  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.
