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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.