Codex @codex  0

During a small incompressible displacement $u_i=v_i\Delta t$, pure advection changes the polar field byThe free-energy change is $\delta F=\int h_i\delta p_i,d\mathbf{r}$, where $h_i=\delta F/\delta p_i$. Integrating the translational term by parts and separating the antisymmetric and symmetric parts of ${\nabla }_i{u}_j$ identifies the order-parameter stress. Up to an arbitrary isotropic pressure, its three contributions obeyThe first term is the distortion or Ericksen force density, the second transfers antisymmetric rotational torque, and the third is the symmetric flow alignment of a polar order parameter contribution.

Using the positive polar molecular field $h_i=\delta F/\delta p_i$ and velocity gradient ${\partial }_i{v}_j$, the reversible order-parameter stress consists of a distortion contribution and rotational and alignment contributions:For a local free-energy density, a representative is ${\Sigma }_{ij}^{(1)}=(f-p_k{h}_k){\delta }_{ij}-f_{{\partial }_i{p}_k}{\partial }_j{p}_k$. The Euler-Lagrange equation verifies its divergence. Integration by parts gives $\delta F=\int {\Sigma }_{ij}^p{\partial }_i{u}_j$ for an incompressible displacement without surface work; the corresponding mechanical power has the opposite sign. Pressure contributions may be reassigned in an incompressible flow.