Maria Stella Mongiovì

On the modeling of nonlinear interactions in large complex systems

Abstract This work deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the modeling of nonlinear interactions which is one of the most important issues in the mathematical approach to modeling and simulating complex systems, and which includes a learning–hiding dynamics. Applications are focused on the modeling of complex biological systems and on immune competition.

Generalization of Vinen's equation including transition to superfluid turbulence

A phenomenological generalization of the well known Vinen equation for the evolution of vortex line density in superfluid counterflow turbulence is proposed. This generalization includes nonlinear production terms in the counterflow velocity and corrections depending on the diameter of the tube. The equation provides a unified framework for the various phenomena (stationary states and transitions) present in counterflow superfluid turbulence: in fact, it is able to describe the laminar regime, the first-order transition from laminar to turbulent TI state, the two turbulent states, the transition from TI to TII turbulent states, and it yields a slower decay of the counterflow turbulence than the classical local description. Finally, a comparison with the experimental results shows that the contribution of the new terms is prevalent in the laminar and in the turbulent TI regime, while in the fully developed turbulent TII regime the equation reduces to the original Vinen equation.

Vortex length, vortex energy and fractal dimension of superfluid turbulence at very low temperature

By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contribution to the energy per unit length of vortex lines, the fractal dimension will be higher than unity. Finally, for the sake of more concrete illustration, we relate the fractal dimension of the tangle to the scaling exponents of amplitude and wavelength of a cascade of Kelvin waves.

Vortex dynamics in rotating counterflow and plane Couette and Poiseuille turbulence in superfluid helium

An equation previously proposed to describe the evolution of vortex-line density in rotating counterflow turbulent tangles in superfluid helium [Phys. Rev B 69, 094513 (2004)] is generalized to incorporate nonvanishing barycentric velocity and velocity gradients. Our generalization is compared with an analogous approach proposed by Lipniacki [Eur. J. Mech. B Fluids 25, 435 (2006)], and with experimental results by Swanson et al. [Phys. Rev. Lett. 50, 190 (1983)] in rotating counterflow, and it is used to evaluate the vortex density in plane Couette and Poiseuille flows of superfluid helium.

A mathematical model of counterflow superfluid turbulence describing heat waves and vortex-density waves

The interaction between vortex-density waves and high-frequency second sound in counterflow superfluid turbulence is examined, incorporating diffusive and elastic contributions of the vortex tangle. The analysis is based on a set of evolution equations for the energy density, the heat flux, the vortex line density, and the vortex flux, the latter being considered here as an independent variable, in contrast to previous works. The latter feature is crucial in the transition from diffusive to propagative behavior of vortex-density perturbations, which is necessary to interpret the details of high-frequency second sound.

Alternative Vinen Equation and its Extension to Rotating Counterflow Superfluid Turbulence

Abstract Two alternative Vinens evolution equations for the vortex line density L in counterflow superfluid turbulence are physically admissible and lead to analogous results in steady states. In Jou and Mongiovi` [Phys. Rev. B 69 (2004) 094513] the most used of them was generalized to counterflow superfluid turbulence in rotating containers. Here, the analogous generalization for the alternative Vinen equation is proposed. Both generalized Vinens equations are compared with the experimental results, not only in steady states but also in some unsteady situations. From this analysis follows that the stationary solutions of the alternative Vinen equation fit better the experimental data and that its unsteady solutions tend faster to the corresponding final steady-state values than the solutions of the usual Vinens equation.

Vortex density waves and high-frequency second sound in superfluid turbulence hydrodynamics

In this Letter we show that a recent hydrodynamical model of superfluid turbulence describes vortex density waves and their effects on the speed of high-frequency second sound. In this frequency regime, the vortex dynamics is not purely diffusive, as for low frequencies, but exhibits ondulatory features, whose influence on the second sound is here explored.

Thermomechanical effects in the flow of a fluid in porous media

This paper deals with analysis, by methods of extended thermodynamics, of the thermomechanical effects which arise in the flow of a weakly viscous fluid in a porous medium. Under the hypothesis that the fluid fills all the interstices among the powder and that the size of the powder grains and of the interstices is much lower than a suitable characteristic length, linearized field equations are written, which include, in a natural way, terms which take into account the Dufour, Soret, and virtual mass effects. As a limiting case when the evolution time of the heat flux goes to infinite and no entropy flux is carried, the flow of liquid helium II in a porous medium is obtained.

Dissipative terms of thermal nature in the theory of an ideal monoatomic superfluid

A dissipative model of helium II was built up in previous works, using a 13-field extended thermodynamic theory formulated by Liu and Müller. In this work a generalization of such model is presented, where an extended thermodynamics with 14 fields due to Kremer is used. It is shown that the fourteenth field is able to account for the experimental data concerning the second sound attenuation. Further, the proposed theory is able to explain the Osborne experiment. Finally, a comparison with the two-fluid model is performed, emphasizing the different ways in which the dissipative phenomena are explained by the two theories.

Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

Abstract We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.