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 $\sigma /L^2$.

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