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Replacing domain-scale viscous stresses by a local linear drag density $-\bar{\eta }\mathbf{v}$ gives the scales $L_1=(\sigma \rho /{\bar{\eta }}^2{)}^{1/3}$ and $t_1=\rho /\bar{\eta }$. The single-scale force model isFor a regular power-law asymptotic $g\sim C{u}^y$, $g^{\prime }\sim Cy{u}^{y-1}$, $g^{\prime \prime }=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, the two inertial-to-drag ratios scale as $(y-1)/u$ and $y/u$. Thus drag dominates inertia at late time, and drag-capillary balance integrates to $g^3\sim 3(c_s/c_d)u$. This proves the displayed $1/3$ exponent and its independence of mass density. The scaling assumes a homogenized stationary network and dominant advective transport. Diffusive Ostwald ripening can have the same exponent, so its omission must be justified separately; order-parameter mobility can otherwise enter the prefactor.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.

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.

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.