On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension

A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface S with a constant time hyperplane and the "time height" function of S over this hyperplane. We also specialize to the case when S lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod). Exploiting properties of convex hypersurfaces in Euclidean space we write down the Penrose inequality in the Minkowski spacetime of arbitrary dimension n+2 as an inequality for two smooth functions on the sphere. We finally obtain a sufficient condition for the validity of the Penrose inequality in the four dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces.Comment: 25 pages, 2 figures, Late

Geometry of normal graphs in Euclidean space and applications to the Penrose inequality in Minkowski

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of S onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang's condition explicitly in terms of the time height function of S over a hyperplane and the geometry of the projection of S along its past null cone onto this hyperplane. We also include, in an Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.Comment: 15 pages, 1 figure, Late

Kinematic Waves And Governing Equations In Bubble Columns And Three-phase Fluidized Beds

New averaged topological equations (ATEs) were obtained from a geometric description of a dispersed two-phase flow using the derivatives of a multi-step and a multi-delta distribution functions as well as a local volume averaging technique. Similar distribution functions were also defined and combined for the obtention of averaged transport equations for multiphase systems considering relevant interfacial effects. A consistency test performed on the set of averaged transport equations shows that the approach followed renders a self-consistent set of governing equations for a multiphase system with an arbitrary number of phases and relevant interfacial effects. Combination of the ATEs and the averaged mass transport equations was proposed to be the foundations of an averaged kinematics of dispersed two-phase flows. The commonly used drift flux model was in turn analyzed and some of its limitations were established as a consequence of the introduction of the ATEs.;Parameters involved in the ATEs are the propagation speeds and the dimensionless strengths of volume fraction waves and specific interfacial area waves, as well as the averaged mean curvature of the dispersed inclusions. An impedance technique was adopted for the obtention of the propagation speed of volume fraction waves and for the objective characterization of several flow patterns in bubble columns and in three-phase fluidized beds. Two important propagation speeds were found in bubble columns under churn-turbulent operation. The smaller one is closely constant and characteristic of homogeneous bubbling conditions; the other one is characteristic of clusters of big bubbles. In three-phase fluidized beds the existence of only one important propagation speed was detected as well as two flow pattern transitions: one of them between the coalesced and the dispersed bubbling flow patterns and the other between one of these flow patterns and the churn-turbulent flow pattern