Assume late-stage <bicontinuous phase separation> has one characteristic domain size $L(t)$ and statistically self-similar morphology. The <dynamical scaling of binary-fluid coarsening> then estimates
$$
|\mathbf v|\sim\dot L,
\qquad
|\partial_t\mathbf v|\sim\ddot L,
\qquad
|(\mathbf v\mathbin\cdot\nabla)\mathbf v|
\sim\frac{\dot L^2}{L},
\qquad
|\nabla^2\mathbf v|\sim\frac{\dot L}{L^2}.
$$
The interface has <surface tension> $\sigma$. Its curvature pressure is of order $\sigma/L$, so its coarse-grained gradient, equivalently the transverse part of $-\phi\nabla\mu$, scales as $\sigma/L^2$. Diffusive composition transport is neglected because hydrodynamic advection controls the late stage, and $\rho,\eta,\sigma$ are treated as constants.

The pressure cannot simply be discarded: in an <incompressible flow> it cancels the longitudinal part of the other forces and contains both capillary and dynamic contributions. Applying the divergence-free projection to the momentum equation eliminates $\nabla P$ while retaining a solenoidal force with the same scaling. Geometry, projection, signs, and correlations are absorbed into dimensionless order-one coefficients. The result is
$$
\rho\left(\alpha\ddot L+\beta\frac{\dot L^2}{L}\right)
=\eta\gamma\frac{\dot L}{L^2}
+\sigma\frac{\delta}{L^2}.
$$

Solution

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

The dynamical-scaling hypothesis replaces all macroscopic lengths by one scale $L(t)$ and the characteristic velocity by $\dot L$. Then inertial, viscous, and capillary force densities scale respectively as $\rho\ddot L+\rho\dot L^2/L$, $\eta\dot L/L^2$, and $\sigma/L^2$.


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