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

Abstract

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’.