= Solution
The field $\phi$ 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 energy>
$$
F=\int\left[\frac a2\phi^2+\frac b4\phi^4+\frac\kappa2|\nabla\phi|^2\right]d\mathbf r,
$$
so $\mu=a\phi+b\phi^3-\kappa\nabla^2\phi$. 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> $-\phi\nabla\mu$ 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, $-\phi\nabla\mu=\mu\nabla\phi-\nabla(\phi\mu)$ 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, giving
$$
\boxed{\frac{d}{dt}\left[F+\frac\rho2\int|\mathbf v|^2\,d\mathbf r\right]=-\int\left[M|\nabla\mu|^2+\eta|\nabla\mathbf v|^2\right]d\mathbf r\le0.}
$$
Thus 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.
Back to article page
