# Stability of a dry patch in a viscous flowing film

Julien Sebilleau, Luc Lebon, Laurent Limat

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Julien Sebilleau, Luc Lebon, Laurent Limat. Stability of a dry patch in a viscous flowing film. The European Physical Journal. Special Topics, 2009, 166 (1), pp.139-142. 10.1140/epjst/e2009-00894-x. hal-00357824

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# Stability of a dry patch in a viscous flowing film

J. Sébilleauª, L. Lebon, and L. Limat

Laboratoire Matière et Systèmes Complexes, Université Paris Diderot, UMR 7057,
10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France

**Abstract.** We investigate the dewetting of liquid films flowing down an incline. At low flow rate we observe the formation of stationary dry patches edged with a liquid rim. Their shape can be predicted by a simple model in which the rim weight is balanced by surface tension. Above a critical flow rate per unit length Γc of typical scale Uclc (Uc capillary velocity, lc capillary length), these dry patches cannot remain stationary and are swept away. An improved model taking into account capillary effects linked to contact line curvature, hydrostatic pressure in the film and inertial effects predicts this loss of stability in good agreement with experiments for sufficiently high viscosity values.

## 1 Introduction

When a liquid (viscosity η, mass density ρ, surface tension γ) flows down an inclined substrate, under partial wetting conditions, dry patches can spontaneously open. Such dry patches must be avoided, or at least forced to heal, in many processes (heat exchangers, coating flows) and, thus, their shape and stability have motivated several studies [1–8]. The first reproducible experiments were performed only recently by Podgorski et al. [3–5] who proposed a simple model, consistent with their own data, for sufficiently high liquid viscosity [3]. In this approach, the equilibrium shape results from a balance between surface tension and the weight of the liquid rim edging the dry patch. After some calculations, the radius of curvature at the apex R (see fig. 1) is given by:

$$R = m \frac{(1 - \cos \theta)^4 l_c^2 U_c}{\theta - \sin \theta \cos \theta \Gamma \sin \alpha} \quad (1)$$

where Γ is the flow rate per unit length, α the plate inclination, Uc = γ/η the capillary velocity, lc = √γ/ρg the capillary length, θ the contact angle and m a numerical form factor (of order 0, 2 and 0, 3) whose precise value depends on the rim geometry and details of the flow within the rim. The law (1) is in good agreement with experiments, but this model fails to describe several observations. First, the patch shape has always a solution at any flow rate, while the experience shows that there is a critical flow rate above which any dry patches should disappears. Also, the dependance of R upon plate inclination presents discrepancies to the expected scaling R ~ 1/(Γsin α), when one looks to the low value of α [5]. Therefore we discuss here a more complex model, taking into account several effects initially neglected: capillary effects linked to contact line curvature, hydrostatic pressure in the film, and inertial effects. We compare its predictions to both available data obtained by Porgorski on a 20 cP oil and new measurements performed by us for another liquid viscosity (10 cP). Our model is in good agreement with these experiments and gives reasonable theoretical predictions of the critical flow rate, at least in this range of viscosity.

<footnote>ª e-mail: sebilleau@pmmh.espci.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Structure of the flow and the dry patch. (a) Schematic of the flow; (b) section of the liquid rim; (c) mass conservation.</figcaption></figure>

## 2 Improved model

The notations, similar to those used in [3] are presented in fig. 1. Our modelisation is based on three basic ingredients:

1. We assume that the liquid flow structure along the rim looks like a Poiseuille flow with a zero shear stress at the free surface. We also assume that the rim cross section is circular. This leads to a mean velocity (averaged over the cross section) just identical to [3–5]:

$$ \bar{U} = m \frac{(1 - \cos \theta)^2}{\theta - \sin \theta \cos \theta} \frac{U_c S(\psi)}{l_c^2} \sin \alpha \cos \psi \quad (2) $$

where $S(\psi)$ is the cross-section of the rim.

2. The mechanical equilibrium of the rim, in a direction normal to contact line, leads to:

$$ \begin{aligned} \gamma(1 - \cos \theta) = & \rho g S(\psi) \sin \alpha \sin \psi \left( 1 - \frac{S^{1/2}}{R(\psi)} \frac{\sin \theta}{(\theta - \sin \theta \cos \theta)^{1/2}} \right) \\ & + \gamma \frac{S^{1/2}}{R(\psi)} (\theta - \sin \theta \cos \theta)^{1/2} + \frac{\rho g}{2} \cos \alpha h_f^2 + \frac{2}{15} \left( \frac{g \sin \alpha \sin \psi}{\nu} \right)^2 h_f^5 \end{aligned} \quad (3) $$

in which $h_f$ is the mean film thickness and the rim weight $\rho g S \sin \alpha \sin \psi$ is now completed with contact line curvature effects, hydrostatic pressure term proportional to $\rho g \cos \alpha$ and downward momentum flux carried by the film. This last term was missing up to now in any of the approaches proposed to improve Podgorski’s model [4,5].

3. The mass conservation imposes:

$$ \Gamma x = S \bar{U} \quad (4) $$

After some calculations, the dimensionless radius of curvature $R^* = R/l_c$ and flow rate $\Gamma^* = \Gamma/U_c l_c$ are linked by an implicit relation:

$$ \begin{aligned} 1 = & h(\theta) R^{*1/2} (\Gamma^* \sin \alpha)^{1/2} - \frac{\sin \theta (1 - \cos \theta)^{1/2}}{(\theta - \sin \theta \cos \theta)^{1/2}} h(\theta)^{3/2} \Gamma^{*3/4} R^{*-1/4} \sin \alpha^{1/4} \\ & + \frac{(\theta - \sin \theta \cos \theta)^{1/2}}{\sin \alpha^{1/4} (1 - \cos \theta)^{1/2}} h(\theta)^{1/2} R^{*-3/4} \Gamma^{*1/4} \\ & + \frac{3^{2/3}}{1 - \cos \theta} \left( \cos \alpha \left( \frac{\Gamma^*}{\sin \alpha} \right)^{2/3} + \frac{2}{5} \frac{\rho^{1/2} \gamma^{3/2}}{\eta^2 g^{1/2}} \left( \frac{\Gamma^*}{\sin \alpha} \right)^{5/3} \right) \end{aligned} \quad (5) $$

where:

$$ h(\theta) = \frac{(\theta - \sin \theta \cos \theta)^{1/2}}{m^{1/2}(1 - \cos \theta)^2}. $$

Its numerical resolution is shown on fig. 2 for several plate inclinations $\alpha$ ($\theta = 52^\circ$ and $m = 0, 23$). This model predicts two possible solutions for the radius of curvature $R^*$ for a given flow
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** (a) Improved model for the evolution of the radius of curvature at the apex with the flow rate; (b) stability analysis for $\alpha = 90^\circ$. The bold line corresponds to initial model of [3].</figcaption></figure>

rate $\Gamma^*$, the smallest being unstable, and the other scaling roughly as $1/(\Gamma \sin \alpha)$ in agreement with [3], but with a shift that depends on the plate inclination. To establish the stability of the solutions, we have considered a dry patch with an out of equilibrium radius of curvature and have evaluated if the extra force necessary to keep mechanical equilibrium was tending to open or to close the patch. A typical result of this analysis is presented figure 2(b). Finally, let us note the main result observed on figure 2: there exists a critical flow rate above which no stationary dry patch can exist, both the unstable and stable solutions disappearing at the same time.

## 3 Comparison with experiments

We have compared our model with experimental results performed on two silicone oils (PDMS): Rhodorsil 47V20 ($\nu = 18, 25 \text{ mm}^2 \cdot \text{s}^{-1}$, $\rho = 0,95 \text{ g} \cdot \text{cm}^{-3}$) and Huber oil ($\nu = 11, 33 \text{ mm}^2 \cdot \text{s}^{-1}$, $\rho = 1,05 \text{ g} \cdot \text{cm}^{-3}$). These oils have an advancing contact angle of $\theta_a \approx 52^\circ$ and a receding

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Comparison between the improved model and experiments for the evolution of the radius of curvature at the apex. (a) Rhodorsil 47V20 oil; (b) Huber oil.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Comparison between the theoretical and experimental critical flow rate. (a) Rhodorsil 47V20 oil; (b) Huber oil.</figcaption></figure>

contact angle $\theta_r \simeq 45^\circ$ on the partial wetting area (of typical size 20 cm × 20 cm) of a glass plate coated with a fluoroploymer (FC725) (for more detail see [3]). The radius of curvature (measured here for progressively increasing flow rate) is in very good agreement with the model provided that one identifies $\theta$ to the advancing contact angle, and that one considers sufficiently inclined plates (see figure 3). We can also open dry patches at a given flow rate. The critical flow rate is then the lower flow rate at which any dry patches disappears. The theoretical and experimental critical flow rates (again defined here for progressively increasing flow rates) are also compared for both oils on figure 4, and a good agreement is observed if one considers the difficulty of such wetting experiments involving a continuous flow. Note that the experimental values of the critical flow rate are systematically smaller than their theoretical equivalent, which corresponds to the fact that the dry patch becomes unstable before the two stable and unstable solutions meet, typically for a critical value of the apex curvature of order $3l_c$. This important result remains to be explained.

## 4 Conclusion

We have shown that though the simple model derived initially by Podgorski et al. (balance between rim weight and surface tension effects), gives the right tendencies and good orders of magnitude for the evolution of the dry patch curvature at its apex, it is necessary to take into account more subtle effects (contact line curvature, hydrostatic and inertia effects) to capture the existence of a critical flow rate, the detail of the dependence upon plate slope and the loss of stability of the solutions. Our improved model is in good agreement with our data in the range of viscosity $\eta = 10–20 cP$, and should presumably hold also for higher viscosities. For lower viscosity values, preliminary experiments and calculations, suggest that the flow along the rim crosses over to an inertia dominated flow. A new model describing this in more details is under way.

## References

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4. T. Podgorski, Thèse de doctorat, Université Paris 6, 2000

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