A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations

TL;DR

This paper establishes a regularity criterion for solutions of the 3D Cahn-Hilliard-Navier-Stokes equations, linking solution smoothness to an energy norm, and supports this with direct numerical simulations of fluid instabilities.

Contribution

The authors prove a new regularity criterion for the 3D CHNS system based on an energy norm, extending the Beale-Kato-Majda theorem to this coupled system.

Findings

The energy norm $E_{ ext{infty}}$ remains bounded in simulations.

Numerical results support the theoretical regularity criterion.

Simulations include gravity-driven instability and forced turbulence.

Abstract

We consider the 3D Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter $\phi$ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the $3D$ incompressible Euler equations [Beale et al. Commun. Math. Phys., Commun. Math. Phys., ${\rm 94}$, $ 61-66 ({\rm 1984})$]. By taking an $L^{\infty}$ norm of the energy of the full binary system, designated as $E_{\infty}$, we have shown that $\int_{0}^{t}E_{\infty}(\tau)\,d\tau$ governs the regularity of solutions of the full 3D system. Our direct numerical simulations (DNSs), of the 3D CHNS equations, for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with $128^3$ to $512^3$ collocation points and over the duration of our DNSs, confirm that $E_{\infty}$ remains bounded as far as our computations allow.