Puzzling Bubble Rise Speed Increase in Dense Granular Suspensions

We present an anomalous experimental observation on the rising speed of air bubbles in a Hele-Shaw cell containing a suspension of spherical, neutrally buoyant, non-Brownian particles. Strikingly, bubbles rise faster in suspensions as compared to particle-less liquids of the same effective viscosity. By carefully measuring this bubble speed increase at various particle volume fraction and via velocity field imaging, we demonstrate that this strange bubble dynamics is linked to a reduction in the bulk dissipation rate. A good match between our experimental data and computations based on a Suspension Balance Model (SBM) illustrates that the underlying mechanism for this dissipation-rate deficit is related to a nonuniform particle distribution in the direction perpendicular to the channel walls due to shear-induced particle migration.

Figure 1

Rise of an air bubble in a Hele-Shaw cell filled with a viscous suspension composed of a water and UCON™ mixture and polystyrene (PS) particles. Schematic of the (a) front view and (b) side view of the setup. (c) Normalized suspension viscosity $\widehat{\eta }(\varphi )\equiv {\eta }_s({\varphi }_0)/{\eta }_0$. Symbols represent experimental data and the green region is Eq. (1) with ${\varphi }_c\in [0.55-0.62]$. (d) Time-lapse imaging of rising air bubbles ($d_b=12.5\pm 0.1\mathrm{mm}$) in a suspension ($d_p=230\mu \mathrm{m}$) for different bulk packing fractions ${\varphi }_0$.

Figure 2

Raw experimental datasets for different particle-less water and UCON mixtures (inset) collapse onto a unique master curve when plotted as normalized bubble rise velocity $v_b/v_0^{\star }$ vs the normalized diameter $d_b/h$. The $y$ axis error bars are computed through the fluctuations of $v_b$ in time.

Figure 3

(a) Comparison of bubble rise velocity $v_b$ between a liquid with particles and a pure liquid of same bulk viscosity, 0.53 Pa.s showing that bubble rises faster in the former. Note that a plateau can be observed if the velocity is divided by the aspect ratio $L/D$ [Eq. (2)]. (b) Normalized bubble velocity $v_b/v_s^{\star }$ vs normalized diameter $d_b/h$ illustrates the anomalous bubble rise speed increase at various ${\varphi }_0$. Empirical fits (dashed-dotted lines) are used to calculate $v_b/v_s^{\star }$ when $d_b\gg h$, whose value is then plotted in (c) as a function of suspension volume fraction ${\varphi }_0$ for both $d_p=230$ and $80\mu \mathrm{m}$. Symbols represent experimental data and the green region Eq. (4) using a volume fraction profile given by Eq. (5) with ${\varphi }_c\in [0.55-0.62]$.

Figure 4

Magnitude of the local velocity around a bubble of size $d_b=25\mathrm{mm}$ for various packing fraction ${\varphi }_0$. Dotted line represents the theoretical dipolar field, $(v_b{d}_b/4)r^{-2}$. The $y$ axis error bars are computed from standard error over different temporal realizations of $|\tilde{\mathbf{v}}/v_b|$. Inset: instantaneous velocity field from PIV measurements (${\varphi }_0=0.10$, $d_p=230\mu \mathrm{m}$).

Figure 5

Local dissipation rate ${\mathcal{D}}_v(\widehat{y})={\eta }_s(\widehat{y}){|\partial \mathbf{v}/\partial \widehat{y}|}^2$ across the channel for a suspension with (continuous line) and without (dashed line) particle migration. Here, ${\mathcal{D}}_v^0(\widehat{y}=1)$ is the dissipation rate at the wall in the case of uniform particle distribution $\varphi (\widehat{y})={\varphi }_0$. Clearly, ${\mathcal{D}}_v(\widehat{y})$ is smaller in the case with shear-induced migration. Inset: corresponding velocity (red) and volume fraction (black) profiles.