# Liquid Fraction Effect on Foam Flow through a Local Obstacle

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Procedure and Data Processing

#### 2.2. Computer Method for Liquid Fraction Calculation

## 3. Results

#### Velocity Field

## 4. Conclusions

- An increase in water content decreases the effect of a negative wake for the foam flowing around an obstacle, with full suppression of the effect in bubble flows;
- For drier foams, an increase in the obstacle height leads to an increase in the effect of a negative wake;
- The permeability of the local obstacle decreases significantly with an increase in the liquid fraction in the studied range of parameters and drops to a value of nearly zero for the liquid fraction above $5.8\times {10}^{-2}$. The existence of a critical liquid fraction corresponding to the near-zero permeability factor at a permeable constriction is important for efficient flow control in applications related to the production of polymer materials based on aqueous foams.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Experimental setup: a Hele-Shaw cell consisting of two glass plates with a gap of $G=1.15$ mm, the tank (in situ foam generator) attached to the bottom plate with holes for air and fluid injection, the light source below the cell and the camera above it; the air from the compressor is partially used for pressing the glass plates; compressed air passing filters enters the air flow regulator and the tank via the needle. The water pump controls the soap solution flow rate. (

**b**) Raw image of the foam flow zoomed in on the permeable obstacle area (obstacle diameter $a=30$ mm) and (

**c**) its binarization, with the red circle corresponding to the position of the permeable obstacle.

**Figure 2.**Method used to find the critical distance ${L}_{c}$ before rearrangement (or T1 event) occurs: (

**a**) selection of a bubble and all its neighbors, which are located at a distance less than $2{d}_{mean}$ from its center; (

**b**) an example of sorted distances between the foam cells in the ascending order, the red star indicates ${\widehat{L}}_{c}$—the critical distance plus border width; (

**c**) histogram of bubble areas superposed with a Gaussian fit (Normal distribution with $s=12.28\phantom{\rule{0.277778em}{0ex}}{\mathrm{mm}}^{2}$ and $\sigma =0.59\phantom{\rule{0.277778em}{0ex}}{\mathrm{mm}}^{2}$).

**Figure 3.**Original images of the foam (on the left) and longitudinal velocity component averaged over 125 s with subtracted mean velocity $u-{U}_{mean}$ (color) together with the velocity vector field $\mathbf{U}-{U}_{mean}$ (on the right) for various liquid fraction $\mathsf{\Phi}$: (

**a**,

**b**) $3.8\times {10}^{-3}$, (

**c**,

**d**) $1.0\times {10}^{-2}$, (

**e**,

**f**) $3.7\times {10}^{-2}$ and (

**g**,

**h**) $5.8\times {10}^{-2}$.

**Figure 4.**Analytical calculation of a two-dimensional potential flow passing a circular obstacle with the longitudinal velocity $u-{U}_{mean}$ in color and the vector field $\mathbf{U}-{U}_{mean}$ shown with black arrows.

**Figure 5.**Profiles of the normalized longitudinal velocity $\widehat{u}$ along the x-axis for obstacle heights $H/G$ of (

**a**) 0.35, (

**b**) 0.6 and (

**c**) 0.78 and liquid fractions of $\mathsf{\Phi}=3.8\times {10}^{-3}$ (blue), $4.7\times {10}^{-3}$ (green), $1.0\times {10}^{-2}$ (red), $3.7\times {10}^{-2}$ (yellow) and $5.8\times {10}^{-2}$ (purple).

**Figure 6.**Maximum normalized longitudinal velocity ${\widehat{u}}^{*}$ with respect to the foam liquid fraction for $H/G=0.35,0.6$ and 0.78.

**Figure 7.**Profiles of the normalized longitudinal velocity $\widehat{u}$ along the y-axis for the obstacle height $H/G=0.78$ and the liquid fractions $\mathsf{\Phi}=3.8\times {10}^{-3}$ (blue), $4.7\times {10}^{-3}$ (green), $1.0\times {10}^{-2}$ (red), $3.7\times {10}^{-2}$ (yellow) and $5.8\times {10}^{-2}$ (purple).

**Figure 8.**Permeability factor Q calculated using Equation (2) with respect to the normalized obstacle height $H/G$ for liquid fractions $\mathsf{\Phi}=3.8\times {10}^{-3}$ (purple), $4.7\times {10}^{-3}$ (yellow), $1.0\times {10}^{-2}$ (red), $3.7\times {10}^{-2}$ (green) and $5.8\times {10}^{-2}$ (blue) and the data from [24] for $\mathsf{\Phi}=7.0\times {10}^{-3}$ (black triangles) and $1.4\times {10}^{-2}$ (black squares).

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**MDPI and ACS Style**

Stennikova, O.; Shmakova, N.; Carrat, J.-B.; Ermanyuk, E.
Liquid Fraction Effect on Foam Flow through a Local Obstacle. *Polymers* **2022**, *14*, 5307.
https://doi.org/10.3390/polym14235307

**AMA Style**

Stennikova O, Shmakova N, Carrat J-B, Ermanyuk E.
Liquid Fraction Effect on Foam Flow through a Local Obstacle. *Polymers*. 2022; 14(23):5307.
https://doi.org/10.3390/polym14235307

**Chicago/Turabian Style**

Stennikova, Oksana, Natalia Shmakova, Jean-Bastien Carrat, and Evgeny Ermanyuk.
2022. "Liquid Fraction Effect on Foam Flow through a Local Obstacle" *Polymers* 14, no. 23: 5307.
https://doi.org/10.3390/polym14235307