Mathematical problems in kinetic and soft matter theories
Research focus
The Peer review has evaluated this group as Excellent
Kinetic theories · Existence of weak global solutions of the Boltzmann equation, with particular attention to models without angle cutoff, conservation of energy, and the trend to equilibrium. · Analysis of flows of rarefied gases between almost parallel plates, by the Reynolds equation. Solution of the Couette-Poiseuille problem by the Boltzmann equation. Important for applications to MEMS (Micro Electro-Mechanical Systems). · Analysis of the 3d Rayleigh-Benard convection in a rarefied gas, with particular attention to the comparison between the kinetic an the Navier-Stokes solutions. · Development and application of deterministic and stochastic methods for the numerical solution of kinetic equations for dilute and dense gases. Applications are mainly oriented toward the study of low Mach number flows in MEMS/NEMS, the structures of the vaporliquid interface and of the Knudsen layer in evaporation/ condensation flows occurring in two-phase systems. Kinetic theory of dense fluids is applied to the study of nonequilibrium flows in nanosized channels. Soft matter theories 1. The soft matter group is mostly focused on the research in liquid crystals theory. Dynamics of defects, determination of complex equilibrium shapes of smectic capillaries, and the detailed analysis of the boundary layer structures that weaken the external anchoring are among the topics we deal with. We also aim at determining the stationary shapes that optimize the curvature-energy functional in lipid vesicles, possibly hosting rigid inclusions. The shape perturbations determined by the inclusions induce an energy increase that in turn creates a mediated interaction between the inclusions themselves. 2. The adopted techniques include the variational analysis of the free energy functional, and the multi-scale perturbative analysis of singular partial differential equations. Moreover, when dealing with the stationary dynamics of defects, Fourier-transform techniques allow to obtain explicit analytical expressions of travelling nematic patters. Free-boundary and shape-optimization problems come into play when the equilibrium shapes of either liquid crystal samples or biological vesicles are determined. 101 3. There are several international collaborations active in this field. Among these, the closest collaborations link members of our group to the University of Southampton (Professor Tim Sluckin), to the University of Cambridge (Professor Eugene M. Terentjev), and to the University of Minneapolis (Professor Maria Carme Calderer). A new PhD grant, awarded by the Italian University and Research Ministry, allows to open a new and interesting research line devoted to the biomedical applications of liquid crystal elastomers. This research will be carried on in collaboration with the Cambridge group. Fluid dynamics and Magnetohydrodynamics 1. Numerical simulation of turbulent complex flows. Large Eddy Simulation (LES) has been used. In particular we developed a finite difference code allowing simulations in both compressible and incompressible regimes in both Cartesian and cylindrical geometries. Applications include the numerical simulation of reacting flows and the study of formation and melting of ice in oceans, lakes and arctic rivers. In the former problem we consider the study of flame stability and we plan to extend the numerical code to complex geometry. In the latter problem we numerically integrate the evolution equations in the Boussinesq limit. Our aim is to predict how the physical parameters control the process of formation and melting of ice. Further perspectives will include the effects of salinity. Scientific contacts: Prof. Peter Wadhams, DAMTP (Cambridge, UK) and Dr. Piero Olla, CNR-ISAC (Cagliari, Italy) Contracts: 6th EC Framework Programme integrated infrastructure initiative HYDRALAB III: "Understanding the impact of a Reduced Ice Cover in the Arctic Ocean". 2. Oscillations of rotating fluids, namely inertial modes. They play an important role in astrophysics and geophysics and in some engineering problems like the stability of artificial satellites. The ill-posed nature of the asymptotic problem yield several kinds of singularities as viscosity vanishes. In this project we investigate numerically the linear eigenvalue problem and we seek an analytical description of the asymptotic spectrum.
Dipartimento di afferenza
Docenti afferenti
Full Professors
Paolo Biscari
Carlo Cercignani
Aldo Frezzotti
Maria Lampis
Associate Professors
Lorenzo Valdettaro
Assistant Professors
Antonella Abbā
Paolo Barbante