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

Because $\varphi$ is a conserved composition density, an infinitesimal material displacement $\mathbf{r}\mapsto \mathbf{r}+\mathbf{u}$ changes it at fixed position byThe 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,The same free-energy change written in terms of the stress tensor isSince $\mathbf{u}$ is arbitrary,This is the Korteweg force density of a diffuse-interface mixture.