for as many surfactants as the free interface, the lower limit $L_{out}(t) = \frac{1}{2}L_{in}$ to the case where the flux of surfactants from the meniscus is zero. All this is detailed in subsection 3.3.

**In our range of measurement, all velocities and chemistries are on the lower bound, suggesting that the flux of surfactants from the meniscus is null in all cases: $j_m = 0$**.

This is the key information we needed to further advance on the theoretical descriptions of flows in sheared foams. This is also the main experimental result of this thesis work.

### Important remarks

* Our result is verified in quite a narrow range of velocities of deformations, from $V_1 = 10$ to $50$ mm/s, corresponding to strain rates of $\dot{\epsilon} = 2 - 10$ Hz given our geometry. At lower strain rates, we lose the invariance by translation along $z$ before being able to measure $L_{in}$ and $L_{out}$ simultaneously. Higher strain rates cannot achieve reliable measurements, either because the motors are limited at $V_1 = 50$ mm/s or because smaller amplitudes of deformations make it impossible to measure $L_{in}$ and $L_{out}$.

* This result is heavily chemistry dependent to the best of our knowledge, as all hidden time variables allowing/excluding the existence of a $j_m$ are adsorption, desorption or diffusion times of surfactants. Notably, even changing the direction of deformation, and thus the sign of the potential $j_m$, could lead to very different results under the same deformation conditions. Adsoption and desorption dynamics are known to be very different. This is a bit explored in section 3.10, where we look at other deformations, unfortunately without achieving measuring the interface balance as we did here.

* The balance computed with the corrected $\epsilon_{1,b}$ shown in Fig.3.39B is a bit below the lower bound of our model. This is puzzling, as it would suggest a $j_m$ of the opposite sign (from the interface to the meniscus in this case, which is excluded). It might be caused by the fact that we neglect the exchanges in the dynamic menisci, the $\Delta\gamma^{in/out}$ of Fig.3.5 which have been neglected and which correspond both to a $\Delta\Gamma^{in/out}$ not taken into account in the surfactant balance. This assumption is more discussed in [4], and could explain this discrepancy.

## 3.10 Other deformations

### 3.10.1 “Push 1”: Potential influence of gravity

In the “Push 1” experiments the geometry of deformation remains exactly the same as in “Push 3” but the roles of the different films are interchanged so that we look at the same system with the direction of gravity changed. The protocol is shown in Fig.3.40. As discussed in subsections 0.2.4 and 0.3.5, what we expect gravity to change here is:

* Giving different initial thicknesses for the compressed/stretched films. Figs.3.6.5 D-F show how different are the films whose roles have been interchanged between “Push3” and “Push1”.
<figure><img src="image_105.png" /><figcaption>Figure 3.40: Protocol for the “Push 1” experiment. The prepared film is now film 1, where we photobleach 10 dots. The angle followed during the main deformation becomes $\theta_1(t)$. This protocol corresponds exactly to the “Push 3” protocol with interchanged roles for the films to probe the effect of gravity.</figcaption></figure>

* Redistribute pieces of films as we deform them, the thicker ones being advected down, the thinner one advected up.

This second point is quite important, as we want to prepare a thick Frankel film in the plane of film 1. Stretching a vertical film means having a Frankel film at both the free meniscus and the motor at the top, and this last one, thicker than the central part, is not in a stable position with respect to gravity (cf. 0.3.5). Thus, the initial recirculations in film 1 induced by the preparation step excluded the least mobile interfaces (i.e. with dodecanol), because the top Frankel film falls slower in the form of chimneys, making the prepared state not reproducible. Without dodecanol, the top thick patches are evacuated faster, seemingly along the solid frame, and a reproducible prepared state is possible. The only change compared to the “Push 3” protocol is a shift of 0.5 s of the preparation time to $t_0 = 28.5$ s.

Besides, we managed to photobleach dots in the prepared film, but tracking them and the edge of the prepared film throughout the main deformation at $t_1$ was a problem, leading to a big experimental noise hindering the measurement of $L_{in}(t)$.

For the sake of measuring the tensions however, the measurement remained possible by tracking the angle between film 2 and 3 in this case. We plot in Fig.3.41 the film tension ratio between the compressed film and the stretched film for both “Push 3” (in blue, $\sigma_3/\sigma_1$) and “Push 1” (in red, $\sigma_1/\sigma_3$). The chemistry is with [DOH] = 0 and a velocity of deformation $V_1 = 35$ mm/s. A few observations:

* The preparation of film 1 in “Push 1”, although identical to the preparation of film 3 in “Push 3”, has a slightly lower amplitude.

* “Push 1” has larger initial slips in $t_0 = 28.5$ s and $t_1 = 32$ s.

* The measurement is less reproducible, leading to a bigger experimental error.

* The slopes in time of the ratios are the same past the slip in $t_1 = 32$ s.

* The tension ratio relaxes towards 1 after the deformation in the same amount of time.
<figure><img src="image_105.png" /><figcaption>Figure 3.40: Protocol for the “Push 1” experiment. The prepared film is now film 1, where we photobleach 10 dots. The angle followed during the main deformation becomes $\theta_1(t)$. This protocol corresponds exactly to the “Push 3” protocol with interchanged roles for the films to probe the effect of gravity.</figcaption></figure>

* Redistribute pieces of films as we deform them, the thicker ones being advected down, the thinner one advected up.

This second point is quite important, as we want to prepare a thick Frankel film in the plane of film 1. Stretching a vertical film means having a Frankel film at both the free meniscus and the motor at the top, and this last one, thicker than the central part, is not in a stable position with respect to gravity (cf. 0.3.5). Thus, the initial recirculations in film 1 induced by the preparation step excluded the least mobile interfaces (i.e. with dodecanol), because the top Frankel film falls slower in the form of chimneys, making the prepared state not reproducible. Without dodecanol, the top thick patches are evacuated faster, seemingly along the solid frame, and a reproducible prepared state is possible. The only change compared to the “Push 3” protocol is a shift of 0.5 s of the preparation time to $t_0 = 28.5$ s.

Besides, we managed to photobleach dots in the prepared film, but tracking them and the edge of the prepared film throughout the main deformation at $t_1$ was a problem, leading to a big experimental noise hindering the measurement of $L_{in}(t)$.

For the sake of measuring the tensions however, the measurement remained possible by tracking the angle between film 2 and 3 in this case. We plot in Fig.3.41 the film tension ratio between the compressed film and the stretched film for both “Push 3” (in blue, $\sigma_3/\sigma_1$) and “Push 1” (in red, $\sigma_1/\sigma_3$). The chemistry is with [DOH] = 0 and a velocity of deformation $V_1 = 35$ mm/s. A few observations:

* The preparation of film 1 in “Push 1”, although identical to the preparation of film 3 in “Push 3”, has a slightly lower amplitude.

* “Push 1” has larger initial slips in $t_0 = 28.5$ s and $t_1 = 32$ s.

* The measurement is less reproducible, leading to a bigger experimental error.

* The slopes in time of the ratios are the same past the slip in $t_1 = 32$ s.

* The tension ratio relaxes towards 1 after the deformation in the same amount of time.
<figure><img src="image_106.png" /><figcaption>Figure 3.41: Comparison of the tension ratio in the “Push 3” (Blue) and “Push 1” (Red) experiments for the same chemistry [DOH] = 0 and velocity of main deformation $V_1$ = 35 mm/s. The tension ratio is $\sigma_3/\sigma_1$ for “Push 3” and $\sigma_1/\sigma_3$ for “Push 1”. The inset is a zoom on the main deformation at $t_1$ = 32 s.</figcaption></figure>

The main conclusion of this comparison is that gravity does not seem to play a major role for the rheology of our elementary foam, although its centimetric dimensions raised the question. However, it poses a lot of major experimental problems for the sake of tracking the patches of exchanged films and we cannot conclude on the robustness of our quantification of the interface exchanges under changing gravity.

## 3.10.2 “Pull 1”: Same symmetry, opposite directions for the imposed velocities

The “Pull 1” experiment aims at looking at the interface exchanges with the same symmetry as “Push 3”, but with minus signs on the imposed velocities. To do so, we designed the protocol detailed in Fig. 3.42. The main fundamental difference with “Push 3” is that the preparation does not respect the axial symmetry of the main deformation as it still bears on the compressed film 3.

We thus manage to photobleach in film 3 as we previously did for “Push 3”, and to track simultaneously the columns of dots and the prepared Frankel film as the film is compressed. A zoom on the region of film 3 neighbouring the free meniscus right before the deformation is shown
<figure><img src="image_106.png" /></figure>

Figure 3.41: Comparison of the tension ratio in the “Push 3” (Blue) and “Push 1” (Red) experiments for the same chemistry [DOH] = 0 and velocity of main deformation $V_1$ = 35 mm/s. The tension ratio is $\sigma_3/\sigma_1$ for “Push 3” and $\sigma_1/\sigma_3$ for “Push 1”. The inset is a zoom on the main deformation at $t_1$ = 32 s.

The main conclusion of this comparison is that gravity does not seem to play a major role for the rheology of our elementary foam, although its centimetric dimensions raised the question. However, it poses a lot of major experimental problems for the sake of tracking the patches of exchanged films and we cannot conclude on the robustness of our quantification of the interface exchanges under changing gravity.

## 3.10.2 “Pull 1”: Same symmetry, opposite directions for the imposed velocities

The “Pull 1” experiment aims at looking at the interface exchanges with the same symmetry as “Push 3”, but with minus signs on the imposed velocities. To do so, we designed the protocol detailed in Fig. 3.42. The main fundamental difference with “Push 3” is that the preparation does not respect the axial symmetry of the main deformation as it still bears on the compressed film 3.

We thus manage to photobleach in film 3 as we previously did for “Push 3”, and to track simultaneously the columns of dots and the prepared Frankel film as the film is compressed. A zoom on the region of film 3 neighbouring the free meniscus right before the deformation is shown
<figure><img src="image_107.png" /><figcaption>Figure 3.42: Protocol for the “Pull 1” experiment. The prepared film is film 3, where we photobleach 16 dots. Compared to “Push 3”, we only swap the direction of deformation film 2. The relevant angle to track $\theta$ is the semiangle between film 2 and 3.</figcaption></figure>

in Fig.3.43A with the arrows showing the direction of deformations for each film. Fig.3.43B shows the raw data of the distances between the matching coloured lines in A and the edge of the free meniscus as the film is compressed. Also, we show the length of outgoing Frankel film $L_{Fr,1}$ detected in film 1. The repeatability of these experiments is limited at longer times by something that we did not face with “Push 3”: Marginal regeneration. In Fig.3.43C, we show how this phenomenon takes place, with the semicircular thin patched going out of the free meniscus. For more details about it, cf. subsection 0.3.4.

**Actually, it is retrospectively surprising that no marginal regeneration takes place in “Push 3”! This even constitutes a result.** The main difference in the case of “Pull 1” is that the compressed film has one frustrated interface and one free, whereas for “Push 3” both are free to slide on the free meniscus. This requires more investigation.

In the meantime, marginal regeneration is a source of low reproducibility for our experiments, and “Pull 1” yields noisy deformations $\epsilon_3$ for the compressed film, as shown in Fig.3.44, inducing an even bigger error on the quantity of ingoing interface $L_{in}$ which we did not manage to compute. Besides, computing the ratio of tension becomes tricky here, as the preparation step breaks the symmetry between film 2 and 3 compared to the main deformation. This would require to compute separately $\sigma_2$ and $\sigma_3$ by computing another angle in the problem, which is achievable but not done yet.

This is as far as we could go on the “Pull 1” experiments on that date, and more refined specific data processing could lead to better computations of the deformation and significant quantities of exchanged interface. For now, the $L_{in}$'s I tried to compute were so noisy that I do not dare showing them in this manuscript.
<figure><img src="image_107.png" /><figcaption>Figure 3.42: Protocol for the “Pull 1” experiment. The prepared film is film 3, where we photobleach 16 dots. Compared to “Push 3”, we only swap the direction of deformation film 2. The relevant angle to track $\theta$ is the semiangle between film 2 and 3.</figcaption></figure>

in Fig.3.43A with the arrows showing the direction of deformations for each film. Fig.3.43B shows the raw data of the distances between the matching coloured lines in A and the edge of the free meniscus as the film is compressed. Also, we show the length of outgoing Frankel film $L_{Fr,1}$ detected in film 1. The repeatability of these experiments is limited at longer times by something that we did not face with “Push 3”: Marginal regeneration. In Fig.3.43C, we show how this phenomenon takes place, with the semicircular thin patched going out of the free meniscus. For more details about it, cf. subsection 0.3.4.

**Actually, it is retrospectively surprising that no marginal regeneration takes place in “Push 3”! This even constitutes a result.** The main difference in the case of “Pull 1” is that the compressed film has one frustrated interface and one free, whereas for “Push 3” both are free to slide on the free meniscus. This requires more investigation.

In the meantime, marginal regeneration is a source of low reproducibility for our experiments, and “Pull 1” yields noisy deformations $\epsilon_3$ for the compressed film, as shown in Fig.3.44, inducing an even bigger error on the quantity of ingoing interface $L_{in}$ which we did not manage to compute. Besides, computing the ratio of tension becomes tricky here, as the preparation step breaks the symmetry between film 2 and 3 compared to the main deformation. This would require to compute separately $\sigma_2$ and $\sigma_3$ by computing another angle in the problem, which is achievable but not done yet.

This is as far as we could go on the “Pull 1” experiments on that date, and more refined specific data processing could lead to better computations of the deformation and significant quantities of exchanged interface. For now, the $L_{in}$'s I tried to compute were so noisy that I do not dare showing them in this manuscript.
<figure><img src="image_108.png" /><figcaption>Figure 3.43: Positions of the edge of prepared Frankel film and the photobleached columns. **A:** Zoom near the meniscus in film 3. The vertical bright fringe on the middle left is the free meniscus, film 3 is on its right. The red line localises the detected edge of prepared Frankel film, whose distance from the edge of free meniscus is $L_{Fr,3}$, the green and yellow lines localise the two columns of photobleached dots, whose distances from the edge of meniscus are $L_{c,1}$ and $L_{c,2}$. **B:** Time evolution of the distances defined in **A** as well as $L_{Fr,1}$ (blue) the outgoing length of interface in film 1. The main deformation happens at $t_1 = 32$ s, where film 3 is compressed and film 1 is stretched. **C:** Towards the end of the deformation, the marginal regeneration instability appears, breaking our invariance by translation and thus stopping the measurement of the exchanged quantities of interface.</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>Figure 3.44: Example given on the chemistry [DOH] = 35 mg/L and a velocity of main deformation $V_1$ = 50 mm/s for the deformations of film 3 and 1. It was not possible to compute the corrected $\epsilon_1$ in this deformation, and we show here the deformation equivalent to the $\epsilon_{1,a}$ of “Push 3”.</figcaption></figure>
<figure><img src="image_108.png" /><figcaption>Figure 3.43: Positions of the edge of prepared Frankel film and the photobleached columns. **A:** Zoom near the meniscus in film 3. The vertical bright fringe on the middle left is the free meniscus, film 3 is on its right. The red line localises the detected edge of prepared Frankel film, whose distance from the edge of free meniscus is $L_{Fr,3}$, the green and yellow lines localise the two columns of photobleached dots, whose distances from the edge of meniscus are $L_{c,1}$ and $L_{c,2}$. **B:** Time evolution of the distances defined in **A** as well as $L_{Fr,1}$ (blue) the outgoing length of interface in film 1. The main deformation happens at $t_1 = 32$ s, where film 3 is compressed and film 1 is stretched. **C:** Towards the end of the deformation, the marginal regeneration instability appears, breaking our invariance by translation and thus stopping the measurement of the exchanged quantities of interface.</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>Figure 3.44: Example given on the chemistry [DOH] = 35 mg/L and a velocity of main deformation $V_1$ = 50 mm/s for the deformations of film 3 and 1. It was not possible to compute the corrected $\epsilon_1$ in this deformation, and we show here the deformation equivalent to the $\epsilon_{1,a}$ of “Push 3”.</figcaption></figure>
## 3.11 Conclusions of Chapter 3

* The fluid mechanics behind the visco-elasticity of elementary liquid foams is not completely settled.

* The elastic contributions are well understood and originate from the finite-sized reservoir effects of the foam films. This was the subject of chapter 1.

* The viscous contributions are suspected to originate from a very localised sheared zone near the meniscus, of size $l \sim 100$ µm. The existence of this zone is imposed by the presence of a stagnation point at the interface of the elementary foam, and a theoretical model already exists to describe it [4].

* The flow field in this zone is fully determined by the Marangoni stresses at the interfaces, which are set by the dynamics of exchange of the surfactants in this zone.

* The quantity of surfactants exchanged with the meniscus was a free parameter of the model which could lead to very different predictions for the dissipation occurring in the elementary foam.

* By looking simultaneously at the quantities of surfactants going in and out of the meniscus during the deformation of the foam, I was able to measure the nullity of the quantity of surfactants exchanged with the meniscus.

* This result is robust even if we vary the velocity of deformation or the chemistry of the foam. I need now to put it back into the theoretical model to close it, and to compare it to the experimental rheological data I also measured during my experimental campaign.
## 3.11 Conclusions of Chapter 3

* The fluid mechanics behind the visco-elasticity of elementary liquid foams is not completely settled.

* The elastic contributions are well understood and originate from the finite-sized reservoir effects of the foam films. This was the subject of chapter 1.

* The viscous contributions are suspected to originate from a very localised sheared zone near the meniscus, of size $l \sim 100$ µm. The existence of this zone is imposed by the presence of a stagnation point at the interface of the elementary foam, and a theoretical model already exists to describe it [4].

* The flow field in this zone is fully determined by the Marangoni stresses at the interfaces, which are set by the dynamics of exchange of the surfactants in this zone.

* The quantity of surfactants exchanged with the meniscus was a free parameter of the model which could lead to very different predictions for the dissipation occurring in the elementary foam.

* By looking simultaneously at the quantities of surfactants going in and out of the meniscus during the deformation of the foam, I was able to measure the nullity of the quantity of surfactants exchanged with the meniscus.

* This result is robust even if we vary the velocity of deformation or the chemistry of the foam. I need now to put it back into the theoretical model to close it, and to compare it to the experimental rheological data I also measured during my experimental campaign.
Conclusion

The common denominator to all my thesis work is the study of in-plane flows within foam films. This is a hydrodynamicist's point of view to help better understanding the ageing and rheology of liquid foams, starting from a single film and scaling up to what we called an elementary liquid foam. The subjects I covered in my Introduction are part of the work of a whole (welcoming) community which has developed the field of foam film hydrodynamics for decades now.

My three chapters are three modest stones that I have the chance to place at three distinct points of this edifice.

Chapter 1 is about the elasticity of a single foam film. Because the bulk is very thin, it is a finite-sized reservoir for the sake of populating the water-air interfaces sandwiching it. This is responsible for an effective elastic contribution under deformation which has already been predicted and measured by Prins et al. [5] in 1967 in the submicellar regime. Here, I provided an extension of this model to the micellar regime in presence of poorly soluble surfactants. I also compared this new thermochemical model to experimental data from Isabelle's group, and found that we could predict the elasticity of the different solutions based on their chemistry. The results were published in *Soft Matter* [6].

Chapter 2 is about a novel measurement of a line tension force $T \sim 10^{-10}$N in the plane of foam films. It originates from the thickness gradients present in the film along a contour and sets it in motion in order to minimise its perimeter. These motions are observed during the marginal regeneration instability for instance, where thin patches take a circular shape because of this force, and are advected up in the film, thus being an important mechanism for its ageing. Another example is the use made of in-plane motions of foam films as a proxy for atmospheric turbulent flows, where this line tension could represent a non-negligible force potentially harming the analogy. I supervised the experimental campaigns where we created the foam film subject to this line tension. We looked at a configuration where we were able to simultaneously measure the force and monitor the relaxation of the in-plane motions. Our force measurement was validated by a quantitative agreement with the witnessed relaxation, which I modelled analytically knowing that the damping mechanism was air friction. The results have been accepted by *Physical Review Letters* by the time I write this manuscript [99].

Chapter 3 is about the main experimental contribution of my thesis. The effective viscosity of an elementary liquid foam is thought to originate from a very localised sheared zone near the meniscus linking the foam films. The exchanges of surfactants there, between the interfaces, the bulk and the meniscus, entirely determine the shear flow responsible for the mechanical dissipation. Adrien Bussonnière and Isabelle Cantat developed a theoretical model for all that [4], but one
Conclusion

The common denominator to all my thesis work is the study of in-plane flows within foam films. This is a hydrodynamicist's point of view to help better understanding the ageing and rheology of liquid foams, starting from a single film and scaling up to what we called an elementary liquid foam. The subjects I covered in my Introduction are part of the work of a whole (welcoming) community which has developed the field of foam film hydrodynamics for decades now.

My three chapters are three modest stones that I have the chance to place at three distinct points of this edifice.

Chapter 1 is about the elasticity of a single foam film. Because the bulk is very thin, it is a finite-sized reservoir for the sake of populating the water-air interfaces sandwiching it. This is responsible for an effective elastic contribution under deformation which has already been predicted and measured by Prins et al. [5] in 1967 in the submicellar regime. Here, I provided an extension of this model to the micellar regime in presence of poorly soluble surfactants. I also compared this new thermochemical model to experimental data from Isabelle's group, and found that we could predict the elasticity of the different solutions based on their chemistry. The results were published in *Soft Matter* [6].

Chapter 2 is about a novel measurement of a line tension force $T \sim 10^{-10}$N in the plane of foam films. It originates from the thickness gradients present in the film along a contour and sets it in motion in order to minimise its perimeter. These motions are observed during the marginal regeneration instability for instance, where thin patches take a circular shape because of this force, and are advected up in the film, thus being an important mechanism for its ageing. Another example is the use made of in-plane motions of foam films as a proxy for atmospheric turbulent flows, where this line tension could represent a non-negligible force potentially harming the analogy. I supervised the experimental campaigns where we created the foam film subject to this line tension. We looked at a configuration where we were able to simultaneously measure the force and monitor the relaxation of the in-plane motions. Our force measurement was validated by a quantitative agreement with the witnessed relaxation, which I modelled analytically knowing that the damping mechanism was air friction. The results have been accepted by *Physical Review Letters* by the time I write this manuscript [99].

Chapter 3 is about the main experimental contribution of my thesis. The effective viscosity of an elementary liquid foam is thought to originate from a very localised sheared zone near the meniscus linking the foam films. The exchanges of surfactants there, between the interfaces, the bulk and the meniscus, entirely determine the shear flow responsible for the mechanical dissipation. Adrien Bussonnière and Isabelle Cantat developed a theoretical model for all that [4], but one
parameter was left free because of a lack of experimental information: **how much surfactants are exchanged with the meniscus? I adapted the experimental setup to perform this measurement and found that there were no exchanges in all the range of strain rates and chemistries I tried.**

The next step of this study is to look back at the theoretical model, setting the flux of surfactants towards the meniscus to zero. This will allow us to predict stress curves $\Delta\sigma = f(U)$ which we can then compare to the unused experimental data measured in my campaigns besides the surfactant exchanges. Notably, I measured the film elasticities with my setup (validated by our previous works in chapter 1) and computed the film deformations. We thus have access to the absolute tensions $\sigma$ of each film of the elementary foam, which will be a valuable piece of knowledge for the comparison with the theoretical predictions.
parameter was left free because of a lack of experimental information: **how much surfactants are exchanged with the meniscus? I adapted the experimental setup to perform this measurement and found that there were no exchanges in all the range of strain rates and chemistries I tried.**

The next step of this study is to look back at the theoretical model, setting the flux of surfactants towards the meniscus to zero. This will allow us to predict stress curves $\Delta\sigma = f(U)$ which we can then compare to the unused experimental data measured in my campaigns besides the surfactant exchanges. Notably, I measured the film elasticities with my setup (validated by our previous works in chapter 1) and computed the film deformations. We thus have access to the absolute tensions $\sigma$ of each film of the elementary foam, which will be a valuable piece of knowledge for the comparison with the theoretical predictions.
# Appendix

## Appendix 1: Derivation of compressible Navier-Stokes equation in 2D

Same derivation than Chapter II of the Landau Lifshitz [108], adapted in 2D. Conservation of momentum in 2D writes:

$$ \rho \frac{D\mathbf{v}}{Dt} = \nabla_s \gamma + \nabla_s \cdot \bar{\tau}_s + \mathbf{f}_{ext} \qquad (3.59) $$

Where $\bar{\tau}_s$ is the rheological stress tensor (NB: compared to section 0.1.4, we dispatched the stress tensor so that $\bar{\sigma}_s = \gamma \bar{I}_s + \bar{\tau}_s$):

$$ \bar{\tau}_s = \eta_s (\overline{\nabla}_s \mathbf{v} + t \overline{\nabla}_s \mathbf{v}) + \mu_s \nabla_s \cdot \mathbf{v} \bar{I}_s \qquad (3.60) $$

Where $\eta_s$ is the surface shear viscosity and $\mu_s$ a coefficient associated to the viscous response to dilation. It is not yet the dilatational surface viscosity though, as we need first to separate among these terms the pure dilatational contribution from the pure shear contributions. To do so, we compute the isotropic part of the tensor:

$$ \bar{\tau} = \frac{\tau_{xx} + \tau_{yy}}{2} = \mu_s \nabla_s \cdot \mathbf{v} + \frac{\eta_s}{2} (2\partial_x v_x + 2\partial_y v_y) = (\mu_s + \eta_s) \nabla_s \cdot \mathbf{v} \qquad (3.61) $$

Thus, let $\kappa_s = \mu_s + \eta_s$ be the surface dilatational viscosity. We can rewrite the tensor $\bar{\tau}_s$ as:

$$ \bar{\tau}_s = \eta_s (\overline{\nabla}_s \mathbf{v} + t \overline{\nabla}_s \mathbf{v} - \nabla_s \cdot \mathbf{v} \bar{I}_s) + \kappa_s \nabla_s \cdot \mathbf{v} \bar{I}_s \qquad (3.62) $$

Its contribution to eq.3.59 can thus be expressed on the x axis:

$$ \nabla_s \cdot \bar{\tau}_s |_x = \partial_x \tau_{xx} + \partial_y \tau_{xy} = \partial_x (\kappa_s \nabla_s \cdot \mathbf{v} + \eta_s (2\partial_x v_x - \partial_x \nabla_s \cdot \mathbf{v})) + \partial_y (\eta_s (\partial_y v_x + \partial_x v_y)) \qquad (3.63) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s (2\partial_{xx}v_x - \partial_x \nabla_s \cdot \mathbf{v} + \partial_{yy}v_x + \partial_{yx}v_y) \qquad (3.64) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s \Delta v_x + \eta_s \left( \frac{\partial_{xx}v_x + \partial_{xy}v_y - \partial_x \nabla_s \cdot \mathbf{v}}{\partial_x \nabla_s \cdot \mathbf{v}} \right) \qquad (3.65) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s \Delta v_x \qquad (3.66) $$

Which can be recast in eq.3.59 as a general vectorial expression on all axis:
# Appendix

## Appendix 1: Derivation of compressible Navier-Stokes equation in 2D

Same derivation than Chapter II of the Landau Lifshitz [108], adapted in 2D. Conservation of momentum in 2D writes:

$$ \rho \frac{D\mathbf{v}}{Dt} = \nabla_s \gamma + \nabla_s \cdot \bar{\tau}_s + \mathbf{f}_{ext} \quad (3.59) $$

Where $\bar{\tau}_s$ is the rheological stress tensor (NB: compared to section 0.1.4, we dispatched the stress tensor so that $\bar{\sigma}_s = \gamma \bar{I}_s + \bar{\tau}_s$):

$$ \bar{\tau}_s = \eta_s (\overline{\nabla}_s \mathbf{v} + {}^t \overline{\nabla}_s \mathbf{v}) + \mu_s \nabla_s \cdot \mathbf{v} \bar{I}_s \quad (3.60) $$

Where $\eta_s$ is the surface shear viscosity and $\mu_s$ a coefficient associated to the viscous response to dilation. It is not yet the dilatational surface viscosity though, as we need first to separate among these terms the pure dilatational contribution from the pure shear contributions. To do so, we compute the isotropic part of the tensor:

$$ \bar{\tau} = \frac{\tau_{xx} + \tau_{yy}}{2} = \mu_s \nabla_s \cdot \mathbf{v} + \frac{\eta_s}{2} (2\partial_x v_x + 2\partial_y v_y) = (\mu_s + \eta_s) \nabla_s \cdot \mathbf{v} \quad (3.61) $$

Thus, let $\kappa_s = \mu_s + \eta_s$ be the surface dilatational viscosity. We can rewrite the tensor $\bar{\tau}_s$ as:

$$ \bar{\tau}_s = \eta_s (\overline{\nabla}_s \mathbf{v} + {}^t \overline{\nabla}_s \mathbf{v} - \nabla_s \cdot \mathbf{v} \bar{I}_s) + \kappa_s \nabla_s \cdot \mathbf{v} \bar{I}_s \quad (3.62) $$

Its contribution to eq.3.59 can thus be expressed on the x axis:

$$ \nabla_s \cdot \overline{\tau}_s |_x = \partial_x \tau_{xx} + \partial_y \tau_{xy} = \partial_x (\kappa_s \nabla_s \cdot \mathbf{v} + \eta_s (2\partial_x v_x - \partial_x \nabla_s \cdot \mathbf{v})) + \partial_y (\eta_s (\partial_y v_x + \partial_x v_y)) \quad (3.63) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s (2\partial_{xx}v_x - \partial_x \nabla_s \cdot \mathbf{v} + \partial_{yy}v_x + \partial_{yx}v_y) \quad (3.64) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s \Delta v_x + \eta_s \left( \frac{\partial_{xx}v_x + \partial_{xy}v_y - \partial_x \nabla_s \cdot \mathbf{v}}{\partial_x \nabla_s \cdot \mathbf{v}} \right) \quad (3.65) $$

$$ = \kappa_s \partial_x \nabla_s \cdot \mathbf{v} + \eta_s \Delta v_x \quad (3.66) $$

Which can be recast in eq.3.59 as a general vectorial expression on all axis:
$$ \rho \frac{D\mathbf{v}}{Dt} = \nabla_s \gamma + \kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}) + \eta_s \Delta_s \mathbf{v} + \mathbf{f}_{ext} \quad (3.67) $$

Which is a general 2D Navier-Stokes equation when associated with a surface conservation law.

NB: This computation deviates from the 3D one when we compute the isotropic part of $\overline{\tau}$ to find the dilatational viscosity (or second viscosity), as a 3D tensor would mean hitting a factor $(\mu + \frac{2}{3}\eta)$ instead. This defines $\kappa = \mu + \frac{2}{3}\eta$ the second bulk viscosity to inject in the expression of $\overline{\tau}$ and the end of the computation leads to a term $(\kappa + \frac{1}{3}\eta)\nabla(\nabla \cdot \mathbf{v})$ for the viscous forces associated with dilation.

## Appendix 2: Limit of high elasticities for the in-plane motions of a foam film

The fact that the film elasticity $E_f$ dominates even for small deformations is a key ingredient in all the models we will develop in the frame of this thesis work. Most of the time, we will even assume the film to be incompressible, as in the following sections. However, before taking this limit, it has to be noted that the first limit of very high elasticities for a deformable film yields interesting results for the description of the kinematics of the film in-plane motions. In fact we will see here that we can decompose our velocity field in an irrotational component and an incompressible component, the former being simple, expressed at the lower order with elasticity, the latter being the solution of a more simple equation at higher order.

In the general case, the uniformity of film tension found in subsection 0.2.4 can be written:

$$ \nabla_s \sigma^f = 0 \quad (3.68) $$

If we consider a film formed of two flat symmetrical interfaces, thus removing capillary stress from the equation, the film tension $\sigma^f$ is then “just” twice the surface tension of both interfaces. This encompasses the elastic and viscous responses of the film, and the uniformity equation 3.68 can be rewritten as twice eq.10 where the exterior forces $\mathbf{f}_{g,s}$ have been discarded and surface tension expressed with $E_f = 2E_M$ (cf. eq.11) in the insoluble limit in the form $\frac{d\gamma}{dt} = E_f(\nabla_s \cdot \mathbf{v}_s)$, yielding:

$$ 0 = E_f \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_s) dt + 2\kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}_s) + 2\eta_s \Delta_s \mathbf{v}_s \quad (3.69) $$

Where $E_f = 2E_M$ is the film elasticity, which is uniform and independent of time in the present case.

We first start by doing a Helmholtz-Hodge decomposition of our vector field: $\mathbf{v}_s = \mathbf{v}_{irr} + \mathbf{v}_{inc}$, where $\mathbf{v}_{irr} = \nabla_s \phi$ is an irrotational velocity field written as a gradient and $\mathbf{v}_{inc} = \nabla_s \times \psi^1$ is

<footnote n="1">¹In 2D, the curl of a vector is usually a scalar. The Helmholtz decomposition is generalised in 2D with an alternative definition of this operator which takes a scalar as input and outputs a vector [109]: $(\nabla_s \times) : \psi(x, y) \rightarrow (-\partial_y\psi, \partial_x\psi)$. Thus, we will implicitly use both definitions $\nabla_s \times$ and $\nabla_s \times \psi$ with or without the bold nabla depending on the kind of 2D curl.</footnote>
$$ \rho \frac{D\mathbf{v}}{Dt} = \nabla_s \gamma + \kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}) + \eta_s \Delta_s \mathbf{v} + \mathbf{f}_{ext} \quad (3.67) $$

Which is a general 2D Navier-Stokes equation when associated with a surface conservation law.

NB: This computation deviates from the 3D one when we compute the isotropic part of $\overline{\tau}$ to find the dilatational viscosity (or second viscosity), as a 3D tensor would mean hitting a factor $(\mu + \frac{2}{3}\eta)$ instead. This defines $\kappa = \mu + \frac{2}{3}\eta$ the second bulk viscosity to inject in the expression of $\overline{\tau}$ and the end of the computation leads to a term $(\kappa + \frac{1}{3}\eta)\nabla(\nabla \cdot \mathbf{v})$ for the viscous forces associated with dilation.

## Appendix 2: Limit of high elasticities for the in-plane motions of a foam film

The fact that the film elasticity $E_f$ dominates even for small deformations is a key ingredient in all the models we will develop in the frame of this thesis work. Most of the time, we will even assume the film to be incompressible, as in the following sections. However, before taking this limit, it has to be noted that the first limit of very high elasticities for a deformable film yields interesting results for the description of the kinematics of the film in-plane motions. In fact we will see here that we can decompose our velocity field in an irrotational component and an incompressible component, the former being simple, expressed at the lower order with elasticity, the latter being the solution of a more simple equation at higher order.

In the general case, the uniformity of film tension found in subsection 0.2.4 can be written:

$$ \nabla_s \sigma^f = 0 \quad (3.68) $$

If we consider a film formed of two flat symmetrical interfaces, thus removing capillary stress from the equation, the film tension $\sigma^f$ is then “just” twice the surface tension of both interfaces. This encompasses the elastic and viscous responses of the film, and the uniformity equation 3.68 can be rewritten as twice eq.10 where the exterior forces $\mathbf{f}_{g,s}$ have been discarded and surface tension expressed with $E_f = 2E_M$ (cf. eq.11) in the insoluble limit in the form $\frac{d\gamma}{dt} = E_f(\nabla_s \cdot \mathbf{v}_s)$, yielding:

$$ 0 = E_f \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_s) dt + 2\kappa_s \nabla_s (\nabla_s \cdot \mathbf{v}_s) + 2\eta_s \Delta_s \mathbf{v}_s \quad (3.69) $$

Where $E_f = 2E_M$ is the film elasticity, which is uniform and independent of time in the present case.

We first start by doing a Helmholtz-Hodge decomposition of our vector field: $\mathbf{v}_s = \mathbf{v}_{irr} + \mathbf{v}_{inc}$, where $\mathbf{v}_{irr} = \nabla_s \phi$ is an irrotational velocity field written as a gradient and $\mathbf{v}_{inc} = \nabla_s \times \psi^1$ is

<footnote n="1">¹In 2D, the curl of a vector is usually a scalar. The Helmholtz decomposition is generalised in 2D with an alternative definition of this operator which takes a scalar as input and outputs a vector [109]: $(\nabla_s \times) : \psi(x, y) \rightarrow (-\partial_y\psi, \partial_x\psi)$. Thus, we will implicitly use both definitions $\nabla_s \times$ and $\nabla_s \times \psi$ with or without the bold nabla depending on the kind of 2D curl.</footnote>
an incompressible velocity field written as a 2D curl.

This is where we use our hypothesis that film elasticity is bigger than any other force: $E_f \gg \frac{\kappa_s}{\tau_{def}}$, $\frac{\eta_s}{\tau_{def}}$ to write at lowest order only the first term of the right-hand side of eq.3.69:

$$0 = \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_s) dt = \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_{irr}) dt \quad (3.70)$$

As $\mathbf{v}_{irr}$ is the non-vanishing component of $\mathbf{v}_s$ when we take the divergence. We also need to anticipate that $\mathbf{v}_{irr}$ will also have a higher order component and we write $\mathbf{v}_{irr} = \mathbf{v}_{irr,0} + \mathbf{v}_{irr,1}$ with $\mathbf{v}_{irr,0}$ being the lower order term, and by differentiating eq.3.70 with respect to time:

$$\nabla_s (\nabla_s \cdot \mathbf{v}_{irr,0}) = 0 \implies \nabla_s \cdot \mathbf{v}_{irr,0} = K(t) \quad (3.71)$$

This leads directly to a Poisson equation for the velocity potential $\Delta\phi = K(t)$ where the source term is necessarily the deformation rate $K(t) = \dot{\epsilon}$ in order to respect area conservation in the problem. To be solved, this part of the velocity field requires us to know the imposed velocity at the edges of the film, as the Poisson problem can be solved using Neumann boundary conditions for $\phi$, that is knowing $\frac{\partial\phi}{\partial n_s} \cdot \mathbf{n}_s = \mathbf{v}_s|_{\text{edge}}$ (where $\mathbf{n}_s$ is the normal vector to the edge of the foam film seen as a 2D domain). This is exactly what we know in practice when we deform a foam film, and we have a complete set of equations for $\mathbf{v}_{irr}$:

$$\left\{ \begin{array}{l} \frac{\partial \phi}{\partial n_s} \cdot \mathbf{n}_s \text{ is known at the edges} \\ \Delta \phi = \dot{\epsilon} \end{array} \right. \quad (3.72)$$

This set gives a unique solution $\mathbf{v}_{irr,0}$ which can be easily found in practice for simple geometries of deformation. Now that this component is found, we go at higher order in eq.3.69:

$$0 = E_f \nabla_s \int_t (\nabla_s \cdot (\mathbf{v}_{irr,1} + \mathbf{v}_{inc})) dt + 2\eta_s \Delta_s \mathbf{v}_{inc} \quad (3.73)$$

Where the terms of $\mathbf{v}_{irr}$ linked to the intrinsic viscosities $\kappa_s$, $\eta_s$ vanish after some reorganisations using $\Delta_s = \nabla_s \cdot \nabla_s \cdot (-\nabla_s \times \nabla_s \times)$. On the other hand, the incompressibility of $\mathbf{v}_{inc}$ allows us to write the set of equations as:

$$\left\{ \begin{array}{l} \nabla_s \cdot \mathbf{v}_{inc} = 0 \\ \nabla_s \gamma_{inc} + \eta_s \Delta_s \mathbf{v}_{inc} = 0 \end{array} \right. \quad (3.74)$$

Where $\gamma_{inc} = E_f \int_t \nabla_s \cdot \mathbf{v}_{irr,1} dt$ is a Lagrange multiplier of a 2D Stokes equation for $\mathbf{v}_{inc}$, enforcing the incompressibility condition. Thus we do not need to make the higher order field $\mathbf{v}_{irr,1}$ explicit to compute the dominant term $\mathbf{v}_{inc}$ of $\mathbf{v}_s \simeq \mathbf{v}_{inc} + \mathbf{v}_{irr,0}$.

Thus, even if a foam film is being stretched, meaning the velocity at its edges is non-zero, we can still compute the total velocity field $\mathbf{v}_s = \mathbf{v}_{inc} + \mathbf{v}_{irr}$. This is thanks to the film elasticity being dominant for any small deformation, allowing us to decompose the problem on different orders, the lower order giving the compressible field $\mathbf{v}_{irr}$ taking into account the global deformation of the foam film, the higher order giving the incompressible field $\mathbf{v}_{inc}$ which is a solution of a 2D Stokes equation.
an incompressible velocity field written as a 2D curl.

This is where we use our hypothesis that film elasticity is bigger than any other force: $E_f \gg \frac{\kappa_s}{\tau_{def}}$, $\frac{\eta_s}{\tau_{def}}$ to write at lowest order only the first term of the right-hand side of eq.3.69:

$$0 = \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_s) dt = \nabla_s \int_t (\nabla_s \cdot \mathbf{v}_{irr}) dt \quad (3.70)$$

As $\mathbf{v}_{irr}$ is the non-vanishing component of $\mathbf{v}_s$ when we take the divergence. We also need to anticipate that $\mathbf{v}_{irr}$ will also have a higher order component and we write $\mathbf{v}_{irr} = \mathbf{v}_{irr,0} + \mathbf{v}_{irr,1}$ with $\mathbf{v}_{irr,0}$ being the lower order term, and by differentiating eq.3.70 with respect to time:

$$\nabla_s (\nabla_s \cdot \mathbf{v}_{irr,0}) = 0 \implies \nabla_s \cdot \mathbf{v}_{irr,0} = K(t) \quad (3.71)$$

This leads directly to a Poisson equation for the velocity potential $\Delta\phi = K(t)$ where the source term is necessarily the deformation rate $K(t) = \dot{\epsilon}$ in order to respect area conservation in the problem. To be solved, this part of the velocity field requires us to know the imposed velocity at the edges of the film, as the Poisson problem can be solved using Neumann boundary conditions for $\phi$, that is knowing $\frac{\partial\phi}{\partial n_s} \cdot \mathbf{n}_s = \mathbf{v}_s|_{\text{edge}}$ (where $\mathbf{n}_s$ is the normal vector to the edge of the foam film seen as a 2D domain). This is exactly what we know in practice when we deform a foam film, and we have a complete set of equations for $\mathbf{v}_{irr}$:

$$\left\{ \begin{array}{l} \frac{\partial \phi}{\partial n_s} \cdot \mathbf{n}_s \text{ is known at the edges} \\ \Delta \phi = \dot{\epsilon} \end{array} \right. \quad (3.72)$$

This set gives a unique solution $\mathbf{v}_{irr,0}$ which can be easily found in practice for simple geometries of deformation. Now that this component is found, we go at higher order in eq.3.69:

$$0 = E_f \nabla_s \int_t (\nabla_s \cdot (\mathbf{v}_{irr,1} + \mathbf{v}_{inc})) dt + 2\eta_s \Delta_s \mathbf{v}_{inc} \quad (3.73)$$

Where the terms of $\mathbf{v}_{irr}$ linked to the intrinsic viscosities $\kappa_s$, $\eta_s$ vanish after some reorganisations using $\Delta_s = \nabla_s \cdot \nabla_s \cdot (-\nabla_s \times \nabla_s \times)$. On the other hand, the incompressibility of $\mathbf{v}_{inc}$ allows us to write the set of equations as:

$$\left\{ \begin{array}{l} \nabla_s \cdot \mathbf{v}_{inc} = 0 \\ \nabla_s \gamma_{inc} + \eta_s \Delta_s \mathbf{v}_{inc} = 0 \end{array} \right. \quad (3.74)$$

Where $\gamma_{inc} = E_f \int_t \nabla_s \cdot \mathbf{v}_{irr,1} dt$ is a Lagrange multiplier of a 2D Stokes equation for $\mathbf{v}_{inc}$ enforcing the incompressibility condition. Thus we do not need to make the higher order field $\mathbf{v}_{irr,1}$ explicit to compute the dominant term $\mathbf{v}_{inc}$ of $\mathbf{v}_s \simeq \mathbf{v}_{inc} + \mathbf{v}_{irr,0}$.

Thus, even if a foam film is being stretched, meaning the velocity at its edges is non-zero, we can still compute the total velocity field $\mathbf{v}_s = \mathbf{v}_{inc} + \mathbf{v}_{irr}$. This is thanks to the film elasticity being dominant for any small deformation, allowing us to decompose the problem on different orders, the lower order giving the compressible field $\mathbf{v}_{irr}$ taking into account the global deformation of the foam film, the higher order giving the incompressible field $\mathbf{v}_{inc}$ which is a solution of a 2D Stokes equation.
Note that this result remains valid even if we keep exterior forces $f_{ext,s}$ such as air friction, as long as they remain of higher order than the film elasticity $E_f$. We only need to add them to eq.3.74 when we need to solve for the incompressible field.

# Appendix 3: Compressible correction for the flow field of the line tension experiment

In section 2.4 of Chapter 2, we suppose the incompressibility of the interfaces based on the fact that the area $A$ of the thin film is almost constant in time, as recalled with Fig.3.45C. However, it undergoes a slight compression, even though the motors have stopped moving.

<figure><img src="image_110.png" /><figcaption>Figure 3.45: Recalling the notations and some experimental data from chapter 2 section 3.4: **A** Notations for the deformable frame; **B** Semi-length $L$ of the arena of thin film throughout its relaxation; **C** Area $A$ of the thin film during the experiment.</figcaption></figure>

This is due to some Frankel film extraction, which happens mostly during the stretching to create the thick film of area $A_2 = aw - A$ (as shown in Fig.3.45A), but which also continues after the deformation for $t > 0$. To try and take this into account for the sake of describing the velocity field in the plane of the foam film, we use the development for the limit of high elasticities led in the Appendix 2. This latter established that we can decompose the velocity field into two components: a compressible one corresponding to a pure stretching, and an incompressible one corresponding to the resolution of a 2D Stokes equation with the exterior forces. **The goal of this Appendix is to apply this decomposition to our concrete problem and to validate that the compressible term can effectively be discarded in our case, as we assumed in section 2.4.**

This decomposition is conditioned by the fact that the in-plane forces are much lower than the film elasticity. In the case of chapter 2, these are the exterior friction forces and the line tension.

First, the friction in this case is dominated by air friction (cf. 2.4.4, and is of the order of magnitude: $f_g \sim \eta_g U/\delta \sim 10^{-4}$Pa with $\eta_g \sim 10^{-5}$Pa·s the air viscosity, $U \sim 10^{-2}$m/s the typical velocity for the in-plane motions and $\delta \sim 10^{-3}$m the typical width of the sheared air layer (all recalled from 2.4.3). Second, the line tension forces are of the order $T/R^2 \sim 10^{-3}$Pa with $T \sim 10^{-9}$N the line tension and $R^2 \sim 10^{-3}$m the typical curvature radius of the line of tension.
Note that this result remains valid even if we keep exterior forces $f_{ext,s}$ such as air friction, as long as they remain of higher order than the film elasticity $E_f$. We only need to add them to eq.3.74 when we need to solve for the incompressible field.

# Appendix 3: Compressible correction for the flow field of the line tension experiment

In section 2.4 of Chapter 2, we suppose the incompressibility of the interfaces based on the fact that the area A of the thin film is almost constant in time, as recalled with Fig.3.45C. However, it undergoes a slight compression, even though the motors have stopped moving.

<figure><img src="image_110.png" /><figcaption>Figure 3.45: Recalling the notations and some experimental data from chapter 2 section 3.4: **A** Notations for the deformable frame; **B** Semi-length *L* of the arena of thin film throughout its relaxation; **C** Area *A* of the thin film during the experiment.</figcaption></figure>

This is due to some Frankel film extraction, which happens mostly during the stretching to create the thick film of area $A_2 = aw - A$ (as shown in Fig.3.45A), but which also continues after the deformation for $t > 0$. To try and take this into account for the sake of describing the velocity field in the plane of the foam film, we use the development for the limit of high elasticities led in the Appendix 2. This latter established that we can decompose the velocity field into two components: a compressible one corresponding to a pure stretching, and an incompressible one corresponding to the resolution of a 2D Stokes equation with the exterior forces. **The goal of this Appendix is to apply this decomposition to our concrete problem and to validate that the compressible term can effectively be discarded in our case, as we assumed in section 2.4.**

This decomposition is conditioned by the fact that the in-plane forces are much lower than the film elasticity. In the case of chapter 2, these are the exterior friction forces and the line tension.

First, the friction in this case is dominated by air friction (cf. 2.4.4, and is of the order of magnitude: $f_g \sim \eta_g U/\delta \sim 10^{-4}$Pa with $\eta_g \sim 10^{-5}$Pa·s the air viscosity, $U \sim 10^{-2}$m/s the typical velocity for the in-plane motions and $\delta \sim 10^{-3}$m the typical width of the sheared air layer (all recalled from 2.4.3). Second, the line tension forces are of the order $T/R^2 \sim 10^{-3}$Pa with $T \sim 10^{-9}$N the line tension and $R^2 \sim 10^{-3}$m the typical curvature radius of the line of tension.
These forces are to be compared to the elastic forces at stake while deforming the foam film globally by stretching it and inducing film extraction. This yields forces of the order of $E_f/L \sim 1$ Pa with $E_f \sim 10^{-2}$N/m the film elasticity and $L \sim 10^{-2}$m the typical length of the foam film. The elastic forces dominate all the others and thus the decomposition of Appendix 2 can be applied.

Note that in the following, time may appear as it is a constant of our experimental problem. Our model though is independent of time, and $t$ acts as a dummy variable listing all the resolutions of the flow field at each time step.

## Compressible component for the uniaxial stretching

Let us first start by computing the compressible velocity field using the set of eq. 3.72. In the specific case of the experiment of chapter 2, the foam film is rectangular and went through a uniaxial deformation which is well known. After the motor has stopped, some Frankel film are still being extracted at the four edges. We make the assumption that the tension is uniform along the four menisci at the straight edges, and the tension being uniform in the film, the velocity of Frankel film extraction $V_{Fr}$ is uniform along the rectangular frame, leading to the explicit conditions for the velocity potential $\phi$:

$$ \left\{ \begin{aligned} v_0 &= \nabla \phi \\ \partial_x \phi(x=0) &= V_{Fr} ; \quad \partial_x \phi(x=w) = -V_{Fr} \\ \partial_y \phi(y=0) &= V_{Fr} ; \quad \partial_y \phi(y=a) = V_{mot} - V_{Fr} \\ \Delta \phi &= \frac{1}{A_2} \frac{dA_2}{dt}(t) \end{aligned} \right. \qquad (3.75) $$

Where $V_{Fr}$ is known by looking at the compression state of the film with the last equation. $V_{mot}$ is the velocity of the motor at the time at which we solve the system, and for our measurable range $t > 0$: $V_{mot} = 0$. The system has a solution for a compressible velocity field $v_0$ that is:

$$ \left\{ \begin{aligned} v_{0,x} &= V_{Fr} + \frac{x}{w}(-2V_{Fr}) \\ v_{0,y} &= V_{Fr} + \frac{y}{a}(V_{mot} - 2V_{Fr}) \end{aligned} \right. \qquad (3.76) $$

This velocity field must now be associated with the incompressible flow which we compute in section 2.4.

## Superposition with the incompressible flow field and importance of this correction for the relaxation dynamics

The computation of the incompressible velocity field $v_{inc}$ is given equivalently in subsection 2.4.2, eq. 2.33, by considering a $\gamma_{inc}$ that takes into account the compressible flow field, or by Appendix 2, eq. 3.74, by adding $f_g$ the air friction and the line tension integral as higher order forces in the system. They both give:

$$ \left\{ \begin{aligned} \nabla_s \cdot v_{inc} &= 0 \\ \eta_s \Delta_s v_{inc} + \nabla_s \gamma_{inc} + f_g &= \oint_{C^*} T \kappa(\mathbf{r}') \delta_{C^*}(\mathbf{r} - \mathbf{r}') \mathbf{n}(\mathbf{r}') dr' \end{aligned} \right. \qquad (3.77) $$
These forces are to be compared to the elastic forces at stake while deforming the foam film globally by stretching it and inducing film extraction. This yields forces of the order of $E_f/L \sim 1$ Pa with $E_f \sim 10^{-2}$N/m the film elasticity and $L \sim 10^{-2}$m the typical length of the foam film. The elastic forces dominate all the others and thus the decomposition of Appendix 2 can be applied.

Note that in the following, time may appear as it is a constant of our experimental problem. Our model though is independent of time, and $t$ acts as a dummy variable listing all the resolutions of the flow field at each time step.

## Compressible component for the uniaxial stretching

Let us first start by computing the compressible velocity field using the set of eq.3.72. In the specific case of the experiment of chapter 2, the foam film is rectangular and went through a uniaxial deformation which is well known. After the motor has stopped, some Frankel film are still being extracted at the four edges. We make the assumption that the tension is uniform along the four menisci at the straight edges, and the tension being uniform in the film, the velocity of Frankel film extraction $V_{Fr}$ is uniform along the rectangular frame, leading to the explicit conditions for the velocity potential $\phi$:

$$ \left\{ \begin{aligned} v_0 &= \nabla \phi \\ \partial_x \phi(x=0) &= V_{Fr} ; \quad \partial_x \phi(x=w) = -V_{Fr} \\ \partial_y \phi(y=0) &= V_{Fr} ; \quad \partial_y \phi(y=a) = V_{mot} - V_{Fr} \\ \Delta \phi &= \frac{1}{A_2} \frac{dA_2}{dt}(t) \end{aligned} \right. \qquad (3.75) $$

Where $V_{Fr}$ is known by looking at the compression state of the film with the last equation. $V_{mot}$ is the velocity of the motor at the time at which we solve the system, and for our measurable range $t > 0$: $V_{mot} = 0$. The system has a solution for a compressible velocity field $v_0$ that is:

$$ \left\{ \begin{aligned} v_{0,x} &= V_{Fr} + \frac{x}{w}(-2V_{Fr}) \\ v_{0,y} &= V_{Fr} + \frac{y}{a}(V_{mot} - 2V_{Fr}) \end{aligned} \right. \qquad (3.76) $$

This velocity field must now be associated with the incompressible flow which we compute in section 2.4.

## Superposition with the incompressible flow field and importance of this correction for the relaxation dynamics

The computation of the incompressible velocity field $v_{inc}$ is given equivalently in subsection 2.4.2, eq.2.33, by considering a $\gamma_{inc}$ that takes into account the compressible flow field, or by Appendix 2, eq.3.74, by adding $f_g$ the air friction and the line tension integral as higher order forces in the system. They both give:

$$ \left\{ \begin{aligned} \nabla_s \cdot v_{inc} &= 0 \\ \eta_s \Delta_s v_{inc} + \nabla_s \gamma_{inc} + f_g &= \oint_{C^*} T \kappa(\mathbf{r}') \delta_{C^*}(\mathbf{r} - \mathbf{r}') \mathbf{n}(\mathbf{r}') dr' \end{aligned} \right. \qquad (3.77) $$
This problem is solved in section 2.4.4 in a form that allows us to get an expression for the time evolution of $L$ with eq.2.46. With the compressible component, we have:

$$ \dot{L}(t) = -\frac{T(t)}{4R(t)\eta_{air}} - \frac{2L(t)}{w}V_{Fr}(t) \qquad (3.78) $$

Where the first term of the right hand-side is given by eq.2.46 and the second term is the compressible contribution. Experimentally, we define $U^{exp} = \dot{L}$ in section 2.4.4, and here we also define a corrected velocity:

$$ U_{corr}^{exp}(t) = U^{exp}(t) + \frac{2L(t)}{w}V_{Fr}(t) \qquad (3.79) $$

Which now takes into account the compression of the thin film as more thick film is extracted. This is the corrected velocity that may be taken into account for the sake of quantifying the relaxation dynamics associated with the line tension forces.

We compute the ratio $U_{corr}^{exp}/U^{exp}$ and plot it in Fig.3.46. Mind that $U^{exp} = \dot{L} < 0$ and $V_{Fr} > 0$, hence the ratio below one. **We directly see that it is a correction of 15% at best, which is a useful correction in general, but given the present experimental uncertainties associated with the line tension measurement (of the order of 50%...), it has been discarded for the sake of simplicity in the story of chapter 2.**

<figure><img src="image_111.png" /><figcaption>Figure 3.46: Ratio of the corrected velocity $U_{corr}^{exp}$ by the experimental velocity $U^{exp}$ retained in chapter 2. The correction is of the order of 15%.</figcaption></figure>

# Appendix 4: 2D Stokes equation and derivation of its Green function

Let us consider a 3D viscous fluid of viscosity $\eta$, to which we apply a force perpendicular to a segment localized by the vector $r_1 = (x_1, y_1)$ in the $(Oxy)$ plane, and by the $z$-scores $z_1 \in [-M, M]$. We denote $f^{lin}$ the force per unit length applied on a part of the segment, and $u$ the unit vector of the direction of the force. $M$ is arbitrarily large. The Stokes equations governing the motion and the pressure field is then:
This problem is solved in section 2.4.4 in a form that allows us to get an expression for the time evolution of $L$ with eq.2.46. With the compressible component, we have:

$$ \dot{L}(t) = -\frac{T(t)}{4R(t)\eta_{air}} - \frac{2L(t)}{w}V_{Fr}(t) \quad (3.78) $$

Where the first term of the right hand-side is given by eq.2.46 and the second term is the compressible contribution. Experimentally, we define $U^{exp} = \dot{L}$ in section 2.4.4, and here we also define a corrected velocity:

$$ U_{corr}^{exp}(t) = U^{exp}(t) + \frac{2L(t)}{w}V_{Fr}(t) \quad (3.79) $$

Which now takes into account the compression of the thin film as more thick film is extracted. This is the corrected velocity that may be taken into account for the sake of quantifying the relaxation dynamics associated with the line tension forces.

We compute the ratio $U_{corr}^{exp}/U^{exp}$ and plot it in Fig.3.46. Mind that $U^{exp} = \dot{L} < 0$ and $V_{Fr} > 0$, hence the ratio below one. **We directly see that it is a correction of 15% at best, which is a useful correction in general, but given the present experimental uncertainties associated with the line tension measurement (of the order of 50%...), it has been discarded for the sake of simplicity in the story of chapter 2.**

<figure><img src="image_111.png" /><figcaption>Figure 3.46: Ratio of the corrected velocity $U_{corr}^{exp}$ by the experimental velocity $U^{exp}$ retained in chapter 2. The correction is of the order of 15%.</figcaption></figure>

# Appendix 4: 2D Stokes equation and derivation of its Green function

Let us consider a 3D viscous fluid of viscosity $\eta$, to which we apply a force perpendicular to a segment localized by the vector $r_1 = (x_1, y_1)$ in the (Oxy) plane, and by the $z$-scores $z_1 \in [-M, M]$. We denote $f^{lin}$ the force per unit length applied on a part of the segment, and $u$ the unit vector of the direction of the force. $M$ is arbitrarily large. The Stokes equations governing the motion and the pressure field is then:
$$ \Delta \mathbf{v}(\mathbf{r}) - \nabla \frac{p(\mathbf{r})}{\eta} = -\frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \delta_{3D}(\mathbf{r} - \mathbf{r}_1) dz_1 \mathbf{u} \quad (3.80) $$

$$ \nabla \cdot \mathbf{v} = 0 \quad (3.81) $$

The dirac delta function $\delta_{3D}$ is defined so that its integration over the volume of definition $\mathcal{R}^3$ is unity. In this case, the Green function of the Stokes equation is integrated over the segment and yields:

$$ \mathbf{v}(\mathbf{r}) = +\frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \overline{\overline{\mathbf{G}}}_{3D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \, dz_1 \quad (3.82) $$

Where $\mathbf{G}_{3D}$, the Oseen tensor, is defined as:

$$ G_{3D,ij}(\mathbf{r}) = \frac{1}{8\pi} \left( \frac{\delta_{ij}}{|\mathbf{r}|} + \frac{r_i r_j}{|\mathbf{r}|^3} \right) \quad (3.83) $$

Where $\delta_{ij}$ is the Kronecker symbol. We now seek an analogy between this 3D mathematical problem in order to solve for a point force 2D problem whose Stokes equations can be written:

$$ \Delta \mathbf{v}(\mathbf{r}) + \nabla \frac{\gamma(\mathbf{r})}{\eta_s} = -\frac{T}{\eta_s} \delta_{2D}(\mathbf{r} - \mathbf{r}_1)\mathbf{u} \quad (3.84) $$

$$ \nabla \cdot \mathbf{v} = 0 \quad (3.85) $$

Where $\frac{\gamma}{\eta_s} = -\frac{p}{\eta}$ defines by analogy a surface tension and a surface shear viscosity, $T$ is a force, and the new dirac function $\delta_{2D}$ is now defined over the $(Oxy)$ plane. The mathematical analogy between the problems can be written so that the Green function of the 2D problem is defined as:

$$ \frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \overline{\overline{\mathbf{G}}}_{3D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \, dz_1 = \frac{T}{\eta_s} \overline{\overline{\mathbf{G}}}_{2D}(\mathbf{r}_{\parallel} - \mathbf{r}_{1,\parallel}) \cdot \mathbf{u} \quad (3.86) $$

The $\parallel$ index symbolizing the fact that the vector is projected in the $(Oxy)$ plane. Computing the integral for each coordinate and identifying the terms for the tensor $\overline{\overline{\mathbf{G}}}_{2D}$, we get for $\mathbf{r} \in \mathcal{R}^2 = (Oxy)$:

$$ G_{2D,ij}(\mathbf{r}) = \frac{1}{4\pi} \left( \delta_{ij} \ln\left(\frac{2M}{|\mathbf{r}|}\right) + \frac{r_i r_j}{|\mathbf{r}|^2} \right) \quad (3.87) $$

Which is the solution to eq. 3.84 yielding the 2D velocity field:

$$ \mathbf{v}(\mathbf{r}) = +\frac{T}{\eta_s} \overline{\overline{\mathbf{G}}}_{2D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \quad (3.88) $$
$$ \Delta \mathbf{v}(\mathbf{r}) - \nabla \frac{p(\mathbf{r})}{\eta} = -\frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \delta_{3D}(\mathbf{r} - \mathbf{r}_1) dz_1 \mathbf{u} \quad (3.80) $$

$$ \nabla \cdot \mathbf{v} = 0 \quad (3.81) $$

The dirac delta function $\delta_{3D}$ is defined so that its integration over the volume of definition $\mathcal{R}^3$ is unity. In this case, the Green function of the Stokes equation is integrated over the segment and yields:

$$ \mathbf{v}(\mathbf{r}) = +\frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \overline{\overline{\mathbf{G}}}_{3D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \, dz_1 \quad (3.82) $$

Where $\mathbf{G}_{3D}$, the Oseen tensor, is defined as:

$$ G_{3D,ij}(\mathbf{r}) = \frac{1}{8\pi} \left( \frac{\delta_{ij}}{|\mathbf{r}|} + \frac{r_i r_j}{|\mathbf{r}|^3} \right) \quad (3.83) $$

Where $\delta_{ij}$ is the Kronecker symbol. We now seek an analogy between this 3D mathematical problem in order to solve for a point force 2D problem whose Stokes equations can be written:

$$ \Delta \mathbf{v}(\mathbf{r}) + \nabla \frac{\gamma(\mathbf{r})}{\eta_s} = -\frac{T}{\eta_s} \delta_{2D}(\mathbf{r} - \mathbf{r}_1)\mathbf{u} \quad (3.84) $$

$$ \nabla \cdot \mathbf{v} = 0 \quad (3.85) $$

Where $\frac{\gamma}{\eta_s} = -\frac{p}{\eta}$ defines by analogy a surface tension and a surface shear viscosity, $T$ is a force, and the new dirac function $\delta_{2D}$ is now defined over the $(Oxy)$ plane. The mathematical analogy between the problems can be written so that the Green function of the 2D problem is defined as:

$$ \frac{f^{\text{lin}}}{\eta} \int_{-M}^{M} \overline{\overline{\mathbf{G}}}_{3D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \, dz_1 = \frac{T}{\eta_s} \overline{\overline{\mathbf{G}}}_{2D}(\mathbf{r}_{\parallel} - \mathbf{r}_{1,\parallel}) \cdot \mathbf{u} \quad (3.86) $$

The $\parallel$ index symbolizing the fact that the vector is projected in the $(Oxy)$ plane. Computing the integral for each coordinate and identifying the terms for the tensor $\overline{\overline{\mathbf{G}}}_{2D}$, we get for $\mathbf{r} \in \mathcal{R}^2 = (Oxy)$:

$$ G_{2D,ij}(\mathbf{r}) = \frac{1}{4\pi} \left( \delta_{ij} \ln\left(\frac{2M}{|\mathbf{r}|}\right) + \frac{r_i r_j}{|\mathbf{r}|^2} \right) \quad (3.87) $$

Which is the solution to eq. 3.84 yielding the 2D velocity field:

$$ \mathbf{v}(\mathbf{r}) = +\frac{T}{\eta_s} \overline{\overline{\mathbf{G}}}_{2D}(\mathbf{r} - \mathbf{r}_1) \cdot \mathbf{u} \quad (3.88) $$
# Appendix 5: A more general equation of motion for foam films with any field of thickness h

In a more general case than presented in subsection 2.4.1, the thickness gradients are not localised along a contour. We can still write the equation of motion for any thickness field $h$ in the foam film. We can do so by adding directly the viscous stress tensor $\overline{\sigma}_{visc}$ for two viscous 2D interfaces to the capillary stress tensor $\overline{\sigma}_{cap}$ computed in eq. 2.7 of section 2.1:

$$ \overline{\sigma}_{visc} = 2\eta_s (\overline{\nabla}v + t \overline{\nabla}v) \qquad (3.89) $$

Where $v$ is the 2D velocity field shared by both interfaces, as we put ourselves in the case where only plug flows are allowed as discussed in 0.2.1 (potentially extensional plug flows though). Also, the factor 2 in $\overline{\sigma}_{visc}$ is here to take into account the two interfaces.

We now write the conservation of momentum in the film neglecting its inertia:

$$ 0 = \nabla \cdot (\overline{\sigma}_{cap} + \overline{\sigma}_{visc}) + f_{ext} \qquad (3.90) $$

Where $f_{ext} = 2f_g$ accounts for the exterior forces acting on the film, which in the present case are only $2f_g$ the friction forces of the two interfaces on the neighbouring gas phase. Taking the divergence of both the capillary and viscous stress tensors, we have:

$$ 2\eta_s \Delta v + 2\nabla\gamma + 2\gamma_0 h \nabla(\Delta h) + 2f_g = 0 \qquad (3.91) $$

The first two terms correspond to the terms of a 2D Stokes equation in which the role of the classical pressure gradient in 3D is held by the Marangoni stress here in 2D. The third term comes from the divergence of the capillary stress tensor alone, where a simplification occurs between $\nabla \cdot \overline{\sigma}_{cap}^*$ and $\nabla \cdot (\sigma^f \overline{I}_2)$ with the terms depending on $h$. Note that we can write equivalently with both notations $\nabla \delta \gamma$ as in eq. 2.7 or $\nabla \gamma$ as in eq. 2.32, as only the deviation from the equilibrium value matters here.

This equation is the general equation of in-plane motion of a foam film with any thickness field $h$ and two interfaces of surface viscosity $\eta_s$ subject to the friction of the neighbouring gas phase. It has been already derived by Bruinsma et al. [64] using another approach.

The specific case of eq. 2.32 where we defined a line tension can be recovered by injecting a thickness field $h$ whose gradient is non-zero in very localised regions only. If we do so (as we did in section 2.4.1), we need to be careful as not only the third term of eq. 3.91 matters in the non-zero gradient of thickness zones, but also $\nabla \gamma$ as discussed in section 2.1.2.
Appendix 5: A more general equation of motion for
foam films with any field of thickness h

In a more general case than presented in subsection 2.4.1, the thickness gradients are not localised along a contour. We can still write the equation of motion for any thickness field $h$ in the foam film. We can do so by adding directly the viscous stress tensor $\overline{\sigma}_{visc}$ for two viscous 2D interfaces to the capillary stress tensor $\overline{\sigma}_{cap}$ computed in eq.2.7 of section 2.1:

$$ \overline{\sigma}_{visc} = 2\eta_s (\overline{\nabla} v + t \overline{\nabla} v) \qquad (3.89) $$

Where $v$ is the 2D velocity field shared by both interfaces, as we put ourselves in the case where only plug flows are allowed as discussed in 0.2.1 (potentially extensional plug flows though). Also, the factor 2 in $\overline{\sigma}_{visc}$ is here to take into account the two interfaces.

We now write the conservation of momentum in the film neglecting its inertia:

$$ 0 = \nabla \cdot (\overline{\sigma}_{cap} + \overline{\sigma}_{visc}) + f_{ext} \qquad (3.90) $$

Where $f_{ext} = 2f_g$ accounts for the exterior forces acting on the film, which in the present case are only $2f_g$ the friction forces of the two interfaces on the neighbouring gas phase. Taking the divergence of both the capillary and viscous stress tensors, we have:

$$ 2\eta_s \Delta v + 2\nabla\gamma + 2\gamma_0 h \nabla(\Delta h) + 2f_g = 0 \qquad (3.91) $$

The first two terms correspond to the terms of a 2D Stokes equation in which the role of the classical pressure gradient in 3D is held by the Marangoni stress here in 2D. The third term comes from the divergence of the capillary stress tensor alone, where a simplification occurs between $\nabla \cdot \overline{\sigma}_{cap}^*$ and $\nabla \cdot (\sigma^f \overline{I}_2)$ with the terms depending on $h$. Note that we can write equivalently with both notations $\nabla \delta \gamma$ as in eq.2.7 or $\nabla \gamma$ as in eq.2.32, as only the deviation from the equilibrium value matters here.

This equation is the general equation of in-plane motion of a foam film with any thickness field $h$ and two interfaces of surface viscosity $\eta_s$ subject to the friction of the neighbouring gas phase. It has been already derived by Bruinsma et al. [64] using another approach.

The specific case of eq.2.32 where we defined a line tension can be recovered by injecting a thickness field $h$ whose gradient is non-zero in very localised regions only. If we do so (as we did in section 2.4.1), we need to be careful as not only the third term of eq.3.91 matters in the non-zero gradient of thickness zones, but also $\nabla \gamma$ as discussed in section 2.1.2.
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**Titre** : Mouvements aux interfaces de mousses liquides élémentaires

**Mots clés** : films de savon, mousses liquides, surfactants, rhéologie de surface, tension de ligne, interfaces liquides

**Résumé** : Les modèles physiques visant à prédire les propriétés mécaniques et rhéologiques des mousses liquides sont encore en développement. Une échelle locale pertinente pour attaquer ce problème du point de vue hydrodynamique est la mousse liquide élémentaire : quelques films de savon reliés entre eux par un ou deux ménisques. Ces travaux portent sur les écoulements dans de tels systèmes, plus précisément sur les mouvements dans les plans de ses différents films de savon. Il comporte essentiellement trois contributions.

La première porte sur le comportement élastique individuel des films sous étirement. Nous avons étendu un modèle connu à une gamme de chimie plus large et plus commune, et avons pu mesurer les modules élastiques effectifs de nos solutions.

La seconde porte sur la mise en évidence et la quantification d'une tension de ligne d'origine purement capillaire dans le plan d'un film de savon d'épaisseur hétérogène. En modélisant les mouvements de relaxation dans le plan du film de savon, nous validons cette mesure inédite par un accord quantitatif avec la dynamique observée.

Enfin, la troisième contribution porte sur les échanges de surfactants entre films de savon voisins lorsque la mousse élémentaire est soumise à une contrainte mécanique. Le résultat principal de cette partie, et de cette thèse, est que le ménisque séparant les films n'intervient pas dans cet échange. Ceci est une information importante qui permet de fermer un modèle théorique préexistant qui vise à prédire la viscosité effective des mousses liquides.

**Title** : Interfaces in motion in elementary liquid foams

**Keywords** : soap films, liquid foams, surfactants, surface rheology, line tension, liquid interfaces

**Abstract** : Models aiming to predict the mechanical and rheological properties of liquid foams are still under development. A relevant local scale to tackle this issue from a hydrodynamic perspective is the elementary liquid foam: a few foam films connected by one or two menisci. This work focuses on flows in such systems, specifically on movements within the planes of their foam films. It comprises three main contributions.

The first focuses on the individual elastic behaviour of films under stretching. We extended a known model to a broader and more common chemical range and were able to measure the effective elastic moduli of our solutions.

The second highlights and quantifies a line tension of purely capillary origin within the plane of a foam film with heterogeneous thickness. By modelling the relaxation movements within the plane of the foam film, we validate this novel measurement through a quantitative agreement with the observed dynamics.

Finally, the third contribution addresses surfactant exchanges between foam films when the elementary foam undergoes mechanical constraint. The main result of this part, and of this thesis, is that the meniscus separating the films does not play any role in this exchange. This is crucial information to close an existing theoretical model aiming to predict the effective viscosity of liquid foams.
**Titre** : Mouvements aux interfaces de mousses liquides élémentaires

**Mots clés** : films de savon, mousses liquides, surfactants, rhéologie de surface, tension de ligne, interfaces liquides

**Résumé** : Les modèles physiques visant à prédire les propriétés mécaniques et rhéologiques des mousses liquides sont encore en développement. Une échelle locale pertinente pour attaquer ce problème du point de vue hydrodynamique est la mousse liquide élémentaire : quelques films de savon reliés entre eux par un ou deux ménisques. Ces travaux portent sur les écoulements dans de tels systèmes, plus précisément sur les mouvements dans les plans de ses différents films de savon. Il comporte essentiellement trois contributions.

La première porte sur le comportement élastique individuel des films sous étirement. Nous avons étendu un modèle connu à une gamme de chimie plus large et plus commune, et avons pu mesurer les modules élastiques effectifs de nos solutions.

La seconde porte sur la mise en évidence et la quantification d'une tension de ligne d'origine purement capillaire dans le plan d'un film de savon d'épaisseur hétérogène. En modélisant les mouvements de relaxation dans le plan du film de savon, nous validons cette mesure inédite par un accord quantitatif avec la dynamique observée.

Enfin, la troisième contribution porte sur les échanges de surfactants entre films de savon voisins lorsque la mousse élémentaire est soumise à une contrainte mécanique. Le résultat principal de cette partie, et de cette thèse, est que le ménisque séparant les films n'intervient pas dans cet échange. Ceci est une information importante qui permet de fermer un modèle théorique préexistant qui vise à prédire la viscosité effective des mousses liquides.

**Title** : Interfaces in motion in elementary liquid foams

**Keywords** : soap films, liquid foams, surfactants, surface rheology, line tension, liquid interfaces

**Abstract** : Models aiming to predict the mechanical and rheological properties of liquid foams are still under development. A relevant local scale to tackle this issue from a hydrodynamic perspective is the elementary liquid foam: a few foam films connected by one or two menisci. This work focuses on flows in such systems, specifically on movements within the planes of their foam films. It comprises three main contributions.

The first focuses on the individual elastic behaviour of films under stretching. We extended a known model to a broader and more common chemical range and were able to measure the effective elastic moduli of our solutions.

The second highlights and quantifies a line tension of purely capillary origin within the plane of a foam film with heterogeneous thickness. By modelling the relaxation movements within the plane of the foam film, we validate this novel measurement through a quantitative agreement with the observed dynamics.

Finally, the third contribution addresses surfactant exchanges between foam films when the elementary foam undergoes mechanical constraint. The main result of this part, and of this thesis, is that the meniscus separating the films does not play any role in this exchange. This is crucial information to close an existing theoretical model aiming to predict the effective viscosity of liquid foams.