= Solution
For a power law $f(u)=C u^p$, restoring dimensions gives
$$
L(t)=C\eta^{2-3p}\rho^{p-1}\sigma^{2p-1}t^p.
$$
Independence of <mass density> requires $p-1=0$. Hence <viscous hydrodynamic coarsening> has
$$
\boxed{f(u)\sim C_vu,\qquad L(t)\sim C_v\frac\sigma\eta t.}
$$
At small $u$, the capillary and viscous terms are both of order $u^{-2}$, while the convective inertial term is of order $u^{-1}$ and the acceleration of the leading linear law is zero. Thus the leading force balance is viscous resistance against capillary driving, giving $C_v=c_s/c_v=-\delta/\gamma>0$ in the signed convention of the preceding part.
Equivalently, the domain-scale <Reynolds number> is $\rho L\dot L/\eta\sim C_v^2u$, so inertia is negligible as $u\to0$. “Small” here refers to time compared with $t_0$ within a late-stage hydrodynamic window: it does not extend the model through the microscopic initial stage, where diffusion or interface formation can dominate. An initial-time offset changes the intercept, not this asymptotic growth exponent.
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