Non-negligible buoyancy effect on bubbles travelling in horizontal microchannels of comparable size at small Bond numbers

When a gas bubble is transported by a liquid flow confined within a horizontal channel of comparable size, buoyancy effects are usually assumed to be negligible if the Bond number of the flow is less than unity. However, recent experimental studies showed that buoyancy may still significantly impact the bubble dynamics even when $\mathrm{Bo}\ll 1$, provided that the flow speed is sufficiently small, such that the viscous and inertial forces are weak. To derive a new criterion to assess the significance of buoyancy on the flow of small bubbles in horizontal microchannels, we have performed systematic numerical simulations using the free software Basilisk, covering a wide range of Bond, capillary and Reynolds numbers, $\mathrm{Bo}=0.004-0.4$, $\mathrm{Ca}=10^{-4}-0.5$ and $\mathrm{Re}=0-100$, and exploring the bubble-to-channel diameter ratios $d_b/D=0.2-0.9$. We demonstrate that the nondimensional group $\mathrm{Bo}{(d_b/D)}^2/\mathrm{Ca}$ is effective in assessing the importance of buoyancy in flows with negligible inertial effects. When $\mathrm{Bo}{(d_b/D)}^2/\mathrm{Ca}<0.1$, buoyancy effects are negligible and the bubble travels along the channel axis. When $\mathrm{Bo}{(d_b/D)}^2/\mathrm{Ca}>10$, buoyancy effects dominate and the bubble travels in the vicinity of the upper wall. For intermediate values, bubbles take equilibrium positions between the channel centre and the wall, and the threshold $\mathrm{Bo}{(d_b/D)}^2/\mathrm{Ca}=1$ is effective in predicting whether bubbles will travel closer to the channel centre or to the wall. The capillary number at the denominator describes the lift force generated by the deformation of the bubble due to viscous shear, which acts towards the channel centre thus opposing buoyancy. Inertial forces significantly alter the shape and equilibrium position of the bubble when the Weber number of the flow exceeds unity and $\mathrm{Bo}/\mathrm{We}<1$. Under the action of inertia, bubbles of size comparable to the channel diameter become elongated but remain centred. Smaller bubbles migrate towards the channel wall and do not exhibit a preferential near-wall circumferential position.