On the motion of dispersed balls in a potential flow: A kinetic description of the added mass effect

In order to describe the motion of a large number of bubbles in a three-dim ensional flow and to rigorously put into evidence the added mass effect, Ru sso (SIAM J. Appl. Math., 56 (1996), pp. 327-357) and Smereka (J. Fluid Mec h., 248 (1993), pp. 79-112) considered the following very idealized situati on. The bubbles are assumed to be spherical particles of radius a. The exte rnal fluid is simply represented as a dipole approximation of a potential f low. The motion of the bubbles is created only by the pressure forces on th e boundary of each bubble. In this paper we go further, showing that the ex act kinetic equation, of Vlasov type, for the dispersed phase can be exactl y derived and no further approximation is needed to write the mean field eq uation. We also prove that this model preserves the energetic structure of the departing dynamical system. This has effects on the solution computed b y numerical simulation, and we show that the behavior of the particles can be different if a truncation is used.