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

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 $\varphi$ 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., $94$, $61-66(1984)$]. 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.

arXiv comment11 pages, 3 figures

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Cited by in corpus (3)

• The role of BKM-type theorems in $3D$ Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
• Two-dimensional magnetohydrodynamic turbulence with large and small energy-injection length scales
• A model of interacting Navier-Stokes singularities