Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 2

The field $\varphi$ is a compositional order parameter in a binary fluid mixture. Its spatial integral is conserved in a closed sample, so advection and diffusion enter a continuity equation; the material derivative follows the incompressible fluid. The diffusive current $\mathbf{J}=-M\nabla \mu$, with positive order-parameter mobility $M$, moves material down the chemical potential gradient. The chemical potential of a composition field is the functional derivative of the symmetric quartic Landau free energyso $\mu =a\varphi +b{\varphi }^3-\kappa {\nabla }^2\varphi$. Positive $b,\kappa$ stabilize large amplitudes and penalize sharp interfaces; $a<0$ permits phase separation.

The momentum equation is the Navier-Stokes equation with dynamic viscosity $\eta$, constant mass density $\rho$, and the Korteweg force $-\varphi \nabla \mu$ arising from composition stresses. The pressure enforces incompressible flow, $\nabla \cdot \mathbf{v}=0$, and can absorb pure gradient contributions to that thermodynamic force. For example, $-\varphi \nabla \mu =\mu \nabla \varphi -\nabla (\varphi \mu )$ gives an equivalent force after redefining the pressure. These equations constitute deterministic Model H dynamics.

Their dissipative signs can be checked by the energy dissipation identity for Model H. Under periodic or no-flux, no-work boundary conditions, composition advection and capillary work cancel between $F$ and the kinetic energy, givingThus diffusion and viscosity dissipate energy, while the reversible advective coupling transfers it between composition and motion. Thermal noises are omitted at this deterministic hydrodynamic level.

The dynamical scaling of binary-fluid coarsening assumes a statistically self-similar bicontinuous morphology with one growing length $L$, thin interfaces, curvature of order $1/L$, and typical fluid speed of order $\dot{L}$. Inertial acceleration then scales as $\ddot{L}$ and ${\dot{L}}^2/L$, viscous force density as $\eta \dot{L}/L^2$, and capillary force density as $\sigma /L^2$. The pressure is eliminated or estimated together with the interfacial force. The numerical coefficients summarize geometry and projected force directions; this is a scaling closure, not a spatially exact momentum equation.

The approximation further neglects diffusive growth relative to advective growth, assumes fixed volume fraction and material parameters, no external forcing, and no additional relevant macroscopic length. A finite initial domain size, interface width or finite container would introduce further ratios outside the asymptotic scaling regime. Velocity gradients need not stay on the domain scale in a genuinely multiscale turbulent flow, so this single-length argument is not a proof for every hydrodynamic morphology.

With $[\rho ]=\mathrm{M}{\mathrm{L}}^{-3}$, $[\eta ]=\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}$ and $[\sigma ]=\mathrm{M}{\mathrm{T}}^{-2}$, dimensional analysis gives the unique scalesThus, within the stated asymptotic single-scale hypothesis, the hydrodynamic coarsening crossover scales give $L=L_{0}f(u)$, $u=t/t_0$. All four force-density terms have the same dimensional prefactor ${\rho }^2{\sigma }^3/{\eta }^4$. Dividing by it yieldsPrimes denote $d/du$. The PDF includes $\eta \gamma \dot{L}/L^2$; the TeX drops $\eta$. The PDF expression is dimensionally consistent. For a positive growth branch, a resisting viscous term and a driving capillary term have opposite signed coefficients; one may set $\gamma =-c_v<0$, $\delta =c_s>0$. The question calls these coefficients order one, not positive. Treating every term on the printed right-hand side as positive would lose the viscous-capillary balance.

For a power law $f(u)=C{u}^p$, restoring dimensions givesIndependence of mass density requires $p-1=0$. Hence viscous hydrodynamic coarsening hasAt small $u$, the capillary and viscous terms are both of order $u^{-2}$, while the convective inertial term is of order $u^{-1}$ and the acceleration of the leading linear law is zero. Thus the leading force balance is viscous resistance against capillary driving, giving $C_v=c_s/c_v=-\delta /\gamma >0$ in the signed convention of the preceding part.

Equivalently, the domain-scale Reynolds number is $\rho L\dot{L}/\eta \sim C_v^{2}u$, so inertia is negligible as $u\to 0$. “Small” here refers to time compared with $t_0$ within a late-stage hydrodynamic window: it does not extend the model through the microscopic initial stage, where diffusion or interface formation can dominate. An initial-time offset changes the intercept, not this asymptotic growth exponent.

For $f(u)=C{u}^p$, the dimensional expression is again $L=C{\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. Independence of dynamic viscosity requires $2-3p=0$, giving inertial hydrodynamic coarsening:For this law, both inertial terms and the capillary term scale as $u^{-4/3}$, whereas the viscous term scales as $u^{-5/3}$. It is consequently subleading at large $u$. Direct substitution gives the leading amplitude conditionA growing branch requires the indicated ratio to be positive. The inertial combination must be nonzero: if $2\beta =\alpha$ its leading acceleration and convective contributions cancel, so this particular schematic equation cannot realize the generic inertial-capillary balance with that pure power law. The requested scaling assumes no such geometrical degeneracy. This also shows why the coefficients cannot determine a universal numerical prefactor without a morphology calculation. The relative viscous contribution decays as $u^{-1/3}$, consistently with an increasingly large domain-scale Reynolds number.

A stationary microporous network removes fluid momentum through linear drag rather than through a domain-scale viscous stress. With $[\bar{\eta }]=\mathrm{M}{\mathrm{L}}^{-3}{\mathrm{T}}^{-1}$, the single-scale force estimate becomesThis drag-limited hydrodynamic coarsening closure uses a fixed isotropic drag coefficient and domain sizes large compared with the pore scale. The only length and time made from $\rho ,\sigma ,\bar{\eta }$ areIndeed $L_1^3=\sigma t_1^2/\rho$, while the choice of $t_1$ equates inertial and drag prefactors. With $L=L_{1}g(u)$ and $u=t/t_1$, all force terms share $\rho L_1/t_1^2=\bar{\eta }{L}_1/t_1=\sigma /L_1^2$. ThereforeAs before, a parameter-free function of this one argument refers to the asymptotic advective scaling model with fixed morphology and negligible extra scales. Retaining the diffusive mobility or resolved network geometry can introduce additional dimensionless parameters. The original $\eta$ does not remain as an independent viscous coefficient after the stipulated replacement.

Use the intended regular scaling ansatz: $g(u)\sim C{u}^y$, $g^{\prime }(u)\sim Cy{u}^{y-1}$ and $g^{\prime \prime }(u)=Cy(y-1)u^{y-2}+o(u^{y-2})$, with $C>0$, $y>0$. Relative to the drag magnitude $g^{\prime }$, the two inertial contributions obeyThus both inertial-to-drag ratios tend to zero at large $u$, whatever the positive growth exponent; for $y=1$ the leading acceleration term vanishes identically. The long-time drag-limited hydrodynamic coarsening balance is thereforesoThe leading law is independent of mass density. Within the drag-capillary reduction, finite initial data give $L^3(t)-L^3(t_{\ast })=3(c_s/c_d)(\sigma /\bar{\eta })(t-t_{\ast })$.

The derivative hypotheses are important: bare $g(u)\sim u^y$ does not justify differentiating an asymptotic equivalence. For instance, $g(u)=u+\frac{1}{2}{\int }_1^u\sin (s^3)ds$ is smooth and increasing, with $g(u)/u\to 1$, but $g^{\prime \prime }=\frac{3}{2}{u}^2\cos (u^3)$ is not small compared with $g^{\prime }=1+\frac{1}{2}\sin (u^3)$. This is a counterexample to an inference from the bare asymptotic relation, not a solution of the coarsening equation. The physical claim uses the regular self-similar power law, for which the ratio calculation above applies.

This conclusion concerns the model's advective drag-capillary channel. Diffusive Ostwald ripening can also have a $1/3$ growth exponent; equality of exponents does not establish that diffusion is negligible. If it is retained, the prefactor and crossover function may also depend on order-parameter mobility, and one must compare transport mechanisms rather than infer the mechanism from the exponent alone.