= Solution
The thermodynamic <order parameter> is the conserved scalar composition difference $\phi$ 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,
$$
(\partial_t+\mathbf v\mathbin\cdot\nabla)\phi=-\nabla\mathbin\cdot\mathbf J.
$$
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 theory>
$$
F[\phi]=\int d^3\mathbf r
\left(\frac a2\phi^2+\frac b4\phi^4+\frac\kappa2|\nabla\phi|^2\right),
$$
the <functional derivative> is
$$
\mu=\frac{\delta F}{\delta\phi}
=a\phi+b\phi^3-\kappa\nabla^2\phi.
$$
These statements give the advective <Cahn--Hilliard equation>.
Constant mass density and <incompressible flow> require $\nabla\mathbin\cdot\mathbf v=0$. 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> $-\phi\nabla\mu$. The pressure is the <Lagrange multiplier> enforcing incompressibility. Together these equations are <Model H dynamics>; isothermality removes the need for a separate energy equation.
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