= Drag-limited hydrodynamic coarsening
{title2=$L\sim(\sigma t/\bar\eta)^{1/3}$}
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 is
$$
\alpha g''+\beta g'^2/g=-c_dg'+c_s/g^2,\qquad L=L_1g(t/t_1),\quad c_d,c_s>0.
$$
For a regular power-law asymptotic $g\sim Cu^y$, $g'\sim Cyu^{y-1}$, $g''=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, 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 $g^3\sim3(c_s/c_d)u$. 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.
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