Analytical study of the buoyancy-drag equation

The buoyancy-drag equation (BDE) is used for describing the time evolution of the mixing zone, between two fluids, due to RayleighTaylor instabilities in a highly non-linear regime. The BDE is similar to an equation of motion for each fluid and, in addition to the inertial term, it contains a buoyancy term (proportional to the external acceleration experienced by the system) and a friction proportional to the square of the velocity. In this paper, the integrability of the BDE is studied using Lie point symmetries. Two relevant situations are studied analytically. First, for a constant acceleration, the general solution with two arbitrary constants is derived. On the other hand, for accelerations varying like a power of time, we obtain a one-parameter family of solutions. These theoretical results can be compared with the various experimental results published in the current literature.