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 Vf 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 Vt and the Reynolds number Ret, the normalized front velocity Vf∕Vt and dispersion coefficient D∕(Vtd) are observed to scale, respectively, as Ret−3∕4 and Ret−3∕2 for Ret≲1000. Also, Vf increases linearly with tanθ and the ratio (D∕Vf2) remains of the order of (35±10)d∕Vt 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 Vf2 is discussed in terms of the characteristic time for transverse mixing across the flow channel.