Transient buoyancy driven front dynamics in near horizontal tilted tubes.

The front velocity $V_f$ for interpenetrating light and heavy fluids in a long tube tilted at a small angle $\alpha$ from horizontal is studied as a function of the time $t$ and of $\alpha$ for different Atwood numbers $At$. Above a critical angle $\alpha_c$ which decreases as $At$ increases ($\alpha_c = 5^o$ for $At = 4 \times 10^{-3}$), the front velocity is always controlled by inertia and constant with time with $ V_f = C (At gd)^{0.5}$ ($C \simeq 0.7$). At lower angles ($\alpha \le \alpha_c$), the front dynamics is initially inertial and the constant $C$ decreases to $0.5$ as $\alpha \rightarrow 0$. Then, after a distance $x_c$ diverging to infinity as $\alpha \rightarrow \alpha_c$, the front velocity becomes controlled by viscosity and decreases with distance towards a value $V_{f\infty}$ decreasing with $\alpha$. For an horizontal tube, in the viscous regime, the front velocity decreases to zero as $t^{-0.5}$ ($V_{f\infty} = 0$); the mean concentration profile $C(x,t)$ (reflecting the height of the interface) is self-similar and a function of the reduced variable $x/t^{0.5}$. The variations of $V_f$ with time are well described by a simple equation including the height of the front as a control parameter.