The Cahn–Hilliard–Navier–Stokes framework for multiphase fluid flows: laminar, turbulent and active

The Cahn–Hilliard–Navier–Stokes (CHNS) partial differential equations (PDEs) provide a powerful framework for the study of the statistical mechanics and fluid dynamics of multiphase fluids. We provide an introduction to the equilibrium and non-equilibrium statistical mechanics of systems in which coexisting phases, distinguished from each other by scalar order parameters, are separated by an interface. We then introduce the coupled CHNS PDEs for two immiscible fluids and generalisations for (i) coexisting phases with different viscosities, (ii) CHNS with gravity, (iii) three-component fluids and (iv) the CHNS for active fluids. We discuss mathematical issues of the regularity of solutions of the CHNS PDEs. Finally we provide a survey of the rich variety of results that have been obtained by numerical studies of CHNS-type PDEs for diverse systems, including bubbles in turbulent flows, antibubbles, droplet and liquid-lens mergers, turbulence in the active-CHNS model and its generalisation that can lead to a self-propelled droplet.