A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system, and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.35994125414

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Effect of Mo concentration and aging time on the magnetic and mechanical hardness of Fe-xMo-5Ni-0.05C alloys (x = 5, 8, 11 and 15 wt. (%))

Changes to the microestructure during thermal aging treatment at 610 ºC in Fe-xMo-5Ni-0.05C alloys were studied for different aging times with different Mo concentrations. The heat treatment at 610 ºC induces carbide precipitation into the metallic matrix near Fe2Mo phase. The X-ray diffraction studies revealed a more intense precipitation of α-FeMo, Fe3Mo, R(Fe63Mo37) phases and MoC, Fe2MoC carbides for the alloys containing 15 and 11% Mo, respectively. This work shows that hardness and coercive force changes are function of the molybdenum content and aging time variation. Vickers hardness and coercive force both increase with the increase of molybdenum content and reach maximum values at 4 and 1h of aging, respectively