title: Regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations.
abstract: We consider the three-dimensional (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 ϕ 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 [J. T. Beale, T. Kato, and A. J. Majda, Commun. Math. Phys. 94, 61 (1984)CMPHAY0010-361610.1007/BF01212349]. By taking an L^{∞} norm of the energy of the full binary system, designated as E_{∞}, we have shown that ∫_{0}^{t}E_{∞}(τ)dτ 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_{∞} remains bounded as far as our computations allow.
authors.affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
authors.forename: John D
authors.initials: JD
authors.lastname: Gibbon
authors.affiliation: Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, 560 012, India.
authors.forename: Nairita
authors.initials: N
authors.lastname: Pal
authors.affiliation: Department of Physics, University of Rome 'Tor Vergata', 00133 Rome, Italy.
authors.forename: Anupam
authors.initials: A
authors.lastname: Gupta
authors.affiliation: Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, 560 012, India.
authors.forename: Rahul
authors.initials: R
authors.lastname: Pandit