= Solution
The <dynamical scaling of binary-fluid coarsening> assumes a statistically self-similar bicontinuous morphology with one growing length $L$, thin interfaces, curvature of order $1/L$, and typical fluid speed of order $\dot L$. Inertial acceleration then scales as $\ddot L$ and $\dot L^2/L$, viscous force density as $\eta\dot L/L^2$, and capillary force density as $\sigma/L^2$. The <pressure> is eliminated or estimated together with the interfacial force. The numerical coefficients summarize geometry and projected force directions; this is a scaling closure, not a spatially exact momentum equation.
The approximation further neglects diffusive growth relative to advective growth, assumes fixed volume fraction and material parameters, no external forcing, and no additional relevant macroscopic length. A finite initial domain size, interface width or finite container would introduce further ratios outside the asymptotic scaling regime. Velocity <gradients> need not stay on the domain scale in a genuinely multiscale turbulent flow, so this single-length argument is not a proof for every hydrodynamic morphology.
With $[\rho]=\mathrm{M\,L^{-3}}$, $[\eta]=\mathrm{M\,L^{-1}\,T^{-1}}$ and $[\sigma]=\mathrm{M\,T^{-2}}$, <dimensional analysis> gives the unique scales
$$
L_0=\frac{\eta^2}{\rho\sigma},\qquad t_0=\frac{\eta^3}{\rho\sigma^2},\qquad \frac{L_0}{t_0}=\frac\sigma\eta.
$$
Thus, within the stated asymptotic single-scale hypothesis, the <hydrodynamic coarsening crossover scales> give $L=L_0f(u)$, $u=t/t_0$. All four force-density terms have the same dimensional prefactor $\rho^2\sigma^3/\eta^4$. Dividing by it yields
$$
\boxed{\alpha f''+\beta\frac{f'^2}{f}=\gamma\frac{f'}{f^2}+\frac\delta{f^2}.}
$$
Primes denote $d/du$. The PDF includes $\eta\gamma\dot L/L^2$; the TeX drops $\eta$. The PDF expression is dimensionally consistent. For a positive growth branch, a resisting viscous term and a driving capillary term have opposite signed coefficients; one may set $\gamma=-c_v<0$, $\delta=c_s>0$. The question calls these coefficients order one, not positive. Treating every term on the printed right-hand side as positive would lose the viscous-capillary balance.
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