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.

Figure 1

Sketch of a confined elongated bubble flowing within a vertical tube and notation used in this work. Regions $AC$ and $DF$ represent the front and rear menisci, respectively; region $CD$ represents the uniform film zone. Points $B$ and $E$, which in the original work of Bretherton [12] defined static menisci regions $AB$ (front) and $EF$ (rear), are not included because under the conditions presently studied the bubble profile does not necessarily end with two static menisci regions.

Figure 2

Examples of bubble shapes extracted from experiments in the $R=1.51\mathrm{mm}$ tube ($\mathrm{Bo}=0.42$) at different liquid flow rates (${\mathrm{Re}}_{\mathrm{l}}\ll 1$). Images contain visualization of the nose and of the central part of the bubble, where the film thickness is uniform.

Figure 3

Comparison of shapes of the bubble front predicted by the theoretical model (solid lines) and shapes extracted from the experiments (symbols). Flow conditions all refer to ${\mathrm{Ca}}_{\mathrm{l}}\approx 0.005$ (${\mathrm{Re}}_{\mathrm{l}}\ll 1$). $x$ and $h$ indicate dimensional quantities.

Figure 4

Comparison of the (a) thickness of the uniform film region and (b) capillary number associated to the bubble velocity given by the model (solid lines) and experiments (symbols), ${\mathrm{Re}}_{\mathrm{l}}\ll 1$. The error bars (in some cases obscured by the solid symbols) refer to the experimental uncertainty measured by taking the average of three measurements for each test condition.

Figure 5

Predictions of the thickness of the uniform film region given by the theoretical model when varying (a) the capillary number of the liquid flow and (b) the Bond number, for ${\mathrm{Re}}_{\mathrm{l}}\ll 1$.

Figure 6

Trends of the bubble capillary number versus Bond number predicted by the model (blue line) when the liquid capillary number is set to a very small value, ${\mathrm{Ca}}_{\mathrm{l}}={10}^{-10}$ (and ${\mathrm{Re}}_{\mathrm{l}}\ll 1$). The black dashed line indicates the results of the theory developed by Bretherton [12] for long bubbles rising in a stagnant liquid [Eq. (24)].

Figure 7

Scaling performance of Eq. (28) versus the theoretical and experimental data (${\mathrm{Re}}_{\mathrm{l}}\ll 1$) for $\mathrm{Bo}<{\mathrm{Bo}}_{\mathrm{cr}}$. The rescaled film thickness $H^{*}$ is defined as $H^{*}=H[3.35{\mathrm{Ca}}_{\mathrm{b}}^{2/3}+{(\sqrt{1-1.12{\mathrm{Bo}}^{1/2}}+\sqrt{1.34}\mathrm{Bo}{\mathrm{Ca}}_{\mathrm{b}}^{1/3})}^{2/3}]$. The rescaled data collapse along a ${\mathrm{Ca}}_{\mathrm{b}}^{2/3}$ line.

Figure 8

Comparison of shapes of the bubble rear predicted as solution of Eq. (32) (solid lines) and shapes extracted from the full numerical simulations (symbols). Flow conditions are ${\mathrm{Ca}}_{\mathrm{l}}=0.00464$ and ${\mathrm{Re}}_{\mathrm{l}}=92.8$. $x\left(h_{\min }\right)$ denotes the axial location where the minimum value of the film thickness is measured. $x$ and $h$ indicate dimensional quantities.

Figure 9

Comparison of the thickness of the uniform film region $H$, bubble to average liquid velocity $U_b/U_l$ and wavelength $\lambda /\ell$ of the ripples appearing at the rear meniscus of the bubble, for three constant values of the liquid capillary number. Full markers identify the results of numerical simulations and solid lines are the predictions of the model, Eq. (32).

Figure 10

Comparison of the (a) thickness of the uniform film region $H$, (b) bubble to average liquid velocity $U_b/U_l$, and (c) wavelength $\lambda /\ell$ of the ripples appearing at the rear meniscus of the bubble obtained as solution of Eq. (32) (solid lines) and numerical simulations (symbols), for ${\mathrm{Ca}}_{\mathrm{l}}=0.00464$ and ${\mathrm{Re}}_{\mathrm{l}}=92.8$. The numerical simulation for $\mathrm{Bo}=5$ gave time-dependent profiles of the bubble rear, the value of the wavelength reported in the figure is an average in time.