Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.Comment: 39 pages, 1 figur

Entanglement and entropy operator for strings in pp-wave time dependent background

In this letter new aspects of string theory propagating in a pp-wave time dependent background with a null singularity are explored. It is shown the appearance of a 2d entanglement entropy dynamically generated by the background. For asymptotically flat observers, the vacuum close to the singularity is unitarily inequivalent to the vacuum at $\tau = -\infty$ and it is shown that the 2d entanglement entropy diverges close to this point. As a consequence, the positive time region is inaccessible for observers in $\tau =-\infty$. For a stationary measure, the vacuum at finite time is seen by those observers as a thermal state and the information loss is encoded as a heat bath of string states.Comment: revtex4, 15 pages, revised version to appear in Physics Letters

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology

Desingularization of vortices for the Euler equation

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation -\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.Comment: 40 page

Produção de biogás por digestão em fase sólida de cama de frango.

RESUMO: A cama de frango é um resíduo da atividade avícola, que é gerado em grandes quantidades, e quando mal manuseado tem elevado potencial poluidor. No entanto, é possível aproveitar este resíduo para geração de biogás, contudo este substrato apresenta desafios devido a baixa umidade e composição química. Diante deste contexto, a digestão em fase sólida se destaca como um processo promissor pelo fato de evitar o manejo ou póstratamento da água derivada do processo. O objetivo do presente estudo foi avaliar a eficiência da produção de biogás de cama de frango em um reator anaeróbio de fase sólida (RDFS) operado em batelada. Ao realizar o processo de digestão em fase sólida, verificouse que este sistema apresentou um rendimento de biogás de 90,30 mLNbiogás.gSVadic -1 ,quando comparado ao valor de referência do teste do potencial bioquímico de biogás que foi de 281 mLNbiogás.gSVadic -1 , a eficiência foi de 32 % após 30 dias de digestão. No entanto, o teor de sólidos totais que o RDFS pode operar é de 30 %, enquanto que a digestão úmida a concentração de sólidos totais é em média de 10%. Neste caso utilizando o RDFS estaremos economizando líquido para diluição do substrato, e este é um sistema robusto, de fácil monitoramento que pode ser utilizado pelos avicultores, podendo ainda ser otimizado para melhoria do sistema e assim aumentar a eficiência de produção de biogás. ABSTRACT: The poultry litter is a residue from poultry activity, which is generated in large quantities and, when poorly handled, presents a high polluting potential. It is possible to take advantage of this residue to generate biogas, however this substrate consists in a challenging material due to its humidity and chemical composition. In this context, solid-state digestion stands out as a promising process because it prevents the handling or post-treatment of water derived from the process. The goal of the present study was to evaluate the efficiency of production of poultry litter biogas in a solid phase anaerobic reactor operated in batch. When carrying out the solid phase digestion process, it was verified that this system presented a biogas production of 90,30 mLNbiogas.gSVadd -1 . When comparing this to the reference value of the biochemical potential of biogas test, which is 281 mLNbiogas.gSVadd -1 , the efficiency attained was 32% at the 30-day of digestion However, the total solids content that the RDFS can operate is 30%, while the wet digestion at the total solids concentration is on average 10%. In this case using the RDFS we will be saving liquid for dilution of the substrate, and this is a robust system, easy to monitor that can be used by poultry farmers and can be optimized for system improvement and thus increase biogas production efficiency

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Regularization of point vortices for the Euler equation in dimension two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page