Dynamical scaling of binary-fluid coarsening

The dynamical-scaling hypothesis replaces all macroscopic lengths by one scale $L(t)$ and the characteristic velocity by $\dot{L}$. Then inertial, viscous, and capillary force densities scale respectively as $\rho \ddot{L}+\rho {\dot{L}}^2/L$, $\eta \dot{L}/L^2$, and $\sigma /L^2$.

Balancing viscous and capillary forces gives $L(t)\sim (\sigma /\eta )t$. This regime is independent of the mass density and applies below the viscous-to-inertial crossover scale.

Balancing inertia and capillarity gives $L(t)\sim (\sigma /\rho {)}^{1/3}{t}^{2/3}$. This regime is independent of viscosity and applies above the viscous-to-inertial crossover scale.

If stirring imposes a time scale $\tau$, replacing time derivatives by $1/\tau$ predicts a steady domain size. Viscous-capillary balance gives $\Lambda \sim \sigma \tau /\eta$, while inertial-capillary balance gives $\Lambda \sim (\sigma {\tau }^2/\rho {)}^{1/3}$.