# Mixtures of foam and paste: suspensions of bubbles in yield stress fluids

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## Abstract

We study the rheological behavior of mixtures of foams and pastes, which can be described as suspensions of bubbles in yield stress fluids. Model systems are designed by mixing monodisperse aqueous foams and concentrated emulsions. The elastic modulus of the bubble suspensions is found to depend on the elastic capillary number \(\textit{Ca}_{_G}\), defined as the ratio of the paste elastic modulus to the bubble capillary pressure. For values of \(\textit{Ca}_{_G}\) larger than \(\simeq 0.5\), the dimensionless elastic modulus of the aerated material decreases as the bubble volume fraction \(\phi \) increases, suggesting that bubbles behave as soft elastic inclusions. Consistently, this decrease is all the sharper as \(\textit{Ca}_{_G}\) is high, which accounts for the softening of the bubbles as compared to the paste. By contrast, we find that the yield stress of most studied materials is not modified by the presence of bubbles. This suggests that their plastic behavior is governed by the plastic capillary number \(\textit{Ca}_{\tau_y}\), defined as the ratio of the paste yield stress to the bubble capillary pressure. At low \(\textit{Ca}_{\tau_y}\) values, bubbles behave as nondeformable inclusions, and we predict that the suspension dimensionless yield stress should remain close to unity, in agreement with our data up to \(\textit{Ca}_{\tau_y}=0.2\). When preparing systems with a larger target value of \(\textit{Ca}_{\tau_y}\), we observe bubble breakup during mixing, which means that they have been deformed by shear. It then seems that a critical value \(\textit{Ca}_{\tau_y}\simeq 0.2\) is never exceeded in the final material. These observations might imply that, in bubble suspensions prepared by mixing a foam and a paste, the suspension yield stress is always close to that of the paste surrounding the bubbles. Finally, at the highest \(\phi \) investigated, the yield stress is shown to increase abruptly with \(\phi \): this is interpreted as a “foamy yield stress fluid” regime, which takes place when the paste mesoscopic constitutive elements (here, the oil droplets) are strongly confined in the films between the bubbles.

## Keywords

Yield stress fluid Bubbles Suspension Foam Emulsion Elastic modulus Yield stress## Notes

### Acknowledgements

We thank Mohammed Bouricha for help on some of the experiments. We acknowledge funding from Saint-Gobain Recherche.

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