The rheology of three-phase suspensions at low bubble capillary number

J. M. Truby*, S. P. Mueller, E. W. Llewellin, H. M. Mader

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

37 Citations (Scopus)


We develop a model for the rheology of a three-phase suspension of bubbles and particles in a Newtonian liquid undergoing steady flow. We adopt an 'effective-medium' approach in which the bubbly liquid is treated as a continuous medium which suspends the particles. The resulting three-phase model combines separate two-phase models for bubble suspension rheology and particle suspension rheology, which are taken from the literature. The model is validated against new experimental data for three-phase suspensions of bubbles and spherical particles, collected in the low bubble capillary number regime. Good agreement is found across the experimental range of particle volume fraction (0 ≤ φp ≲ 0.5) and bubble volume fraction (0 ≤ φb ≲ 0.3). Consistent with model predictions, experimental results demonstrate that adding bubbles to a dilute particle suspension at low capillarity increases its viscosity, while adding bubbles to a concentrated particle suspension decreases its viscosity. The model accounts for particle anisometry and is easily extended to account for variable capillarity, but has not been experimentally validated for these cases.

Original languageEnglish
Article number20140557
Number of pages18
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2173
Early online date27 Nov 2014
Publication statusPublished - Jan 2015


  • rheology
  • particle suspension
  • bubble suspension
  • analogue experiments
  • three phase
  • FLOW

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