Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 3 / b / Solution

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.