A mathematical clue to the separation phenomena on the two-dimensional Navier-Stokes equation

In general, before separating from a boundary, the flow moves toward reverse direction near the boundary against the laminar flow direction. Here in this paper, a clue to such reverse flow phenomena (in the mathematical sense) is observed. More precisely, the non-stationary two-dimensional Navier-Stokes equation with an initial datum having a parallel laminar flow (we define it rigorously in the paper) is considered. We show that the direction of the material differentiation is opposite to the initial flow direction and effect of the material differentiation (inducing the reverse flow) becomes bigger when the curvature of the boundary becomes bigger. We also show that the parallel laminar flow cannot be a stationary Navier-Stokes flow near a portion of the boundary with nonzero curvature

Asymptotically constrained and real-valued system based on Ashtekar's variables

We present a set of dynamical equations based on Ashtekar's extension of the Einstein equation. The system forces the space-time to evolve to the manifold that satisfies the constraint equations or the reality conditions or both as the attractor against perturbative errors. This is an application of the idea by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically stable (i.e., constrained) system for the Einstein equation, adding dissipative forces in the extended space. The obtained systems may be useful for future numerical studies using Ashtekar's variables.Comment: added comments, 6 pages, RevTeX, to appear in PRD Rapid Com