New trends in the Cahn–Hilliard–Navier–Stokes description of binary-fluid mixtures: an overview

We provide an overview of the Cahn-Hilliard-Navier-Stokes (CHNS) framework that has been used to study multi-phase flows. We concentrate on recent advances in modelling active-fluid mixtures and the non-reciprocal CHNS system. We first use the zero-field Ising model to illustrate two-phase coexistence, below the critical temperature, and to recall how equilibrium properties can be studied approximately by using mean-field or Curie-Weiss theory and its gradient expansion, namely, Landau-Ginzburg theory for a scalar order-parameter ϕ, which distinguishes between the two coexisting phases. We then move to the spatio-temporal evolution of ϕ, which requires time-dependent Ginzburg-Landau theory (Model A), if the order parameter is not conserved, or Cahn-Hilliard theory (CH or Model B), if the order parameter is conserved. Next, we consider the combination of CH models with the Navier-Stokes equation for an incompressible fluid; this yields the CHNS system. We give a synoptic view of recent generalizations of this CHNS system that include different active models (e.g. Active Models B) or non-reciprocity. This article is part of the theme issue 'Anti-diffusive transport and layering: from sub-cellular to astrophysical scales'.