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

For a power law $f(u)\sim u^n$,Independence of $\rho$ requires $n=1$, givingThis 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 $u\ll 1$.

Independence of $\eta$ requires $n=2/3$, givingThis 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 $u\gg 1$.

The crossover occurs when $u$ is order one. Hencewith a dimensionless order-one constant $C_X$ determined by the coefficients and morphology.