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Numerical simulations of the bubble-bursting phenomenon in two tandem bubbles at the free surface are conducted to explore the influence of a following bubble behind the bursting bubble on the jet ejection at fixed Bo=0.05 and Oh=0.022. The equivalent radius of the bursting bubble ( RB) is fixed and the configuration of two tandem bubbles is varied systematically by changing the equivalent radius of the following bubble ( RF) and the gap distance between the two bubbles ( L). An increase in the bubble-bubble interactive force (repulsive force) is observed with a decrease of L or an increase of RF. As the repulsive force increases, the velocity of the primary capillary wave (PCW) increases due to the reduced wavelength of the PCW, thus increasing the bursting jet velocity. However, when the repulsive force is sufficiently large, the curvature of the PCW near the bottom of the bursting bubble is reversed, causing a new secondary capillary wave to be generated. An increase in the secondary capillary wavelength with an increase in the force disturbs the self-similar behavior of the interface of the bursting bubble, resulting in a decrease of the bursting jet velocity. In order to scale the bursting jet velocity using RF and L in cases where PCWs are important to induce a bursting jet, a scaling law is formulated by defining the scaling variable φ in terms of RF and L. The proposed scaling law is found to be capable of providing accurate predictions of capillary numbers as a function of φ.

Inspired by the intermittent locomotion of fish schools, numerical simulations are performed with two self-propelled flexible fins in a side-by-side configuration with anti-phase oscillation actuated by laterally constrained heaving motions. For an intermittent swimming gait, one type of the half-tail-beating mode (HT mode) and two types of multiple-tail-beating modes coasting at the smallest (MTS mode) and largest (MTL mode) lateral gap distances are applied. Similar to the continuous-tail-beating mode (CT mode), equilibrium lateral gap distances between two fins with HT and MTL modes exist, whereas two fins with MTS mode do not maintain a lateral equilibrium state. Although the cycle-averaged lateral force acting on two fins with CT and MTL modes is mostly determined by an outward deflected jet and enhanced positive pressure between two fins, an added-mass lateral force related to an asymmetric flapping kinematics by passive flexibility also plays an important role in MTL mode to achieve a stable state with a lateral gap distance smaller than that in CT mode. When the cruising speed or the cycle-averaged input power is identical in a stable state, the cost of transport (COT) for two fins with MTL mode is smaller than that with CT mode due to not only a benefit from the intermittent swimming gait but also an enhanced schooling benefit with a small equilibrium lateral gap distance. The COT for two fins with CT mode is reduced further when the bending rigidity increases, whereas it is opposite with MTL mode.