The flow of liquid (water) and air bubbles has been attracting the interest of many researchers. Experimental investigations of the flow including microbubbles are now well developed regarding engineering applications, such as drag reduction of pipe flow, cleaning of polymer ink and separation of carbon fibers using microbubbles. However, numerical investigations of the liquid flow containing air bubbles are facing difficulty in numerical accuracy and a heavy computational load. Multiphase flow has been computationally studied by the volume of fluid method (VOF), the discrete element method (DEM), the immersed boundary method (IBM), the moving particle semi-implicit method (MPS), and the theoretical model equations taking into account of inertial and added-mass body acceleration forces. Although VOF can be applied to many complex configurations, it needs much CPU time and heavy memory capacity. IBM proposed by Peskin has been largely applied to solve the liquid motion including solid particles. But it cannot be used to solve particle-liquid flow if the particle density is less than 0.3 of the liquid density. MPS also has a problem in a heavy calculation load.
Takemura et al. performed experimental studies of a rising air bubble and a solid-like air bubble near a vertical wall. There are some works which studied multiphase flow including plural bubbles. At low Reynolds numbers, Katz and Meneveau conducted experiments of the interacting air bubbles rising in a quiescent distilled water, and observed the velocities of the two bubbles rising in line. Watanabe and Sanada and Kusuno et al. performed experiments of two rising bubbles in a quiescent silicon oil.
The force-coupling method (FCM) originally proposed by Maxey and Patel, which we call the original FCM (OFCM) in this paper, requires no moving boundaries of particles, but represents the center position of a particle by the point-body force acting from the particle to the liquid.