Mass transfer around bubbles flowing in cylindrical microchannels

This work focuses on the mass transfer around unconfined bubbles in cylindrical microchannels when they are arranged in a train. We characterise how the mass transfer, quantified by the Sherwood number, $Sh$, is affected by the channel and bubble sizes, distance between bubbles, diffusivity, mean flow velocity, deformation of the bubble, the presence of surfactants in the limit of rigid interface and off-centred positions of the bubbles. We analyse the influence of the dimensionless numbers and especially the distance between bubbles and the Péclet number, $Pe$, which we vary over eight decades, identifying five different mass transfer regimes. We show different concentration patterns and the dependence of the Sherwood numbers. These regimes can be classified by either the importance of the diffusion along the streamlines or the interaction between bubbles. For small $Pe$ the diffusion along the streamlines is not negligible as compared to convection, whereas for large $Pe$ convection dominates in the streamlines direction and, thus, crosswind diffusion becomes crucial in governing the mass transfer through boundary layers or the remaining wake behind the bubbles. Interaction of bubbles occurs for very small $Pe$ where the mass transfer is purely diffusive, or for very large $Pe$ where long wakes eventually reach the following bubble. We also observe that the bubble deformability mainly affects the $Sh$ in the regime for very large $Pe$ in which bubbles interaction matters, and also that the rigid interface affects the boundary layer and the remaining wake. The effect of off-centred position of the bubble, determined by the transverse force balance, is also limited to large $Pe$. The boundary layers on rigid bubble surfaces are thicker than those on stress-free bubble surfaces, and thus the mass transfer is weaker. For centred bubbles, the influence of inertia on the mass transfer is negligible. Finally, we discuss the implication of our results on the dissolution of bubbles.