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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.

For $f(u)=C{u}^p$, the dimensional expression is again $L=C{\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. Independence of dynamic viscosity requires $2-3p=0$, giving inertial hydrodynamic coarsening:For this law, both inertial terms and the capillary term scale as $u^{-4/3}$, whereas the viscous term scales as $u^{-5/3}$. It is consequently subleading at large $u$. Direct substitution gives the leading amplitude conditionA growing branch requires the indicated ratio to be positive. The inertial combination must be nonzero: if $2\beta =\alpha$ its leading acceleration and convective contributions cancel, so this particular schematic equation cannot realize the generic inertial-capillary balance with that pure power law. The requested scaling assumes no such geometrical degeneracy. This also shows why the coefficients cannot determine a universal numerical prefactor without a morphology calculation. The relative viscous contribution decays as $u^{-1/3}$, consistently with an increasingly large domain-scale Reynolds number.

For a power law $f(u)\sim u^n$,Independence of $\rho$ requires $n=1$, givingThis is viscous hydrodynamic coarsening. The viscous and capillary terms both scale as $u^{-2}$, whereas the convective inertial term scales as $u^{-1}$. It is therefore subleading for $u\ll 1$.

Independence of $\eta$ requires $n=2/3$, givingThis is inertial hydrodynamic coarsening. Both inertial terms and the capillary term scale as $u^{-4/3}$, while the viscous term scales as $u^{-5/3}$ and is subleading for $u\gg 1$.

The crossover occurs when $u$ is order one. Hencewith a dimensionless order-one constant $C_X$ determined by the coefficients and morphology.