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

The thermodynamic order parameter is the conserved scalar composition difference $\varphi$ of the binary fluid mixture. The momentum density $\rho \mathbf{v}$ is an additional conserved hydrodynamic mode; it is not a symmetry-breaking order parameter, but it must be retained because momentum relaxes only through spatial transport.

Composition conservation gives a continuity equation,The current $\mathbf{J}=-M\nabla \mu$ is the leading isotropic, dissipative constitutive law: it drives material down gradients of the chemical potential. For the Landau-Ginzburg theorythe functional derivative isThese statements give the advective Cahn--Hilliard equation.

Constant mass density and incompressible flow require $\nabla \cdot \mathbf{v}=0$. Momentum conservation gives the Navier-Stokes equation: material acceleration equals the sum of the Newtonian viscous force $\eta {\nabla }^2\mathbf{v}$, the pressure force $-\nabla P$, and the Korteweg force $-\varphi \nabla \mu$. The pressure is the Lagrange multiplier enforcing incompressibility. Together these equations are Model H dynamics; isothermality removes the need for a separate energy equation.

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

SetThen $\dot{L}=(L_0/t_0)f^{\prime }$ and $\ddot{L}=(L_0/t_0^2)f^{\prime \prime }$. Every term in the scaling equation has the common dimensions and magnitude ${\rho }^2{\sigma }^3/{\eta }^4$. Cancelling that factor yields the dimensionless equationor equivalently

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.

For an arrested domain size $\Lambda$, make the prescribed replacementWritinggives the schematic algebraic equationor

For $w\ll 1$, viscous-capillary balance giveswhich is independent of $\rho$. For $w\gg 1$, inertial-capillary balance giveswhich is independent of $\eta$. These are the two limits of the stirring-arrested binary-fluid domain size.

Inertia dominates viscosity whenequivalently when the imposed root-mean-square velocity gradient satisfies ${\tau }^{-1}\ll t_0^{-1}$. In the inertial-capillary regime,The corresponding Reynolds number isThe geometry is therefore not fixed as the stirring rate changes. Slower stirring permits much larger domains, and the growth of ${\Lambda }^2$ more than offsets the reduction of $U/\Lambda$. This is why the inertial regime occurs at low imposed velocity gradient in this self-adjusting coarsening problem.