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

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.

: Experimental data and one standard deviation errors. These data appear in figure 4 of the main text.
If we assume that the errors on η r, * and φ p are normally distributed, we can use each measured or calculated value as the mean, together with the known standard deviation error, to calculate a probability distribution for each value (e.g. figure 1). We can then resample the entire dataset, drawing values at random from each probability distribution, and use these values to calculate a best-fit φ m . Repeating this process a very large number of times (10,000, in our case), results in a distribution of values of φ m (figure 2).
Assuming that this distribution is normal, we can calculate a mean, φ m = 0.5933 (which is almost identical to the value calculated from a least squares fit to the original data, φ m = 0.5934), and a standard deviation, σ m = 0.0177. Thus we have an estimate of the potential error in our value of φ m .  2 Will increasing bubble volume fraction increase or decrease reference viscosity?
It is of interest to know the effect of adding an additional volume of bubbles to an existing suspension. Depending on the intial bubble and particle volume fractions, this may cause an increase in viscosity, a decrease in viscosity, or a decrease followed by an increase. At some point in between, the addition of an infinitesimally small volume of bubbles will cause no change in viscosity; this will be the points at which (from equation 3.2 in the main manuscript) By finding the root of this equation, we can determine the location of the regime divide between bubbles causing an increase in viscosity, and bubbles causing a decrease in viscosity. Expanded, the differential equation becomes (2) and the root is an explicit solution cannot be found for φ * p . Equation 3 is plotted in figure  3. For all suspensions with bubble and particle volume fractions to the left of the black curve, the addition of bubbles will lead to an increase in reference viscosity. An example is shown by the blue trajectory, for a fluid that begins with no suspended particles or bubbles. To the right of the black curve, addition of bubbles will initially lead to a decrease in reference viscosity. An example is shown by the red trajectory, for a fluid with an initial particle volume fraction φ * p = 0.5. If enough bubbles are added that the suspension reaches the black curve, viscosity will begin to increase again. In order to determine the effect of adding some volume of bubbles to a suspension, both the initial bubble volume fraction and particle volume fraction must be known.
The effect of adding (or growing) bubbles, for an initially bubble-free suspension, is demonstrated clearly by our data (figure 4). For a particle-free suspension, addition of bubbles will always lead to a viscosity increase, compared to the bubble-free suspension (equivalent to the blue trajectory in figure 3); on the other hand, the addition of bubbles to a concentrated particle suspension leads to a viscosity decrease (equivalent to the red trajectory in figure 3). The black curve is equation 3, and the point φ * p,crit is the point at which addition of bubbles to an initially bubble-free suspension will switch from increasing to decreasing viscosity. The red and blue trajectories are described in the main text. : Relative viscosity η r, * normalised by the relative viscosity of the suspension with the same φ * p but no bubbles, i.e. η r, * / (η r, * at φ * b = 0), for a particle-free suspension (blue datapoints and model curve) and a particle-rich suspension (red datapoints and model curve).