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The Cahn--Hilliard equation transports a conserved composition field down gradients of its chemical potential: ${\partial }_t\varphi =\nabla \cdot (M\nabla \mu )$ with $\mu =\delta F/\delta \varphi$. Advection adds $\mathbf{v}\cdot \nabla \varphi$ to the left-hand side.

Model H couples a conserved scalar composition field to an incompressible momentum density. It combines the advective Cahn--Hilliard equation with the Navier-Stokes equation and the thermodynamic force density $-\varphi \nabla \mu$.

The positive order-parameter mobility $M$ relates the diffusive transport flux to the chemical potential gradient: $\mathbf{J}=-M\nabla \mu$. Combining this law with local conservation gives the Cahn--Hilliard equation when $\mu$ is the functional derivative of a composition free energy.

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.

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In the modela small Fourier mode has growth rate $\sigma (k)=-\lambda k^2-\gamma k^4$. Thus the straight pattern is linearly stable for $\lambda >0$ and unstable to sufficiently long transverse waves for $\lambda <0$.

When $\lambda <0$, the unstable band is $0<k^2<-\lambda /\gamma$. Differentiating $\sigma$ with respect to $k^2$ gives the fastest growing scale $k^2=-\lambda /(2\gamma )$ and maximum growth ${\lambda }^2/(4\gamma )$. The fourth derivative stabilizes short waves rather than allowing arbitrarily fine structure to grow fastest.

Writing the slope as $p={\varphi }_Y$ givesa conserved Cahn--Hilliard equation for the local orientation. When $\lambda <0$ and $\alpha <0$, its homogeneous bulk potential $W(p)=\lambda p^2/2-\alpha p^4/12$ has two minima at $p=\pm \sqrt{3\lambda /\alpha }$. These are preferred local slopes; boundary conditions and the conserved mean slope still constrain the complete pattern.