General power-law temporal scaling for unequal-size microbubble coalescence

We systematically study the effects of liquid viscosity, liquid density, and surface tension on global microbubble coalescence using lattice Boltzmann simulation. The liquid-gas system is characterized by Ohnesorge number $\text{Oh}\equiv {\eta }_h/\sqrt{{\rho }_h\sigma r_F}$ with ${\eta }_h,{\rho }_h,\sigma$, and $r_F$ being viscosity and density of liquid, surface tension, and the radius of the larger parent bubble, respectively. This study focuses on the microbubble coalescence without oscillation in an Oh range between 0.5 and 1.0. The global coalescence time is defined as the time period from initially two parent bubbles touching to finally one child bubble when its half-vertical axis reaches above 99% of the bubble radius. Comprehensive graphics processing unit parallelization, convergence check, and validation are carried out to ensure the physical accuracy and computational efficiency. From 138 simulations of 23 cases, we derive and validate a general power-law temporal scaling $T^{*}=A_0{\gamma }^{-n}$, that correlates the normalized global coalescence time $\left(T^{*}\right)$ with size inequality $\left(\gamma \right)$ of initial parent bubbles. We found that the prefactor $A_0$ is linear to Oh in the full considered Oh range, whereas the power index $n$ is linear to Oh when $\text{Oh}<0.66$ and remains constant when $\text{Oh}>0.66$. The physical insights of the coalescence behavior are explored. Such a general temporal scaling of global microbubble coalescence on size inequality may provide useful guidance for the design, development, and optimization of microfluidic systems for various applications.

Figure 1

Schematic diagrams for a global coalescence of two unequal-size microbubbles in a square domain periodic in both directions. (a) Initial parent bubbles (solid circles) and final child bubble (dashed-dotted circle). (b) A coalescing child bubble with $D_y$ defined as the half-vertical axis at the center of horizontal axis.

Figure 2

(a) Illustration of the half-vertical axis $\left(D_y\right)$ defined as the vertical distance from the bubble center to the top interface of the bubble. (b) Simulation results (symbols) are well aligned on the half power-law scaling (line) from analytical prediction [6], demonstrating the validity of the LBM simulation.

Figure 3

Power-law temporal scaling of unequal-size microbubble coalescence with three different Oh numbers corresponding to cases 1, 5, and 10 in Table 3. Symbols: numerical results; lines: power-law fitting.

Figure 4

Effects of Oh number on (a) prefactor $A_0$ and (b) power index $n$ with $R^2=0.983$ in the power-law temporal scaling. Symbols: numerical results; lines: fitting outcomes.

Figure 5

General power-law temporal scaling for unequal-size air microbubble coalescence at $\text{Oh}=0.509$ (a), 0.654 (b), and $\text{Oh}=0.8$ (c). Symbols: simulation results. Lines: general power-law [Eq. (10)] prediction.