Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.     

Singularity formation for Prandtl’s equations

Si tratta di una memoria in cui si espongono alcuni progressi matematici recentemente compiuti nello studio delle equazioni di strato limite de Prandtl. Gli argomenti principali trattati sono la derivazione rigorosa delle equazioni stesse, la teoria matematica del problema di valore iniziale associato e la teoria delle soluzioni di similarità.

Abstract

Этa-излoжитeльнaя paбoтa o нeкoтopых мaтeмaтихecких ycпeхaх в изyхeнии ypaвнeний гpaнихнoгo cлoя Пpaндтля. Ocнoвныe paccмaтpывaeмыe вoпpocы: cтpoгий вывoд ypaвнeний, мaтe-мaтихecкaя тeopия cвязaннoй зaдaхи нaхaльнoгo знaхeния, и тeopия пoдoбнocтых pэшeний.

Non-Local Scattering Kernel and the Hydrodynamic Limit

In this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory. We consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall. This results in a scattering kernel which is a non-local generalization of the classical Maxwell scattering kernel. The proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation. We study the hydrodynamic limit performing a Knudsen layer analysis, and derive a new class of (weakly) nonlocal boundary conditions to be imposed to the Navier–Stokes equations.     

Anatomy of a polarization switch of a vertical-cavity semiconductor laser

Using a streak camera we have measured the three Stokes polarization parameters during a polarization switch of a vertical-cavity semiconductor laser. The switch occurs along a corkscrew path on the Poincare sphere and takes on average a few nanoseconds; this value agrees with a theoretical treatment based upon the Fokker-Planck equation.     

Transitions in a stratified Kolmogorov flow

We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is \(Ri=10^{-5}\), and restrict our analysis to the range \(0<Re<30\). The effect of the stabilizing temperature is to shift bifurcation points and to reduce the region of existence of stable drifting states. The flow reaches chaotic configurations through two different routes, one involving drifting states, the other involving a gluing bifurcation. Along the latter route we observe, as the precursor to chaotic states, a period tripling bifurcation.     

Asymptotic Analysis of a Slightly Rarefied Gas with Nonlocal Boundary Conditions

In this paper nonlocal boundary conditions for the Navier–Stokes equations are derived, starting from the Boltzmann equation in the limit for the Knudsen number being vanishingly small. In the same spirit of (Lombardo et al. in J. Stat. Phys. 130:69–82, 2008) where a nonlocal Poisson scattering kernel was introduced, a gaussian scattering kernel which models nonlocal interactions between the gas molecules and the wall boundary is proposed. It is proved to satisfy the global mass conservation and a generalized reciprocity relation. The asymptotic expansion of the boundary-value problem for the Boltzmann equation, provides, in the continuum limit, the Navier–Stokes equations associated with a class of nonlocal boundary conditions of the type used in turbulence modeling.     

On the Prandtl Boundary Layer Equations in Presence of Corner Singularities

In this paper we prove the well-posedness of the Prandtl boundary layer equations on a periodic strip when the initial and the boundary data are not assigned to be compatible.     

Two OPEEs (organic phase enzyme electrodes) used to check the percentage water content in hydrophobic foods and drugs.

The development and optimization of an analytical method using enzymatic biosensors able to operate in organic solvents [organic phase enzyme electrodes (OPEEs)] for the determination of the water content in food fats (butter, margarine) or pharmaceutical or cosmetic ointments is described. The method is based on the increase in enzymatic activity which is related to the increase in the percentage water content in the organic phase into which the biosensor is dipped. The enzymes used to assemble the biosensors were tyrosinase or catalase, the substrates were phenol or p-cresol and tert-butyl hydroperoxide, respectively, and the organic solvents were acetonitrile or dioxane. A gas diffusion amperometric electrode for oxygen measurement was used as electrochemical transducer. The results were compared with those obtained applying the Karl Fischer method to the same food or drug matrices. The correlations among the two methods proved satisfactory, as the difference in the computed values of water content was never higher than 7%. Also, the precision of measurements was acceptable (RSD < 6%) in all the analyses of real matrices.