English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider the 3D Cahn-Hilliard equations coupled to, and driven by, theforced, incompressible 3D Navier-Stokes equations. The combination, known asthe Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statisticalmechanics to model the motion of a binary fluid. The potential development ofsingularities (blow-up) in the contours of the order parameter $\phi$ is anopen problem. To address this we have proved a theorem that closely mimics theBeale-Kato-Majda theorem for the $3D$ incompressible Euler equations [Beale etal. Commun. Math. Phys., Commun. Math. Phys., ${\rm 94}$, $ 61-66 ({\rm1984})$]. By taking an $L^{\infty}$ norm of the energy of the full binarysystem, designated as $E_{\infty}$, we have shown that$\int_{0}^{t}E_{\infty}(\tau)\,d\tau$ governs the regularity of solutions ofthe full 3D system. Our direct numerical simulations (DNSs), of the 3D CHNSequations, for (a) a gravity-driven Rayleigh Taylor instability and (b) aconstant-energy-injection forcing, with $128^3$ to $512^3$ collocation pointsand over the duration of our DNSs, confirm that $E_{\infty}$ remains bounded asfar as our computations allow.

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