ remain fixed, while the Hamaker constant $A_{H}$ is varied in the range $10^{-22}{-}10^{-18}~\text{J}$ , all the other properties of the liquid being set to the values given in § 2.3. The critical thickness $h^{\ast }$ increases with the dimensionless Hamaker constant as a power law (solid line in figure 6 e) with an exponent $\unicode[STIX]{x1D6FC}\approx 0.32$ .

Such a power-law behaviour for the critical thickness as a function of the Hamaker constant had already been predicted by Vrij & Overbeek (Reference Vrij and Overbeek1968) for small horizontal liquid films. Their approach consisted in calculating the film lifetime by summing the time needed for the film to reach a certain thickness and the time required for a van der Waals-driven instability to develop. Under the assumption of flat and rigid liquid/air interfaces, they derived the critical thickness, which was found to vary with the Hamaker constant as $A_{H}^{2/7}$ ( $2/7\approx 0.29$ ), an exponent which is close to our value of 0.32. The difference probably stems from the flow in our continuously stretched film with stress-free interfaces, which is quite different from the Poiseuille flow expected for rigid interfaces.

We finally give arguments regarding how the power-law behaviour of the critical thickness $h^{\ast }$ versus ${\mathcal{A}}^{\prime }$ can be related to the variation of $h^{\ast }$ with $\unicode[STIX]{x1D700}$ . Figure 6(e) shows that the critical thickness follows $h^{\ast }=\unicode[STIX]{x1D6FD}{{\mathcal{A}}^{\prime }}^{\unicode[STIX]{x1D6FC}}$ , where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are a priori functions of $\unicode[STIX]{x1D700}$ and $Ca$ (for example, $\unicode[STIX]{x1D6FC}=0.32$ for $\unicode[STIX]{x1D700}=0.034$ and $Ca=0.47$ ). Considering that film rupture is essentially driven by the Hamaker number ${\mathcal{A}}={\mathcal{A}}^{\prime }/\unicode[STIX]{x1D700}^{3}$ , we can assume that, for a given capillary number $Ca$ , the critical thickness $h^{\ast }$ is essentially a function of ${\mathcal{A}}$ : $h^{\ast }=f({\mathcal{A}})$ . Still for a fixed capillary number, this is equivalent to saying that $h^{\ast }$ depends on $\unicode[STIX]{x1D700}$ only through ${\mathcal{A}}={\mathcal{A}}^{\prime }/\unicode[STIX]{x1D700}^{3}$ , i.e. $h^{\ast }=f({\mathcal{A}})=f({\mathcal{A}}^{\prime }/\unicode[STIX]{x1D700}^{3})=\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D700})\times {{\mathcal{A}}^{\prime }}^{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D700})}$ . This imposes that the exponent $\unicode[STIX]{x1D6FC}$ is independent of $\unicode[STIX]{x1D700}$ and that $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D700})\propto \unicode[STIX]{x1D700}^{-3\unicode[STIX]{x1D6FC}}$ , i.e. $h^{\ast }\propto \unicode[STIX]{x1D700}^{-0.96}$ for $Ca=0.47$ and ${\mathcal{A}}^{\prime }=1.84\times 10^{-14}$ . Figure 6(c) shows that the assumption that $h^{\ast }=f({\mathcal{A}})$ seems reasonable, but is not rigorously true for $\unicode[STIX]{x1D700}\lesssim 0.01$ and $\unicode[STIX]{x1D700}\gtrsim 0.1$ , since the data deviate from the $h^{\ast }\propto \unicode[STIX]{x1D700}^{-0.96}$ behaviour.

The variations of the critical thickness with $Ca$ and $\unicode[STIX]{x1D700}$ point to the fact that $h^{\ast }$ is not an intrinsic property of the liquid but also depends on the experimental conditions and flow dynamics. Numerous studies (see e.g. Manev & Nguyen Reference Manev and Nguyen2005) have been dedicated to the critical thickness of horizontal thin liquid films, from both the experimental and theoretical point of view. The results presented in figure 6(b,d) put into question the applicability of these static investigations to dynamic situations, such as foam generation for instance.

Figure 7.

The dimensionless film break-up height $L^{\ast }$ is plotted (a) as a function of the capillary number $Ca$ for a fixed fibre half-width $H=50~\unicode[STIX]{x03BC}\text{m}$ ( $\unicode[STIX]{x1D700}=0.034$ ) in log-linear and log-log scale (inset), (b) as a function of the aspect ratio $\unicode[STIX]{x1D700}$ for a fixed pulling velocity $U=10~\text{mm}~\text{s}^{-1}$ ( $Ca=0.47$ ) in log-linear and log-log scale (inset), and (c) as a function of the dimensionless Hamaker constant ${\mathcal{A}}^{\prime }=A_{H}/16\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}\ell _{c}^{2}$ for $H=50~\unicode[STIX]{x03BC}\text{m}$ and $U=10~\text{mm}~\text{s}^{-1}$ . The dotted lines represent the static limit given by (3.6).

4.4 Maximal height of the film

As demonstrated in §§ 4.1 and 4.2, the height $L^{\ast }$ reached by the film when the rupture occurs is well defined. In figure 7, we report this break-up height as a function of the capillary number $Ca$ (figure 7 a) and the aspect ratio $\unicode[STIX]{x1D700}$ (figure 7 b) for given liquid properties. For fixed pulling parameters, the effect of the dimensionless Hamaker constant ${\mathcal{A}}^{\prime }$ on the break-up height $L^{\ast }$ is displayed in figure 7(c).

For $Ca\lesssim 10^{-3}$ , the break-up height of the film is equal to the static limit given by (3.6) (dotted line in figure 7 a). Beyond this value, $L^{\ast }$ increases with the capillary number, which is expected since the creation of the film is driven by viscous entrainement. This increase approximately follows a power law of exponent 0.38, as shown in the inset of figure 7(a). Note that, for given liquid properties, the value of the exponent may a priori depend on the aspect ratio $\unicode[STIX]{x1D700}$ .

As displayed in figure 7(b), the break-up height $L^{\ast }$ is also an increasing function of the aspect ratio. This can be understood from the fact that the larger the aspect ratio $\unicode[STIX]{x1D700}$ , the smaller the Hamaker number ${\mathcal{A}}={\mathcal{A}}^{\prime }/\unicode[STIX]{x1D700}^{3}$ and the thinner the film when van der Waals forces come into play and trigger film rupture. The $\unicode[STIX]{x1D700}$ dependency of the break-up height seems to exhibit three different regimes. For $\unicode[STIX]{x1D700}\lesssim 10^{-3}$ , the static limit given by (3.6) (dotted line in figure 7 b) is asymptotically reached. As shown in the inset in figure 7(b), the break-up height $L^{\ast }$ then appears to follow a power-law increase of exponent $1/2$ up to $\unicode[STIX]{x1D700}\sim 3\times 10^{-2}$ . Beyond, it enters a third regime where the break-up height rises more slowly than $\unicode[STIX]{x1D700}^{1/2}$ . This is likely again a signature of the $\unicode[STIX]{x1D700}^{2}$ term in the denominator of the mean curvature (2.23), which becomes non-negligible for $\unicode[STIX]{x1D700}\gtrsim 10^{-1}$ .

Finally, the film break-up height decreases with the dimensionless Hamaker constant, due to the contribution of ${\mathcal{A}}^{\prime }$ to the Hamaker number ${\mathcal{A}}$ . The break-up height turns out to be less sensitive to the Hamaker constant, varied in the range $A_{H}=10^{-22}{-}10^{-18}~\text{J}$ , than to the fibre half-width $H$ or to the pulling velocity $U$ .

Note that solving the system without inertia, i.e. setting $We=0$ in equation (2.22b ), essentially yields the same results. Although it plays virtually no part within the range of parameters explored here, inertia was kept in all simulations since it was found to help the convergence of the numerical solver used in COMSOL.

Figure 8.

Sketch of the experimental set-up used to measure the maximal height of vertical liquid films. A container (1) filled with liquid is bound to a vertical motorised translation plate (2). By lowering the container at a constant velocity $U$ , a thin liquid film (3) is deposited on a fixed frame (4) gridded with fishing lines (5). The break-up of this film is detected using the force transducer (6) to which the frame is attached. The stars indicate the elements of the set-up which are mobile in the reference frame of the laboratory.

5.1 Experimental set-up and protocol

The liquid films are formed on a $2~\text{cm}\times 9~\text{cm}$ rigid acrylonitrile butadiene styrene (ABS) frame covered in cross-linked polyepoxide to avoid liquid penetration. As sketched in figure 8, the frame is gridded with one horizontal and two vertical nylon fishing lines of diameter $D=11$ , 57 or $175~\unicode[STIX]{x03BC}\text{m}$ . The spacing between the vertical fishing lines is equal to $15~\text{mm}$ . The frame is initially immersed in a liquid container, the horizontal fishing line being located at about $1~\text{mm}$ below the surface of the liquid bath. While the frame remains fixed during the whole experiment, the liquid container can be lowered at a constant velocity $0.01~\text{mm}~\text{s}^{-1}\leqslant U\leqslant 10~\text{mm}~\text{s}^{-1}$ using a motorised linear stage (Newport UTS 150 CC) coupled to a motion controller (Newport SMC 100 CC), hence creating a thin liquid film between the fishing lines.

The film lifetime $\unicode[STIX]{x1D70F}_{exp}$ is defined in the experiments as the difference between the time when the horizontal fishing line crosses the surface of the liquid bath and the time when the film breaks. This experimental definition would correspond to an initial meniscus height $L_{0,\mathit{dim}}=0$ in the simulations. The film rupture is detected thanks to the force sensor (HBM, 5 g) on which the frame is attached. When the film bursts, the signal measured by the force transducer undergoes a jump due to the disappearance of the surface tension contribution to the total vertical force exerted on the frame. When this jump is detected, the translation plate automatically stops and comes back to its initial position. This automated set-up allows for statistical measurements of the film lifetime $\unicode[STIX]{x1D70F}_{exp}$ or, equivalently, of the film break-up height $L_{exp}^{\ast }=U\unicode[STIX]{x1D70F}_{exp}$ .

Table 1.

Main physical properties of the silicone oils used in the experiments. The viscosity and surface tension are given at $25\,^{\circ }\text{C}$ and have been measured after the fractionation processes for the $V350$ and $V1000$ oils. The experimental errors are ${\lesssim}0.5\,\%$ for viscosity measurements and $\pm 0.1~\text{mN}~\text{m}^{-1}$ for surface tension measurements. The number average molar mass $M_{n}$ is deduced from the viscosity using the expression $\log \unicode[STIX]{x1D702}=1.00+0.0123\,M_{n}^{1/2}$ (Mark Reference Mark1999).

5.2 Finding a pure liquid

In our experiments, the liquid used to form the films is silicone oil (methyl-terminated polydimethylsiloxane, PDMS, purchased from either Sigma Aldrich or Bluestar silicones), which is often considered as a model Newtonian pure liquid. The physicochemical parameters of the different silicone oils tested in the experiments are summed up in table 1. The viscosities and surface tensions were respectively measured with a Physica MCR 300 rheometer and the rising bubble technique (Tracker, TECLIS), in the temperature range $T=18{-}25\,^{\circ }\text{C}$ . The Hamaker constant is expected to be the same for all the silicone oils used in this study and equal to $A_{H}=4.4\times 10^{-20}~\text{J}$ at $20\,^{\circ }\text{C}$ (Drummond & Chan Reference Drummond and Chan1997).

As the liquid viscosity is varied over more than two orders of magnitude, the other physicochemical parameters, i.e. density and surface tension (see table 1) as well as the Hamaker constant, remain virtually constant. This allows us to consider that, for a given pulling velocity $U$ and a given fibre half-width $H$ , changing the liquid will only affect the viscosity, hence the value of the capillary number $Ca$ , while leaving the other dimensionless parameters $\unicode[STIX]{x1D700}$ , $We$ and ${\mathcal{A}}$ unchanged.

The model we have built in § 2 is valid only for pure liquids, i.e. in the absence of tangential stress at the interfaces. The experimental difficulty is that any small surface tension gradient, due to temperature inhomogeneity or surface active agents for example, will invalidate the stress-free boundary condition at the interfaces (2.5a ). Because of their high surface tension, water surfaces are particularly prone to be contaminated by surface active impurities, which can lead to surface tension gradients. It has been shown that the drainage of a water film between two bubbles can be modelled using a no-slip boundary condition for surface tension gradients as small as $0.1~\text{mN}~\text{m}^{-2}$ (Yaminsky et al. Reference Yaminsky, Ohnishi, Vogler and Horn2010). This is the reason why we turned to silicone oil as a model pure liquid in the first place.

Commercially available silicone oils are actually mixtures of PDMS chains of different lengths (or equivalently of different molar masses), the distribution of which is peaked around the number average molar mass $M_{n}$ . The surface tension of silicone oil increases with the number average molar mass up to $M_{n}\sim 10^{4}~\text{g}~\text{mol}^{-1}$ , where the surface tension saturates at a constant value $\unicode[STIX]{x1D6FE}\approx 21~\text{mN}~\text{m}^{-1}$ (Mark Reference Mark1999). Since short PDMS chains (i.e. with a molar mass less than approximately $10^{4}~\text{g}~\text{mol}^{-1}$ ) have a lower surface tension than longer chains, any spatial heterogeneity in their concentration, either due to the creation of fresh interface or to evaporation, can generate surface tension gradients. This phenomenon was already observed by Bascom, Cottington & Singleterry (Reference Bascom, Cottington and Singleterry1964) to have a strong influence on the spreading of oil on solid surfaces. Similarly, we suspect that traces of short PDMS chains may be sufficient to create interfacial shear stress in our silicone oil films, thus resulting in their overstabilisation.

To avoid this problem, we use a basic fractionation protocol to get rid of the shortest chains in silicone oil. The commercial silicone oil is poured into a separator funnel with about three times as much acetone (in volume). The mixture is vigorously shaken for five minutes and left to decant for several hours. Short PDMS chains are soluble in acetone while longer chains are not, so when the phase separation is complete, the PDMS phase contains less short chains than initially. The PDMS phase is then extracted, left in an oven at $60\,^{\circ }\text{C}$ for at least 20 h and finally put in a vacuum chamber for at least 20 h to remove the remaining acetone.

Figure 9 shows the break-up height of $V350$ silicone oil films as a function of the pulling velocity for a non-fractionated and (up to three times) fractionated oils. While the data corresponding to the non-fractionated oil are scattered, the application of the fractionation protocol described above yields much more reproducible results. Moreover, fractionated oils lead, on average, to shorter films than the non-fractionated oil, supporting the hypothesis of surface tension gradients due to short PDMS chains. The lower the viscosity, the more short chains in the oil. Thus, the fractionation protocol had to be applied three times to the $V350$ oil, but only once to the $V1000$ oil to get reproducible results. No fractionation was needed for the most viscous oils ( $V10000$ , $V12500$ and $V60000$ ).

6.1 Influence of the pulling velocity

Using a nylon fibre of fixed diameter $D=175~\unicode[STIX]{x03BC}\text{m}$ , the influence of the pulling velocity $U$ on the film break-up height is first explored for silicone oils of various viscosities. The film break-up height is measured using the protocol described in § 5.1 and the experiment is repeated at least 20 times for a given velocity and a given liquid viscosity. In the results presented in figure 10, each point is thus an average over at least 20 points and the error bar represents the square root of the quadratic sum of the standard deviation and a systematic error of $\pm 0.5~\text{mm}$ on the position of the liquid bath surface. It can be noted that the standard deviation of the measurements increases with $Ca$ , which may be due to the finite acquisition frequency of the force sensor used to detect film rupture.

Data points were obtained with pulling velocities varying over three orders of magnitude and liquid viscosities varying over two orders of magnitude. They collapse onto a single master curve when plotted versus the capillary number $Ca=\unicode[STIX]{x1D702}\,U/\unicode[STIX]{x1D6FE}$ . Note that the data corresponding to the less viscous oil ( $V350$ ) lie slightly above the others for $Ca\gtrsim 0.1$ , perhaps because of residual surface tension gradients due to short PDMS chains that were not removed by the fractionation protocol.

The experimental data are in good agreement with our model with $2H=175~\unicode[STIX]{x03BC}\text{m}$ (solid line in figure 10) up to $Ca\sim 0.1$ . A small deviation from the theoretical prediction is observed at capillary numbers larger than 0.1, where the films last longer than expected from the model, as will be discussed in § 6.3.

Figure 9.

The experimental break-up height $L_{exp}^{\ast }$ of $V350$ silicone oil films is measured as a function of the pulling velocity $U$ with a nylon fibre of diameter $D=175~\unicode[STIX]{x03BC}\text{m}$ . The various symbols/colours show the data obtained after the application of $n$ fractionations ( $0\leqslant n\leqslant 3$ ) to the silicone oil taken from the bottle. The fractionation protocol, which aims at removing the shortest methyl-terminated polydimethylsiloxane chains, significantly improves the reproducibility of the measurements.

Figure 10.

The break-up height $L_{exp}^{\ast }$ of films made from silicone oils of various viscosities is measured as a function of the pulling velocity $U$ , using the same nylon fibre of diameter $D=175~\unicode[STIX]{x03BC}\text{m}$ . The film break-up height is non-dimensionalised by the capillary length $\ell _{c}=1.49~\text{mm}$ and displayed as a function of the capillary number $Ca=\unicode[STIX]{x1D702}\,U/\unicode[STIX]{x1D6FE}$ . The non-dimensionalised experimental data (symbols) collapse onto a single master curve, which is compared to the prediction of our model (solid line, see also figure 7 a) with $2H=175~\unicode[STIX]{x03BC}\text{m}$ .

Figure 11.

The break-up height $L_{exp}^{\ast }$ of films made from silicone oil is measured as a function of the pulling velocity $U$ , using nylon fibres of various diameters. The experimental data (symbols) are non-dimensionalised as in figure 10 and compared with the predictions of our model (solid lines), where $2H$ has been set equal to the fibre diameter $D$ . The data for $D=11$ and $175~\unicode[STIX]{x03BC}\text{m}$ were obtained using the $V1000$ silicone oil, while the experiments for $D=57~\unicode[STIX]{x03BC}\text{m}$ were carried out with the $V10000$ silicone oil.

6.2 Influence of the fibre diameter

The effect of the diameter $D$ of the horizontal fibre which supports the film is then investigated experimentally. The film break-up height is measured as a function of the capillary number for three different fibre diameters, $D=11$ , 57 and $175~\unicode[STIX]{x03BC}\text{m}$ and the results are shown in figure 11. As in figure 10, each data point is averaged over at least 20 measurements. The theoretical predictions computed from our model, setting $2H=11$ , 57 and $175~\unicode[STIX]{x03BC}\text{m}$ (i.e. $\unicode[STIX]{x1D700}=0.0037$ , 0.019 and 0.059, respectively), are also displayed in figure 11.

For all fibre diameters, the experimental data are in good agreement with the theoretical predictions for $Ca\leqslant 0.1$ . The deviation previously observed at higher capillary numbers for the thickest fibre ( $D=175~\unicode[STIX]{x03BC}\text{m}$ ) is reduced as the fibre diameter diminishes and the data for the thinnest fibres ( $D=57~\unicode[STIX]{x03BC}\text{m}$ and $D=11~\unicode[STIX]{x03BC}\text{m}$ ) stay in line with the model for the whole range of capillary numbers probed in the experiments.

Figure 12.

(a) Sketches of the boundary condition at the fibre in the model (top), assuming a rectangular cross-section of width $2H$ , and in the experiments (bottom), where the fibre has a circular cross-section of diameter $D$ , certainly coated with a liquid layer, leading to an effective fibre width $2H_{eff}>D$ . (b) The sets of experimental data displayed in figures 10 and 11 are replotted together (symbols) and compared with the predictions of the model (dashed lines) using an effective fibre $2H_{eff}=D+2h_{LL}$ to account for the coating of a liquid layer of thickness $h_{LL}$ around the fibre. The solid lines are again the predictions of the model assuming $2H=D$ .