Because $\phi$ is a conserved composition density, an infinitesimal material displacement $\mathbf r\mapsto\mathbf r+\mathbf u$ changes it at fixed position by
$$
\delta\phi=-\nabla\mathbin\cdot(\phi\mathbf u)
=-\mathbf u\mathbin\cdot\nabla\phi-\phi\nabla\mathbin\cdot\mathbf u.
$$
The second term is essential when the deformation is compressible. By the definition of the <chemical potential>, and taking $\mathbf u$ to vanish on the boundary,
$$
\delta F=\int\mu\,\delta\phi\,d\mathbf r
=-\int\mu\nabla\mathbin\cdot(\phi\mathbf u)d\mathbf r
=\int\phi\nabla_j\mu\,u_jd\mathbf r.
$$
The same free-energy change written in terms of the <stress>[stress tensor] is
$$
\delta F=\int\Sigma_{ij}\nabla_i u_jd\mathbf r
=-\int(\nabla_i\Sigma_{ij})u_jd\mathbf r.
$$
Since $\mathbf u$ is arbitrary,
$$
\boxed{\nabla_i\Sigma_{ij}=-\phi\nabla_j\mu.}
$$
This is the <Korteweg force> density of a diffuse-interface mixture.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 1 / a / 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.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 2 / a / Solution

{c}

The Korteweg force is the capillary body-force density generated by composition gradients in a diffuse-interface fluid. Up to a pressure gradient it can be written $-\phi\nabla\mu$, where $\mu$ is the chemical potential.


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 Incoming links: Korteweg force