For a <power law> $f(u)\sim u^n$,
$$
L(t)\sim
t^n\rho^{\,n-1}\sigma^{\,2n-1}\eta^{\,2-3n}.
$$
Independence of $\rho$ requires $n=1$, giving
$$
f(u)\sim u,
\qquad
L(t)\sim\frac{\sigma}{\eta}t.
$$
This 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 $u\ll1$.

Independence of $\eta$ requires $n=2/3$, giving
$$
f(u)\sim u^{2/3},
\qquad
L(t)\sim\left(\frac{\sigma}{\rho}\right)^{1/3}t^{2/3}.
$$
This 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 $u\gg1$.

The crossover occurs when $u$ is order one. Hence
$$
t_X=C_Xt_0
$$
with a dimensionless order-one constant $C_X$ determined by the coefficients and morphology.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 2 / d / Solution

Balancing viscous and capillary forces gives $L(t)\sim(\sigma/\eta)t$. This regime is independent of the mass density and applies below the viscous-to-inertial crossover scale.


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 Incoming links: Viscous hydrodynamic coarsening