Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn-Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524

Saturated hydrocarbon polymeric binder for advanced solid propellant and hybrid solid grains Quarterly report no. 3, 1 May - 31 Jul. 1966

Saturated hydrocarbon polymeric binder for advanced solid propellant and hybrid solid grain

Saturated hydrocarbon polymeric binder for advanced solid propellant and hybrid solid grains Quarterly report no. 2, 1 Feb. - 30 Apr. 1966

Synthesis and analysis of ethylene-neohexene copolymers with other non ketene-imine group free radicals for solid and hybrid grain propellant saturated hydrocarbon binder progra

Polychart contour plotter enables data extrapolation from multiple plotting charts

A polychart contour plotter is used to reduce the data from all 19 antenna pattern charts to a one-chart form

The Detectability of AGN Cavities in Cooling-Flow Clusters

Chandra X-ray Observatory has revealed X-ray cavities in many nearby cooling flow clusters. The cavities trace feedback from the central active galactic nulceus (AGN) on the intracluster medium (ICM), an important ingredient in stabilizing cooling flows and in the process of galaxy formation and evolution. But, the prevalence and duty cycle of such AGN outbursts is not well understood. To this end, we study how the cooling is balanced by the cavity heating for a complete sample of clusters (the Brightest 55 clusters of galaxies, hereafter B55). In the B55, we found 33 cooling flow clusters, 20 of which have detected X-ray bubbles in their ICM. Among the remaining 13, all except Ophiuchus could have significant cavity power yet remain undetected in existing images. This implies that the duty cycle of AGN outbursts with significant heating potential in cooling flow clusters is at least 60 % and could approach 100 %, but deeper data is required to constrain this further.Comment: 4 pages, 2 figures; to appear in the proceedings of "The Monsters' Fiery Breath", Madison, Wisconsin 1-5 June 2009, Eds. Sebastian Heinz & Eric Wilcots; added annotation to the figur

X-ray Supercavities in the Hydra A Cluster and the Outburst History of the Central Galaxy's Active Nucleus

A 227 ksec Chandra Observatory X-ray image of the hot plasma in the Hydra A cluster has revealed an extensive cavity system. The system was created by a continuous outflow or a series of bursts from the nucleus of the central galaxy over the past 200-500 Myr. The cavities have displaced 10% of the plasma within a 300 kpc radius of the central galaxy, creating a swiss-cheese-like topology in the hot gas. The surface brightness decrements are consistent with empty cavities oriented within 40 degrees of the plane of the sky. The outflow has deposited upward of 10^61 erg into the cluster gas, most of which was propelled beyond the inner ~100 kpc cooling region. The supermassive black hole has accreted at a rate of approximately 0.1-0.25 solar masses per year over this time frame, which is a small fraction of the Eddington rate of a ~10^9 solar mass black hole, but is dramatically larger than the Bondi rate. Given the previous evidence for a circumnuclear disk of cold gas in Hydra A, these results are consistent with the AGN being powered primarily by infalling cold gas. The cavity system is shadowed perfectly by 330 MHz radio emission. Such low frequency synchrotron emission may be an excellent proxy for X-ray cavities and thus the total energy liberated by the supermassive black hole.Comment: 8 pages, 3 figures; Submitted to ApJ, revised per referee's suggestion