Structure of solutions of the buoyancy -- drag equation

In this paper, the well-known buoyancy-drag equation (BDE) is studied. This equation describes the non linear regime of Rayleigh -- Taylor instabilities and also the structure of the mixing zone where both fluids are present. Analytical solutions of the BDE are derived for time-dependent accelerations, $\gamma $(t), of the form $\gamma $(t) $\sim $ t$^{n}$ where the exponent n can be positive, negative or zero. It is shown, first, that the width, h(t), of the mixing zone behaves like h$_{n}$(t) $\sim $ t$^{n+2}$ and, second, provided the initial conditions satisfy some constraints, the special solution h$_{n}$(t) is an attractor for t going to infinity. On the other hand, the behavior of the asymtotic solutions for $\gamma $(t) $\sim $ t$^{n}$ is examined in terms of the drag coefficient, C$_{d}$, that is present in the drag force (proportional to the square of the derivative dh/dt) in the right hand side of the BDE. Critical values for this coefficient are derived analytically and it is shown that the asymptotic behaviors are strongly dependent on the value of C$_{d}$. These results are also evidenced from numerical simulations achieved with the CLAWPACK numerical package.