Model H dynamics

Model H couples a conserved scalar composition field to an incompressible momentum density. It combines the advective Cahn--Hilliard equation with the Navier-Stokes equation and the thermodynamic force density $-\varphi \nabla \mu$.

The Korteweg force is the capillary body-force density generated by composition gradients in a diffuse-interface fluid. Up to a pressure gradient it can be written $-\varphi \nabla \mu$, where $\mu$ is the chemical potential.

In bicontinuous phase separation, both fluid phases form interpenetrating connected domains. During late-stage coarsening their morphology is often statistically self-similar after lengths are divided by one characteristic domain size $L(t)$.

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 $\Lambda \sim (\sigma {\tau }^2/\rho {)}^{1/3}$.