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

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 estimatesThe interface has surface tension $\sigma$. Its curvature pressure is of order $\sigma /L$, so its coarse-grained gradient, equivalently the transverse part of $-\varphi \nabla \mu$, 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