= Solution
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>:
$$
\boxed{f(u)\sim C_i u^{2/3},\qquad L(t)\sim C_i\left(\frac\sigma\rho\right)^{1/3}t^{2/3}.}
$$
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 condition
$$
\frac{2(2\beta-\alpha)}9 C_i=\frac\delta{C_i^2},\qquad C_i^3=\frac{9\delta}{2(2\beta-\alpha)}.
$$
A 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>.
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