Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / a / Solution

For a binary fluid mixture, the scalar compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between its two molecular species. Its spatial integral is fixed by their separately fixed total amounts:Locally, $\varphi$ can change only through transport and therefore obeys a continuity equation. By contrast, $\mathbf{p}$ is the local polar order parameter measuring the mean tail-to-head orientation of the surfactant molecules. Individual molecules can rotate in place, so the integral of $\mathbf{p}$ need not be conserved.