Front dynamics and macroscopic diffusion in buoyant mixing in a tilted tube

The buoyancy driven interpenetration of two fluids of different densities has been studied in a long tilted tube in the strong mixing regime for which the mean concentration profile along the tube length satisfies a macroscopic diffusion equation. Variations of the corresponding macroscopic diffusion coefficient D and of the front velocity V_f are studied as a function of the Atwood number At, the viscosity ν, the tube diameter d and the tilt angle θ. Introducing the characteristic inertial velocity V_t and the Reynolds number Re_t, the normalized front velocity V_f/ V_t and dispersion coefficient D/ (V_td) are observed to scale respectively as Re_t^-3/4 and Re_t^-3/2 for Re_t1000. Also, V_f increases linearly with θ and the ratio (D/V_f^2) remains of the order of (35 ±10) d/V_t in a wide range of values of the tilt angle and of the other control parameters. This close relation observed between the variations of D and V_f^2 will be discussed in terms of the characteristic time for transverse mixing across the flow channel.