Rheologica Acta

, Volume 52, Issue 3, pp 237–253 | Cite as

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

  • Michael Kogan
  • Lucie Ducloué
  • Julie Goyon
  • Xavier Chateau
  • Olivier Pitois
  • Guillaume Ovarlez
Original Contribution


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.


Yield stress fluid Bubbles Suspension Foam Emulsion Elastic modulus Yield stress 



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


  1. Ancey C, Jorrot H (2001) Yield stress for particle suspensions within a clay dispersion. J Rheol 45:297–319CrossRefGoogle Scholar
  2. Baravian C, Lalante A, Parker A (2002) Vane rheometry with a large, finite gap. Appl Rheol 12:81–87Google Scholar
  3. Blanc F, Peters F, Lemaire E (2011) Local transient rheological behavior of concentrated suspensions. J Rheol 55:835–854CrossRefGoogle Scholar
  4. Chateau X, Ovarlez G, Luu Trung K (2008) Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids. J Rheol 52:489–506CrossRefGoogle Scholar
  5. Coussot P (2005) Rheometry of pastes, suspensions and granular materials. Wiley, HobokenCrossRefGoogle Scholar
  6. Coussot P, Tabuteau H, Chateau X, Tocquer L, Ovarlez G (2006) Aging and solid or liquid behavior in pastes. J Rheol 50:975–994CrossRefGoogle Scholar
  7. Denkov ND, Tcholakova S, Golemanov K, Ananthpadmanabhan KP, Lips A (2009) The role of surfactant type and bubble surface mobility in foam rheology. Soft Matter 5:3389–3408CrossRefGoogle Scholar
  8. Dormieux L, Kondo D, Ulm FJ (2006) Microporomechanics. Wiley, HobokenCrossRefGoogle Scholar
  9. Dubash N, Frigaard IA (2004) Conditions for static bubbles in viscoplastic fluids. Phys Fluids 16:4319–4330CrossRefGoogle Scholar
  10. Dubash N, Frigaard IA (2007) Propagation and stopping of air bubbles in Carbopol solutions. J Non-Newton Fluid Mech 142:123–134CrossRefGoogle Scholar
  11. Dutta A, Chengara A, Nikolov AD, Wasan DT, Chen K, Campbell B (2004a) Texture and stability of aerated food emulsions–effects of buoyancy and Ostwald ripening. J Food Eng 62:169–175CrossRefGoogle Scholar
  12. Dutta A, Chengara A, Nikolov AD, Wasan DT, Chen K, Campbell B (2004b) Destabilization of aerated food products: effects of Ostwald ripening and gas diffusion. J Food Eng 62:177–184CrossRefGoogle Scholar
  13. Dzuy NQ, Boger DV (1983) Yield stress measurement for concentrated suspensions. J Rheol 27:321CrossRefGoogle Scholar
  14. Frankel NA, Acrivos A (1970) The constitutive equation for a dilute emulsion. J Fluid Mech 44:65–78CrossRefGoogle Scholar
  15. Gadala-Maria F, Acrivos A (1980) Shear-induced structure in a concentrated suspension of solid spheres. J Rheol 24:799–814CrossRefGoogle Scholar
  16. Gandolfo FG, Rosano HL (1997) Interbubble gas diffusion and the stability of foams. J Colloid Interface Sci 194:31–36CrossRefGoogle Scholar
  17. Geiker MA, Brandl M, Thrane L, Nielsen NF (2002a) On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete. Cem Concr Aggr 24:3–6CrossRefGoogle Scholar
  18. Geiker MR, Brandl M, Thrane LN, Bager DH, Wallevik O (2002b) The effect of measuring procedure on the apparent rheological properties of self compacting concrete. Cem Concr Res 32:1791–1795CrossRefGoogle Scholar
  19. Gonnermann HM, Manga M (2007) The fluid mechanics inside a volcano. Annu Rev Fluid Mech 39:321–356CrossRefGoogle Scholar
  20. Goyon J, Bertrand F, Pitois O, Ovarlez G (2010) Shear induced drainage in foamy yield-stress fluids. Phys Rev Lett 128301:104Google Scholar
  21. Griffiths RW (2000) The dynamics of lava flows. Annu Rev Fluid Mech 32:477–518CrossRefGoogle Scholar
  22. Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140CrossRefGoogle Scholar
  23. Keentok M, Milthorpe JF, O’Donovan E (1985) On the shearing zone around rotating vanes in plastic liquids: theory and experiment. J Non-Newton Fluid Mech 17:23–35CrossRefGoogle Scholar
  24. Koczo K, Lobo LA, Wasan DT (1992) Effect of oil on foam stability: aqueous foams stabilized by emulsions. J Colloid Interface Sci 150:492–506CrossRefGoogle Scholar
  25. Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, New YorkGoogle Scholar
  26. Leighton D, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181:415–439CrossRefGoogle Scholar
  27. Ley MT, Folliard KJ, Hover KC (2009) Observations of air-bubbles escaped from fresh cement paste. Cem Concr Res 39:409–416CrossRefGoogle Scholar
  28. Liddell PV, Boger DV (1996) Yield stress measurements with the vane. J Non-Newton Fluid Mech 63:235–261CrossRefGoogle Scholar
  29. Llewellin EW, Mader HM, Wilson SDR (2002) The rheology of a bubbly liquid. P R Soc A 458:987–1016CrossRefGoogle Scholar
  30. Mabille C, Schmitt V, Gorria P, Leal Calderon F, Faye V, Deminière B, Bibette J (2000) Rheological and shearing conditions for the preparation of monodisperse emulsions. Langmuir 16:422–429CrossRefGoogle Scholar
  31. Mahaut F, Chateau X, Coussot P, Ovarlez G (2008a) Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids. J Rheol 52:287–313CrossRefGoogle Scholar
  32. Mahaut F, Mokéddem S, Chateau X, Roussel N, Ovarlez G (2008b) Effect of coarse particle volume fraction on the yield stress and thixotropy of cementitious materials. Cem Concr Res 38:1276–1285CrossRefGoogle Scholar
  33. Mason TG, Bibette J, Weitz DA (1995) Elasticity of compressed emulsions. Phys Rev Lett 75:2051–2054CrossRefGoogle Scholar
  34. Mewis J, Wagner NJ (2012) Colloidal suspension rheology. Cambridge University Press, CambridgeGoogle Scholar
  35. Mason TG, Bibette J, Weitz DA (1996) Yielding and flow of monodisperse emulsions. J Colloid Interface Sci 179:439–448CrossRefGoogle Scholar
  36. Nguyen QD, Boger DV (1985) Direct yield stress measurement with the vane method. J Rheol 29:335–347CrossRefGoogle Scholar
  37. Ovarlez G, Bertrand F, Rodts S (2006) Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging. J Rheol 50:259–292CrossRefGoogle Scholar
  38. Ovarlez G, Rodts S, Ragouilliaux A, Coussot P, Goyon J, Colin A (2008) Wide-gap Couette flows of dense emulsions: local concentration measurements, and comparison between macroscopic and local constitutive law measurements through magnetic resonance imaging. Phys Rev E 036307:78Google Scholar
  39. Ovarlez G, Barral Q, Coussot P (2010) Three-dimensional jamming and flows of soft glassy materials. Nat Mater 9:115–119CrossRefGoogle Scholar
  40. Ovarlez G, Mahaut F, Bertrand F, Chateau X (2011) Flows and heterogeneities with a vane tool: magnetic resonance imaging measurements. J Rheol 5:197–223CrossRefGoogle Scholar
  41. Ovarlez G, Bertrand F, Coussot P, Chateau X (2012) Shear-induced sedimentation in yield stress fluids. J Non-Newton Fluid Mech 42:148–157Google Scholar
  42. Pal R (2004) Rheological constitutive equation for bubbly suspensions. Ind Eng Chem Res 43:5372–5379CrossRefGoogle Scholar
  43. Parsi F, Gadala-Maria F (1987) Fore-and-aft asymmetry in a concentrated suspension of solid spheres. J Rheol 31:725–732CrossRefGoogle Scholar
  44. Phillips RJ, Armstrong RC, Brown RA, Graham AL, Abbott JR (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys Fluids 4:30–40CrossRefGoogle Scholar
  45. Ramamurthy K, Kunhanandan Nambiar EK, Indu Siva Ranjani G (2009) A classification of studies on properties of foam concrete. Cem Concr Compos 31:388–396CrossRefGoogle Scholar
  46. Rust AC, Manga M (2002a) Effects of bubble deformation on the viscosity of dilute suspensions. J Non-Newton Fluid Mech 104:53–63CrossRefGoogle Scholar
  47. Rust AC, Manga M (2002b) Bubble shapes and orientations in low Re simple shear flow. J Colloid Interface Sci 249:476–480CrossRefGoogle Scholar
  48. Salonen A, Lhermerout R, Rio E, Langevin D, Saint-Jalmes A (2012) Dual gas and oil dispersions in water: production and stability of foamulsion. Soft Matter 8:699–706CrossRefGoogle Scholar
  49. Sikorski D, Tabuteau H, de Bruyn J (2009) Motion and shape of bubbles rising through a yield-stress fluid. J Non-Newton Fluid Mech 159:10–16CrossRefGoogle Scholar
  50. Stickel JJ, Powell RL (2005) Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 37:129–149CrossRefGoogle Scholar
  51. Struble LJ, Jiang Q (2004) Effects of air entrainment on rheology. Materials Journal 101:448–456Google Scholar
  52. Turner D, Dlugogorski B, Palmer T (1999) Factors affecting the stability of foamed concentrated emulsions. Colloids Surf A 150:171–184CrossRefGoogle Scholar
  53. Uhlerr PHT, Guo J, Tiu C, Zhang XM, Zhou JZQ, Fang TN (2005) The shear-induced solid-liquid transition in yield stress materials with chemically different structures. J Non-Newton Fluid Mech 125:101–119CrossRefGoogle Scholar
  54. van Aken GA (2001) Aeration of emulsions by whipping. Colloids Surf A 190:333–354CrossRefGoogle Scholar
  55. Vu TS, Ovarlez G, Chateau X (2010) Macroscopic behavior of bidisperse suspensions of noncolloidal particles in yield stress fluids. J Rheol 54:815–833CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Kogan
    • 1
  • Lucie Ducloué
    • 1
  • Julie Goyon
    • 1
  • Xavier Chateau
    • 1
  • Olivier Pitois
    • 1
  • Guillaume Ovarlez
    • 1
  1. 1.Laboratoire Navier (UMR CNRS 8205)Université Paris-EstChamps-sur-MarneFrance

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