= Hydrodynamic coarsening crossover scales
{title2=$L_0=\eta^2/(\rho\sigma),\quad t_0=\eta^3/(\rho\sigma^2)$}
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 $L=L_0f(t/t_0)$ reduces the force-density estimate to $\alpha f''+\beta f'^2/f=\gamma f'/f^2+\delta/f^2$. 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.
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