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 $-\varphi \nabla \mu$.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.

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 $\sigma /L^2$.

Replacing domain-scale viscous stresses by a local linear drag density $-\bar{\eta }\mathbf{v}$ gives the scales $L_1=(\sigma \rho /{\bar{\eta }}^2{)}^{1/3}$ and $t_1=\rho /\bar{\eta }$. The single-scale force model isFor a regular power-law asymptotic $g\sim C{u}^y$, $g^{\prime }\sim Cy{u}^{y-1}$, $g^{\prime \prime }=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, the two inertial-to-drag ratios scale as $(y-1)/u$ and $y/u$. Thus drag dominates inertia at late time, and drag-capillary balance integrates to $g^3\sim 3(c_s/c_d)u$. This proves the displayed $1/3$ exponent and its independence of mass density. The scaling assumes a homogenized stationary network and dominant advective transport. Diffusive Ostwald ripening can have the same exponent, so its omission must be justified separately; order-parameter mobility can otherwise enter the prefactor.

Under single-length dynamical scaling of binary-fluid coarsening, mass density $\rho$, dynamic viscosity $\eta$ and surface tension $\sigma$ determine the displayed unique length and time scales by dimensional analysis. Writing $L=L_{0}f(t/t_0)$ reduces the force-density estimate to $\alpha f^{\prime \prime }+\beta f^{\prime 2}/f=\gamma f^{\prime }/f^2+\delta /f^2$. Resistive and driving terms need opposite signed geometric coefficients. Viscous hydrodynamic coarsening has $f\sim u$ and inertial hydrodynamic coarsening has $f\sim u^{2/3}$. These follow respectively from density-independent and viscosity-independent powers in $L\propto {\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. They require thin interfaces, statistically self-similar bicontinuous domains, dominant advection and no additional relevant macroscopic scales. A degeneracy in the signed inertial coefficients can invalidate the pure-power force balance without invalidating the dimensional calculation.

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