The Frenet-Serret curve analysis is extended from nonnull to null trajectories in a generic spacetime using the Newman-Penrose formalism, recovering old results which are not well known and clarifying the associated Fermi-Walker transport which has been left largely unexplored in the literature. This machinery is then used to discuss null circular orbits in stationary axisymmetric spacetimes using the Kerr spacetime as a concrete example, and to integrate the equations of parallel transport along null geodesics in any spacetime.

The circular motion of spinning massive test particles in the spacetime of a rotating Kerr black hole is investigated in the case in which the components of the spin tensor are allowed to vary along the orbit.

This work is concerned with the semiclassical approximation of the Schro ̈dinger-Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a N-body quantum problem, then K-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for Paulis exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects

A direct numerical simulation of a turbulent flow field with a lattice BGK method is presented. A spatial coarse graining of the numerical results is compared with the expected LBGK dynamics for a flow field on a reduced lattice size. This comparison permits to exhibit subgrid properties of the fluid which are not resolved on the coarse lattice. As expected from existing subgrid models, an effective viscosity can be measured that increases when the lattice is coarse grained. Turbulence models based on an effective viscosity are particularly interesting in a lattice Boltzmann simulation, due to the linearity of the propagation operator.

It is shown that homogeneous Rayleigh-Bénard flow, i.e., Rayleigh-Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full nonlinear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.

The phenomenon of apparent slip in micro-channel flows is analyzed by means of a two-phase mesoscopic lattice Boltzmann model including non-ideal fluid-fluid and fluid-wall interactions. The weakly inhomogeneous limit of this model is solved analytically. The present mesoscopic approach permits to access much larger scales than molecular dynamics, and comparable with those attained by continuum methods. However, at variance with the continuum approach, the existence of a gas layer near the wall does not need to be postulated a priori, but emerges naturally from the underlying non-ideal mesoscopic dynamics. It is therefore argued that a mesoscopic lattice Boltzmann approach with non-ideal fluid-fluid and fluid-wall interactions might achieve an optimal compromise between physical realism and computational efficiency for the study of channel micro-flows.

We present a mesoscopic model of the fluid–wall interactions for flows in microchannel geometries. We define a suitable implementation of the boundary conditions for a discrete version of the Boltzmann equations describing a wall-bounded single-phase fluid. We distinguish different slippage properties on the surface by introducing a slip function, defining the local degree of slip for hydrodynamical fields at the boundaries. The slip function plays the role of a renormalizing factor which incorporates, with some degree of arbitrariness, the microscopic effects on the mesoscopic description. We discuss the mesoscopic slip properties in terms of slip length, slip velocity, pressure drop reduction (drag reduction), and mass flow rate in microchannels as a function of the degree of slippage and of its spatial distribution and localization, the latter parameter mimicking the degree of roughness of the ultra-hydrophobic material in real experiments. We also discuss the increment of the slip length in the transition regime, i.e. at ${O}(1)$ Knudsen numbers. Finally, we compare our results with molecular dynamics investigations of the dependence of the slip length on the mean channel pressure and local slip properties and with the experimental dependence of the pressure drop reduction on the percentage of hydrophobic material deposited on the surface.

The effect of bubbles on fully developed turbulent flow is investigated numerically and experimentally, summarizing the results of our previous papers (Mazzitelli et al., 2003, Physics of Fluids15, L5. and Journal of Fluid Mechanics488, 283; Rensen, J. et al. 2005, Journal of Fluid Mechanics538, 153). On the numerical side, we simulate Navier–Stokes turbulence with a Taylor–Reynolds number of Re?˜60, a large large-scale forcing, and periodic boundary conditions. The point-like bubbles follow their Lagrangian paths and act as point forces on the flow. As a consequence, the spectral slope is less steep as compared to the Kolmogorov case. The slope decrease is identified as a lift force effect. On the experimental side, we do hot-film anemometry in a turbulent water channel with Re? ˜ 200 in which we have injected small bubbles up to a volume percentage of 3%. Here the challenge is to disentangle the bubble spikes from the hot-film velocity signal. To achieve this goal, we have developed a pattern recognition scheme. Furthermore, we injected microbubbles up to a volume percentage of 0.3%. Both in the counter flowing situation with small bubbles and in the co-flow situation with microbubbles, we obtain a less spectral slope, in agreement with the numerical result.

The statistics of Lagrangian particles transported by a three-dimensional fully developed turbulent flow is investigated by means of high-resolution direct numerical simulations. The analysis of single trajectories reveals the existence of strong trapping events vortices at the Kolmogorov scale which contaminates inertial range statistics up to 10 tau(eta). For larger time separations, we find that Lagrangian structure functions display intermittency in agreement with the prediction of the multifractal model of turbulence. The study of two-particle dispersion shows that the probability density function of pair separation is very close to the original prediction of Richardson of 1926. Nevertheless, moments of relative dispersion are strongly affected by finite Reynolds effects, thus limiting the possibility to measure numerical prefactors, such as the Richardson constant g. We show how, by using an exit time statistics, it is possible to have a precise estimation of g which is consistent with recent laboratory measurements.

New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced. Numerical results and efficience tests are presented in order to show the behaviour of FSF in comparison with G, FG and a conservative scheme of second order.

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

We provide a model of traffic flow on networks, starting from the second order model proposed by Aw and Rascle. The existence of solutions is proved for perturbation of equilibria.

We construct a population dynamics model of the competition among immune system cells and generic tumor cells. Then, we apply the theory of optimal control to find the optimal schedule of injection of autologous dendritic cells used as immunotherapeutic agent. The optimization method works for a general ODE system and can be applied to find the optimal schedule in a variety of medical treatments that have been described by a mathematical model.