Dynamics of long gas bubbles rising in a vertical tube in a cocurrent liquid flow

When a confined long gas bubble rises in a vertical tube in a cocurrent liquid flow, its translational velocity is the result of both buoyancy and mean motion of the liquid. A thin film of liquid is formed on the tube wall and its thickness is determined by the interplay of viscous, inertial, capillary and buoyancy effects, as defined by the values of the Bond number ($\mathrm{Bo}\equiv \rho g{R}^2/\sigma$ with $\rho$ being the liquid density, $g$ the gravitational acceleration, $R$ the tube radius, and $\sigma$ the surface tension), capillary number (${\mathrm{Ca}}_{\mathrm{b}}\equiv \mu U_b/\sigma$ with $U_b$ being the bubble velocity and $\mu$ the liquid dynamic viscosity), and Reynolds number (${\mathrm{Re}}_{\mathrm{b}}\equiv 2\rho U_{b}R/\mu$). We perform experiments and numerical simulations to investigate systematically the effect of buoyancy ($\mathrm{Bo}=0\text{–}5$) on the shape and velocity of the bubble and on the thickness of the liquid film for ${\mathrm{Ca}}_{\mathrm{b}}={10}^{-3}\text{–}{10}^{-1}$ and ${\mathrm{Re}}_{\mathrm{b}}={10}^{-2}\text{–}{10}^3$. A theoretical model, based on an extension of Bretherton's lubrication theory, is developed and utilized for parametric analyses; its predictions compare well with the experimental and numerical data. This study shows that buoyancy effects on bubbles rising in a cocurrent liquid flow make the liquid film thicker and the bubble rise faster, when compared to the negligible gravity case. In particular, gravitational forces impact considerably the bubble dynamics already when $\mathrm{Bo}<0.842$, with ${\mathrm{Bo}}_{\mathrm{cr}}=0.842$ being the critical value below which a bubble does not rise in a stagnant liquid in a circular tube. The liquid film thickness and bubble velocity in a liquid coflow may vary by orders of magnitude as a result of small changes of $\mathrm{Bo}$ around this critical value. The reduction of the liquid film thickness for increasing values of the Reynolds numbers, usually observed for ${\mathrm{Re}}_{\mathrm{b}}\lesssim {10}^2$ when $\mathrm{Bo}\ll 1$, becomes more evident at larger Bond numbers. Buoyancy effects also have a significant influence on the features of the undulation appearing near the rear meniscus of the bubble, as they induce a substantial increase in its amplitude and decrease in its wavelength.