= Energy dissipation identity for Model H
{title2=$\dot E=-\int[M|\nabla\mu|^2+\eta|\nabla\mathbf v|^2]$}
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:
$$
\frac{d}{dt}\left[F+\frac\rho2\int|\mathbf v|^2\right]=-\int[M|\nabla\mu|^2+\eta|\nabla\mathbf v|^2]\le0.
$$
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 $\dot F=\int\phi\mathbf v\cdot\nabla\mu-M\int|\nabla\mu|^2$. Dotting the momentum equation with velocity gives $\dot E_{\rm kin}=-\int\phi\mathbf v\cdot\nabla\mu-\eta\int|\nabla\mathbf v|^2$. <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.
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